Hypergeometric Functions over Finite Fields

HYPERGEOMETRIC FUNCTIONS OVER FINITE FIELDS

arXiv:1510.02575v1 [math.NT] 9 Oct 2015

JENNY FUSELIER, LING LONG, RAVI RAMAKRISHNA, HOLLY SWISHER, FANG-TING TU

Abstract. Based on the developments of many people including Evans, Greene, Katz, McCarthy, and Rodriguez-Villegas, we consider period functions for hypergeometric type algebraic varieties over finite fields and consequently study hypergeometric functions over finite fields in a manner that is parallel to the classical hypergeometric functions. Using a comparison between the classical gamma function and its finite field analogue the Gauss sum, we give a systematic way to obtain certain types of transformation and evaluation formulas over finite fields and interpret them geometrically using a Galois representation perspective. As an application, we obtain a few finite field analogues of algebraic hypergeometric identities, quadratic or higher transformation formulas, and evaluation formulas.

Contents 1. Introduction 2. Preliminaries 2.1. Gamma and beta functions 2.2. Gauss and Jacobi sums 2.3. Lagrange Inversion 2.4. A dictionary between the complex and finite field settings 3. Classical hypergeometric functions 3.1. Classical development 3.2. Some properties of hypergeometric functions with n = 1 4. Finite field analogues 4.1. Periods in the finite field setting 4.2. Hypergeometric varieties 4.3. Hypergeometric functions over finite fields 4.4. Comparison with Greene and McCarthy’s versions 5. Galois representation interpretation 5.1. Notation for the mth symbol 5.2. Jacobi sums and Grossencharacters 5.3. Galois interpretation for 1 P0 5.4. Galois interpretation for 2 P1 5.5. Some special cases of 2 P1 -functions 5.6. Galois interpretation for n+1 Fn 5.7. A finite field version of Clausen formula by Evans and Greene 5.8. Analogues of Ramanjuan type formulas for 1/π 5.9. Summary 6. Translation of Some Classical Results 6.1. Kummer’s 24 Solutions 1

2 6 6 8 10 11 11 11 13 16 16 18 18 19 20 20 21 23 24 28 29 29 31 32 33 33

2 JENNY FUSELIER, LING LONG, RAVI RAMAKRISHNA, HOLLY SWISHER, FANG-TING TU

6.2. Pfaff-Saalsch¨ utz evaluation formula 6.3. A few analogues of algebraic hypergeometric formulas 7. Quadratic or higher transformation formulas 7.1. A quadratic transformation formula 7.2. An immediate corollary of the quadratic transformation formula 7.3. The proof of Theorem 2 7.4. Using 2 F1 symmetries to eliminate extra characters 7.5. A cubic 2 F1 formula and a corollary 7.6. Another cubic 2 F1 formula and applications 8. Acknowledgments References

36 37 40 40 46 46 47 48 54 56 56

1. Introduction Based on the seminal works of Evans [21, 23, 24], Greene [30, 31, 32], Katz [35], Koblitz [36], McCarthy [46], Rodriguez-Villegas [49], Stanton [33] et al., the finite field analogues of special functions, in particular generalized hypergeometric functions have been developed theoretically, and computations on the corresponding hypergeometric motives have been implemented in computational packages like Pari and Magma. In this paper, which is currently in preliminary form, we will focus on the finite field analogues of classical transformation and evaluation formulas based on the papers mentioned above, especially [30] by Greene and [46] by McCarthy. To achieve our goals, we modify their notation slightly so that the analysis becomes more parallel to the classical case and the proofs in the complex case can be extended naturally to the finite field setting. In particular, we want to emphasize that the classical n+1 Fn functions, with rational parameters and suitable normalizing factors, can be computed from the periods of hypergeometric varieties of the form y N = xi11 · · · xinn (1 − x1 )j1 · · · (1 − xn )jn (1 − λx1 x2 · · · xn )k , where N, is , jt , k are integers, λ is a fixed parameter, y and xm are variables. It is helpful to distinguish the n+1 Fn functions from the actual periods, in particular when we consider their finite field analogues, denoted here by n+1 Fn . On the level of finite fields, we use two main methods throughout. The first method is calculus-style and is based on the conversion between the classical and the finite field settings which is well-known to experts. In the classical setting, the gamma function satisfies reflection and multiplication formulas (see Theorems 5 and 7), which yield many interesting hypergeometric identities. Roughly speaking, Gauss sums are finite field analogues of the gamma function [24] and they satisfy very similar reflection and multiplication formulas (see (11) and Theorem 8). The technicality in converting identities from the classical setting lies in analyzing the ‘error terms’ associated with Gauss sum computations (which are delta terms here, following Greene’s work [30]). This approach is particularly handy for translating to the finite field setting a classical transformation formula that satisfies the following condition:


3

(∗) it can be proved using only the binomial theorem, the reflection and multiplication formulas, or their corollaries (such as the PfaffSaalsch¨ utz formula). For the second method, we rely on the built-in symmetries for the 2 F1 hypergeometric functions over finite fields, to be introduced below, which are very helpful for many purposes including getting around the error term analysis. Also we want to stay in alignment with geometry by putting the finite field analogues in the explicit context of hypergeometric varieties and their corresponding Galois representations. This gives us important guidelines as the finite field analogues, in particular evaluation formulas such as the Pfaff-Saalsch¨ utz identity, do not look exactly like the classical cases. From a different perspective, hypergeometric functions over finite fields are twisted exponential sums in the sense of [1, 2] and are related to hypergeometric motives [35, 49], abstracted from cohomology groups of the hypergeometric varieties. This direction is pursued by Katz [35] and further developed and implemented by many others including Rodriguez-Villegas [49]. It offers a profound approach to hypergeometric motives and their L-functions by combining both classical and p-adic analysis, arithmetic, combinatorics, computation, and geometry. Our approach to the Galois perspective is derived from our attempt to understand hypergeometric functions over finite fields parallel to the classical setting and Weil’s approach to consider Jacobi sums as Grossencharacters [62]. This is reflected in our notation in §5. This paper is organized as follows. Section 2 contains preliminaries on gamma and beta functions as well as their finite field counterparts, Gauss and Jacobi sums, including a dictionary in §2.4 between objects in the classical and finite field settings. We recall some well-known facts for classical hypergeometric functions in Section 3. We assume in this paper the parameters for all the hypergeometric functions are rational numbers. In Section 4, we introduce the period functions n+1 Pn , and the normalized period functions n+1 Fn . In Section 5, we interpret the n+1 Pn and n+1 Fn functions, in particular the n = 1 case, as traces of Galois representations at Frobenius elements via explicit hypergeometric algebraic varieties. Given a rational number of the form mi , a number field K containing Q(e2πi/m ) and a prime ideal p of K coprime to the discriminant of K, there is a natural map which assigns a multiplicative character ιp ( mi ) to the residue field of p, see (42). It is compatible with field extensions of K. This map, our setting for the finite field period functions, and the dictionary between the two settings give us a natural way to convert the classical hypergeometric functions to the hypergeometric functions over the residue fields. We illustrate the map by the following diagram a1 a2 · · · an+1 ιp (a1 ) ιp (a2 ) · · · ιp (an+1 ) P ; λ → 7 P ; λ; q(p) , n+1 n n+1 n b1 · · · bn ιp (b1 ) · · · ιp (bn ) where K = Q(e2πi/N ), N is the least positive common denominator of all ai ’s and bj ’s, p is any unramified prime ideal of the ring of integers of K, and q(p) denotes the size of the residue field. There is a similar diagram between the hypergeometric functions n+1 Fn and n+1 Fn . Thus when we speak of the finite field analogues of classical peorid (or hypergeometric) functions in this paper we mean the converted functions in the finite residue fields unless we specify otherwise. We show the following.


Theorem 1. Let a, b, c ∈ Q with least common denominator N such that a, b, a − c, b − c ∈ / Z and λ ∈ Q \ {0, 1}. Let K = Q(e2πi/N , λ) with the ring of integers OK and ` be any prime such that Q` (λ, ζN ) is a degree ϕ(N ) extension of Q` (λ). Then there is a 2-dimensional representation π of GK := Gal(K/K) over Q` (λ, ζN ) such that for each unramified prime ideal p of OK so that λ, 1 − λ can be mapped to nonzero elements in the residue field, then π evaluated at the Frobenius conjugacy class at p is an algebraic integer (independent of the choice of `), satisfying ιp (a) ιp (b) (1) Tr π(Frobp ) = −2 P1 ; λ; q(p) . ιp (c) Our proof makes use of generalized Legendre curves which are 1-dimensional hypergeometric varieties. Viewing the finite field period functions as the traces of finite dimensional Galois representations at Frobenius elements is helpful when we translate classical results to the finite field setting. Using Weil’s result in [62] on Jacobi sums being Grossencharacters, one can give similar interpretation for the normalized period functions n+1 Fn . For readers only interested in the finite field analogues, this section can be skipped as our later proofs are mainly about characters over finite fields. In Section 6, also focusing on the n = 1 case, we recall and reinterpret the finite field analogues of Kummer’s 24 relations considered by Greene [30]. These relations give us a lot of built-in 2 F1 symmetries. Then we use the dictionary between the classical and finite field settings to obtain a few finite field analogues of algebraic hypergeometric identities. For example, we recall the following formula (see [54, (1.5.20)] by Slater): for a, z ∈ C and |z| < 1, then √ 1−2a a − 21 a 1+ 1−z . ; z = (2) F 2 1 2 2a When a ∈ Q, this formula is related to a Dihedral covering group. It also satisfies the (∗) condition mentioned above. To see its finite field analogue, it istempting 2

√

to translate the right hand side into the corresponding character A 1+ 21−z using the dictionary. However, Theorem 1 implies one character is insufficient as the corresponding `-adic Galois representations should be 2-dimensional. In fact, we will show (see Theorem 13) that for Fq of odd characteristic, φ the quadratic character, A any character on F× q having order at least 3, and any z ∈ Fq , √ √ A Aφ 1 + φ(1 − z) 1+ 1−z 1− 1−z 2 2 ; z = A + A . 2 F1 2 2 2 A2 Note that the right hand side is well-defined as it takes value zero if 1 − z is not a square in Fq , and we use a bar to denote the complex conjugation. This generalizes a recent result of Tu and Yang in [56]. It implies the following formula corresponding × to a fusion rule for the representations of Dihedral groups: if A, B, AB, AB ∈ Fc q

have order larger than 2, then for z 6= 1 A φA B Bφ ; z 2 F1 ; z 2 F1 A2 B2 " AB φAB AB 2 z = 2 F1 2 F1 2 ; z +B 4 (AB)

# φAB ; z . (AB)2


5

In Section 7, we apply our main technique, that is to obtain analogues of classical formulas satisfying the (∗) condition using the dictionary between two settings, to prove a quadratic formula (Theorem 14) over finite fields. This is equivalent to a quadratic formula in [30]. The proof appears to be technical, but is very straightforward. It demonstrates that under the (∗) assumption our technique together with the explicit analysis on the finite field level have the capacity to produce the finite field analogues which are satisfied by all values in Fq . Based on the analysis of the proof, we will obtain the following more general statement. (In this statement we use variable x.) Theorem 2. Suppose there is a classical quadratic or higher degree formula relating two primitive 2 F1 functions with rational parameters and arguments of the form c1 x and c2 xn1 (1 − c3 x)n2 respectively, where n1 , n2 ∈ Z, n1 > 0 and c1 , c2 , c3 ∈ Q× . If this formula satisfies the (∗) condition, then the finite field analogue of the formula holds with a few extra terms of the form Ca δ(x − a) where a ∈ Fq such that either c1 a or c2 an1 (1 − c3 a)n2 equals 0 or 1 in Fq , or possibly a fixed number, depending on the classical formula, of additional characters of Fq . Here δ(x) = 0 if x 6= 0 and δ(0) = 1. As an example, we consider the following cubic formula ([27, (5.18)]): for a, x ∈ C 3a −3a 3x a −a 27x(1 − x)2 = 2 F1 . (3) 2 F1 1 ; 1 ; 4 4 2 2 It satisfies the (∗) condition and the assumptions of Theorem 2. We obtain the following explicit finite field analogue: Theorem 3. Let q be an odd prime power, A be any character such that A6 is not the trivial character, then for all x ∈ Fq , A 2 F1

A 27x(1 − x)2 ; = 4 φ # " A3 A3 3x − φ(−3)δ(x − 1) − φ(−3)A(−1)δ(x − 1/3). ; 2 F1 4 φ

As a consequence of this cubic formula, we obtain the following evaluation formula: Theorem 4. Assume that q is a prime power with q ≡ 1 mod 6. For A, χ, η3 ∈ × 6 6 3 3 Fc q such that η3 has order 3, and none of A , χ , (Aχ) , (Aχ) being the trivial character, then " # 3 X J(B χ, η3 )J(B χ, η3 ) X J(B χ, B χ) χ 3 A3 A ; = = A(−1) . 3 F2 3 χη3 )J(B, χη3 ) 4 J(η3 χ, χη 3 ) J(B, φ χ 3 3 B =A c B∈F× q B 3 =A3

This is the analogue of the result by Gessel and Stanton [27] which states that for n ∈ N, a ∈ C, B(1 + a + n, 23 )B(1 − a + n, 34 ) 1 + 3a 1 − 3a −n 3 F ; = . 3 2 3 −1 − 3n 4 B(1 + a, n + 23 )B(1 − a, n + 43 ) 2


In these statements we use J( , ) to denote the Jacobi sum and B( , ) for the beta function. See Section 2 for definitions. As we shall see later, the Jacobi sums are the finite field analogues of the beta function. It is worth mentioning that Gessel and Stanton’s proof uses a derivative and cannot be translated directly to the finite field setting. We include this case here to make the following point. The Galois perspective can predict beyond the domain governed by the (∗) condition. Of course, proving these additional predictions will require more techniques. We end this section with another cubic formula for 2 F1 hypergeometric functions satisfying the (∗) condition and the assumptions of Theorem 2. We use it and Theorem 2 to obtain another finite field algebraic hypergeometric identity corresponding to the triangle group (2,3,4), see Theorem 15 and (30) for notation. In summary, our setup and main technique allow us translate many things straightforward and the Galois perspective gives our guidelines to see beyond those satisfying the (∗) condition. We have to be careful: if a result requires additional n n n+1 structures, such as the Pascal triangle identity m + m−1 = m+1 , using the derivative structure, or any proof derived from the Lagrange inversion formula, we cannot expect a direct translation.1 Rather, we can translate any formula satisfying condition (∗) or other results that have a suitable geometric interpretation, such as in terms of periods or Galois representation. In this sense, we will give a finite field analogue of Ramanujan type formulas for 1/π, see §5.8. This paper is an extension of our previous attempts along the line of a hybrid approach using both finite field and Galois methods. In [17], the authors study Jacobian varieties arising from generalized Legendre curves. In [18], the authors use finite field hypergeometric evaluation formulas to compute the local zeta functions of some higher dimensional hypergeometric varieties. This information allows us to explain the geometry behind several supercongruences satisfied by the corresponding truncated hypergeometric functions. We prove these supercongruences using a method in [45] which is based on the local analytic properties of p-adic Gamma functions. 2. Preliminaries In this section, we describe the foundation for the finite field analogue of the hypergeometric function. To do so, we first recall preliminaries from the classical setting. For more details on the classical setting, see [4, 6, 54] and see [23, 24, 30] for the finite field setup. 2.1. Gamma and beta functions. Recall the usual binomial coefficient n! k!(n−k)! . For any a ∈ C and n ∈ N, define the rising factorial by

n k

=

(a)n := a(a + 1) · · · (a + n − 1), 1In [23] by Evans, there is an analogue for the derivative operator over finite fields ([23, (2.8)]) which satisfies an analogue of the Leibniz rule ([23, Theorem 2.2]). However it is not a derivation. In [31], Greene gives the Lagrange inversion formula over finite fields, see Theorem 9 below, and points out its drawbacks. We will come back to this point later.


7

and define (a)0 = 1. Note then, that (4)

(−1)k (−n)k = k!

n . k

We define the gamma function Γ(x) and beta function B(x, y) as follows. Definition 1. For Re(x) > 0, define Z

∞

Γ(x) :=

tx−1 e−t dt.

0

Definition 2. For Re(x) > 0, Re(y) > 0, define Z 1 B(x, y) := tx−1 (1 − t)y−1 dt. 0

By the functional equation of Γ(x), one has Γ(a + n) , Γ(a) which holds for n ∈ Z≥0 , while the gamma and beta functions are related via (5)

(a)n =

(6)

B(x, y) =

Γ(x)Γ(y) . Γ(x + y)

For a proof of (6), which is based on ideas of Euler, see Section 1.1 of [4]. Using the notation of the rising factorial and (4), the binomial theorem says that for a ∈ C and variable z, ∞ ∞ X X (−a)k k a a (−z)k = (7) (1 − z) = z . k k! k=0

k=0

Binomial coefficients, rising factorials, gamma and beta functions are all used frequently in the classical theory of hypergeometric functions. In fact, gamma and beta functions play more primary roles while binomial coefficients and rising factorials make the notation more compact. Also of fundamental importance are specific formulas involving the gamma function. First, we state Euler’s reflection formula. Theorem 5 (Euler’s Reflection Formula). For a ∈ C, π Γ(a)Γ(1 − a) = . sin(πa) The gamma function also satisfies multiplication formulas. Theorem 6 (Legendre’s Duplication formula). For a ∈ C, 1 1/2 2a−1/2 Γ(2a) (2π) =2 Γ(a)Γ a + . 2 In terms of the rising factorial, Legendre’s duplication formula gives (8)

(a)2n = 22n

a a + 1 , 2 n 2 n

∀n ∈ N.

The more general case is Gauss’ multiplication formula, given below. See Section 1.5 of [4] for details.


Theorem 7 (Gauss’ Multiplication formula). If m is a positive integer and a ∈ C, 1 m−1 (m−1)/2 ma−1/2 Γ(ma)(2π) =m Γ(a)Γ a + ···Γ a + . m m In terms of rising factorial, it takes the following form: for any n ∈ N m Y i mn (ma)mn = m a+ m n i=1 We end this subsection with a mention of the Lagrange inversion theorem for formal power series. If f (z) and g(z) are formal power series where g(0) = 0 and g 0 (0) 6= 0, then Lagrange’s inversion theorem gives a way to write f as a power series in g(z). In particular, one can write (9)

f (z) = f (0) +

∞ X

ck g(z)k ,

k=1

where

f 0 (z) kg(z)k and Res denotes the coefficient of 1/z. Lagrange’s original result can be found in [38], and Good [28] provides a simple proof, as well as generalizations to several variables. We discuss finite field analogues of this result in Section 2.3. ck = Resz

2.2. Gauss and Jacobi sums. We now lay the necessary groundwork for the finite field setting based on [21, 22, 24, 30]. We recall analogues of the gamma and beta functions, and consequently the binomial coefficient and the rising factorial, and we also state analogues of the formulas for Γ(x) given in Theorems 5-7. To begin, let Fq be a finite field, where q = pe is a power of an odd prime. Recall × × that χ is a multiplicative character on F× q if χ : Fq → C is a group homomorphism, × and that the set Fc of all multiplicative characters on F× forms a cyclic group of q

q

order q − 1 under composition. Throughout, we use the following notation ε : trivial character,

φ : the quadratic character.

Thus ε(a) = 1 for a 6= 0. We extend characters on F× q to all of Fq by setting χ(0) = 0, including ε(0) = 0.2 × Further, it is useful to define δ on either Fc or F by q

( δ(χ) :=

1 0

if χ = ε, if χ = 6 ε;

q

( 1 δ(x) := 0

if x = 0, if x = 6 0,

respectively following [30]. This definition will allow us to describe formulas which hold for all characters, without having to separate cases involving trivial characters. × Let ζp be a primitive pth root of unity, and A, B ∈ Fc q . We define the Gauss sum and Jacobi sum respectively as follows. X X (10) g(A) := A(x)ζptr(x) , J(A, B) := A(x)B(1 − x), x∈F× q

x∈Fq

2A different choice would be ε(0) = 1. However, we would prefer to work with functions that are multiplicative for all elements in Fq . Thus ε(0) = 0 is preferred.


9

where tr(x) is the trace of x from Fq to Fp . Notice that g(ε) = −1. While in general the Gauss sums depend on the choice of ζp , the Jacobi sums do not. For a more detailed introduction to characters, see Chapter 8 of [34] and [9]. We note that although [34] contains proofs of most results we give throughout the rest of this subsection, care must be taken for cases involving the trivial character ε, which is defined to be 1 at 0 in this reference. The Gauss sum is the finite field analogue of the gamma function, while the Jacobi sum is the finite field analogue of the beta function [24]. With this in mind, we observe that the following are finite field analogues of Euler’s reflection formula: (11)

g(A)g(A) = qA(−1) − (q − 1)δ(A)

(12)

1 A(−1) q − 1 = + δ(A). q q g(A)g(A)

Note that (12) can be observed from (11) by multiplying both right hand sides together. Furthermore, since g(ε) = −1, we can write (13) (14)

qA(−1) + (q − 1)δ(A) g(A) A(−1)g(A) q − 1 1 = − δ(A). q q g(A)

g(A) =

The finite field analogues of the multiplication formulas for Γ(x) given in Theorems 6 and 7 are the Hasse-Davenport Relations. The general version given below is a finite field analogue of Theorem 7. Theorem 8 (Hasse-Davenport Relation [39]). Let m ∈ N and q = pe be a prime × power with q ≡ 1 mod m. For any multiplicative character ψ ∈ Fc q , we have Y Y m −m g(χ). g(χψ) = −g(ψ )ψ(m ) c c χ∈F× χ∈F× q q χm =ε χm =ε Taking m = 2 and denoting φ as the character of order 2, one arrives at the analogue of Legendre’s duplication formula, Theorem 6: (15)

g(A)g(φA) = g(A2 )g(φ)A(4).

Additionally, one can relate Gauss sums and Jacobi sums as follows: g(A)g(B) + (q − 1)B(−1)δ(AB). g(AB) We note some special cases for Jacobi sums below. We have (16)

J(A, B) =

J(χ, χ) = −χ(−1) + (q − 1)δ(χ) J(χ, ε) = −1 + (q − 1)δ(χ). We now are ready to define finite field analogues for the rising factorial and the binomial coefficient.3 We let 3Note that our binomial coefficient differs from Greene’s by a factor of −q.

10JENNY FUSELIER, LING LONG, RAVI RAMAKRISHNA, HOLLY SWISHER, FANG-TING TU

g(Aχ) (A)χ := , g(A) A := −χ(−1)J(A, χ). χ

(17) (18)

× Then we see directly that for A, χ1 , χ2 ∈ Fc q ,

(A)χ1 χ2 = (A)χ1 (Aχ1 )χ2 , which is the analogue of the identity (a)n+m = (a)n (a + n)m , for integers n, m ≥ 0. The multiplication formula in Theorem 8 can be written in this notation as Y (Am )χm = χ(mm ) (Aη)χ . c η∈F× q η m =ε

Now we give a proof of the following useful fact about Jacobi sums, see (2.6) of n n Greene [30], which is the analogue of m = n−m . A Lemma 1. For any characters A, B on F× q , J(A, B) = A(−1)J(A, BA), i.e. B = A AB in the binomial notation. Proof. We observe by (16), (13), and (14), that J(A, B)

= = =

g(A)g(B) + (q − 1)B(−1)δ(AB) g(AB) g(A) qB(−1) + (q − 1)δ(B) + (q − 1)B(−1)δ(AB) g(B) g(AB) g(A)qB(−1) AB(−1) q−1 g(AB) − δ(AB) g(B) q q +(q − 1)δ(B) + (q − 1)B(−1)δ(AB) g(A)g(AB) + (q − 1)δ(B) g(B)

=

A(−1)

=

A(−1)J(A, AB).

The above proof demonstrates that one strategy for using the basic properties of Gauss and Jacobi sums is to keep track of the delta terms. 2.3. Lagrange Inversion. In [31], Greene gave a finite field analogue of the Lagrange inversion formula (see (9)). We will state the case when the basis of complex × valued functions on the finite field is made up of all multiplicative characters in Fc , q

together with δ(x) as follows: Theorem 9 ([31] Theorem 2.7). Let p be an odd prime and q = pe . Suppose f : Fq → C and g : Fq → Fq . Then X X X f (y) = δ(g(x)) f (y) + cχ χ(g(x)), y∈Fq y∈Fq c × χ∈ Fq g(y)=g(x)

g(y)=0


11

where cχ =

1 X f (y)χ(g(y)). q−1 y∈Fq

We will use this version in Section 4 to develop the finite field analogue hypergeometric function. 2.4. A dictionary between the complex and finite field settings. We now list a dictionary we will use for convenience. It is well-known to experts. Let N ∈ N, and a, b ∈ Q with common denominator N . n∈Z

↔

1 N

a=

i N,

b=

j N

trivial character ε

↔ a primitive character ηN of order N j × i ↔ A, B ∈ Fc q , A = ηN , B = ηN

a

↔

A(x)

a+b

↔

A(x)B(x) = AB(x)

a+b

↔

A·B

−a

↔

A

Γ(a)

↔

g(A)

(a)n = Γ(a + n)/Γ(a) R1 dx 0

↔

(A)χ = g(Aχ)/g(A) P

x x

Γ(a)Γ(1 −

a) = sinπaπ , m Y mn

(ma)mn = m

i=1

a∈ /Z i a+ m n

↔ ↔ ↔

x∈F

g(A)g(A) = A(−1)q, A 6= ε m Y i (Am )χm = χ(mm ) (Aηm )χ i=1

R1 Remark 1. It seems like Greene used q x∈F for the finite field analogue of 0 ·dx P in his work so that the ‘volume’ 1q x∈F 1 = 1. This can be seen from his choice of the binomial coefficient over finite fields which has an additional factor of q in the denominator compared to our version. Our choice above makes the point counting formula more natural, but does not affect the normalized period functions to be introduced later. P 1

3. Classical hypergeometric functions 3.1. Classical development. We now recall the basic development of the classical (generalized) hypergeometric functions. See [4, 6, 54] for a thorough treatment. The classical (generalized) hypergeometric functions n+1 Fn with complex parameters a1 , . . . , an+1 , b1 , . . . , bn , and argument z are defined by ∞ X a1 a2 · · · an+1 (a1 )k (a2 )k · · · (an+1 )k k (19) ; z := z . n+1 Fn (b1 )k · · · (bn )k k! b1 · · · bn k=0

They satisfy order n + 1 ordinary Fuchsian differential equations and the following inductive integral relation of Euler when Re(bn ) > Re(an+1 ) > 0, (see equation


(2.2.2) of [4]) (20)

a1

··· ···

an+1 −1 ; z = B(an+1 , bn − an+1 ) n+1 Fn bn Z 1 a1 a2 · · · tan+1 −1 (1 − t)bn −an+1 −1 · n Fn−1 · b1 · · · 0 a2 b1

an bn−1

; zt dt.

To relate these functions to periods of algebraic varieties, we first observe that by (4) and the binomial theorem, we may define for general a ∈ C, (21)

1 P0 [a; z]

:= (1 − z)−a =

∞ X (a)k k=0

k!

z k = 1 F0 [a; z].

Here we are using the notation 1 P0 to indicate a relationship to periods of algebraic varieties. In particular, when a, b, c ∈ Q the integral Z 1 a b ; z := tb−1 (1 − t)c−b−1 1 P0 [a; zt] dt 2 P1 c 0 Z 1 = tb−1 (1 − t)c−b−1 (1 − zt)−a dt 0

can be realized as a period of a corresponding generalized Legendre curve [5, 17, 63]. To relate it to 2 F1 , one uses (20) to see that when Re(c) > Re(b) > 0, 2 P1

a

Z 1 b ;z = tb−1 (1 − t)c−b−1 · 1 P0 [a; zt]dt c 0 Z 1 = tb−1 (1 − t)c−b−1 · 1 F0 [a; zt]dt 0 a b = B(b, c − b) · 2 F1 ; z . c

Inductively one can define the (higher) periods n+1 Pn similarly by a1 a2 · · · an+1 (22) n+1 Pn ; z := b1 · · · bn Z 1 a1 a2 · · · an an+1 −1 bn −an+1 −1 t (1 − t) ; zt dt. n Pn−1 b1 · · · bn−1 0 Again using the beta function, one can show that when Re(bi ) > Re(ai+1 ) > 0 for each i ≥ 1, a1 a2 · · · an+1 (23) n+1 Fn ; z = b1 · · · bn n Y a1 a2 · · · an+1 B(ai+1 , bi − ai+1 )−1 · n+1 Pn ;z . b1 · · · bn i=1

By definition (19) any given n+1 Fn function satisfies two nice properties: 1) The leading coefficient is 1; 2) The roles of upper entries ai (resp. lower entries bj ) are symmetric.


13

Clearly, the n+1 Pn period functions do not satisfy these properties in general. The hypergeometric functions can thus be viewed as periods that are ‘normalized’ so that both properties 1) and 2) are satisfied. 3.2. Some properties of hypergeometric functions with n = 1. For an excellent exposition of the topic, please see [10]. 3.2.1.

The hypergeometric function 2 F1

a

b ; z is a solution of the Hypergeoc

metric Differential Equation (24)

HDE(a, b; c; z) : z(1 − z)F 00 + [(a + b + 1)z − c]F 0 + abF = 0,

which is an order-2 Fuchsian differential equation with 3 regular singularities at 0, 1, and ∞, see [65]. 3.2.2. • Around z = 0, if c ∈ / Z, the solution space of HDE(a, b; c; z) is spanned by a b 1+a−c 1+b−c 1−c ; z and z ; z . 2 F1 2 F1 c 2−c • Around z = 1, if a + b − c ∈ / Z, one has the two independent solutions a b F ; 1 − z 2 1 1+a+b−c and (1 − z)c−a−b 2 F1

c−a

c−b ; 1−z . 1+c−a−b

• Around z = ∞, if a − b ∈ / Z, one has the two independent solutions a 1+a−c b 1+b−c z −a 2 F1 ; 1/z and z −b 2 F1 ; 1/z . 1+a−b 1+b−a 3.2.3. For a given HDE(a, b; c; z), the group of symmetries S4 acts projectively on the vector space of solutions. This group can be regarded as an extension of S3 , acting as permutations of the singular points 0, 1, ∞, by the Klein 4-group, which corresponds to the following transformations due to Pfaff: a b a c−b z −a F ; z = (1 − z) F ; 2 1 2 1 z−1 c c c−a b z −b , = (1 − z) 2 F1 ; c z−1 which can be obtained from the definition and a change of variables, and Euler’s formula: a b c−a c−b c−a−b (25) ; z =(1 − z) ; z , 2 F1 2 F1 c c which can be deduced from the Pfaff transformations.


3.2.4. Together, these give Kummer’s 24 solutions to the differential equation HDE(a, b; c; z) describing the analytic continuations of hypergeometric functions. For more details, please see [4], [54]. For example, the following equation expresses a b the analytic continuation of 2 F1 ; z near ∞, (see the second part of Theorem c 2.3.2 in [4]): a b a a−c+1 Γ(c)Γ(b − a) (26) 2 F1 ; z = · (−z)−a 2 F1 ; 1/z Γ(c − a)Γ(b) c a−b+1 b b−c+1 Γ(c)Γ(a − b) −b ; 1/z . + · (−z) 2 F1 Γ(c − b)Γ(a) b−a+1 3.2.5. For classical hypergeometric functions, there are many evaluation formulas useful for obtaining new transformation formulas, computing periods, or proving supercongruences [44, 45, 55, et al.]. They are obtained in various ways [4, 27, 54]. Firstly, one has the Gauss summation formula obtained by the beta integral: a b B(b, c − a − b) (27) ; 1 = , Re(c − a − b) > 0. 2 F1 B(b, c − b) c Next the Kummer evaluation theorem says B(1 + c − b, 2c + 1) c b (28) F . ; −1 = 2 1 B(1 + c, 2c + 1 − b) 1+c−b The useful Pfaff-SaalSch¨ utz formula (see [4, Thm. 2.2.6]) follows from comparing coefficients on both sides of (25). It says that for a positive integer n and a, b, c ∈ C, a b −n (c − a)n (c − b)n (29) ; 1 = . 3 F2 (c)n (c − a − b)n c 1+a+b−n−c 3.2.6. There are many quadratic or higher order transformation formulas for 2 F1 (and a few for 3 F2 functions). An example is the following quadratic formula [4, Thm. 3.1.1]. c 1+c c b −4z −c 2 2 −b F . ; z = (1 − z) F ; 2 1 2 1 c−b+1 c − b + 1 (1 − z)2 These formulas are obtained in various ways, see [29, 57, 60]. 3.2.7. Associated to a HDE(a, b; c; z) is a degree-2 monodromy representation of the fundamental group π1 (CP 1 \ {0, 1, ∞}, t0 ) where t0 ∈ CP 1 \ {0, 1, ∞} is determined up to conjugation in GL2 (C) [65]. When a, b, c ∈ Q, the image of the monodromy representation, called the monodromy group of HDE(a, b, c; z), is a triangle group of the form (30)

(e1 , e2 , e3 ) := hx, y | xe1 = y e2 = (xy)e3 i ,

where ei ∈ N∪{∞} is computable from a, b, c, and we can assume that e1 ≤ e2 ≤ e3 . Suppose c, a + b − c, a − b 6∈ Z. The monodromy group corresponding to a b ; z is GL2 (C)-conjugate to the triangle group generated by 2 F1 c 1 0 M0 = 0 e2πi(1−c)


M1 = A

1 0

where

0

e2πi(c−a−b)

A−1 , M∞ = B

e2πia 0

0

e2πib

15

B −1 ,

!

A

B=

Γ(c)Γ(c−a−b) Γ(c)Γ(a+b−c) Γ(c−a)Γ(c−b) Γ(a)Γ(b) = Γ(2−c)Γ(c−a−b) , Γ(2−c)Γ(a+b−c) Γ(1−a)Γ(1−b) Γ(a−c+1)Γ(b−c+1) ! Γ(c)Γ(b−a) Γ(c)Γ(a−b) Γ(c−a)Γ(b) Γ(c−b)Γ(a) (−1)c−1 Γ(2−c)Γ(a−b) (−1)c−1 Γ(2−c)Γ(b−a) Γ(1−a)Γ(b−c+1) Γ(a−c+1)Γ(1−b)

are given by the relations (26) and

(31)

2 F1

a

b a b Γ(c)Γ(c − a − b) ; z = F ; 1 − z 2 1 Γ(c − a)Γ(c − b) c a+b+1−c c−a c−b Γ(c)Γ(a + b − c) c−a−b (1 − z) + ; 1−z . 2 F1 Γ(b)Γ(a) 1+c−a−b

Example 1. For a = 1/4, b = 3/4, c = 1/2, the monodromy matrices are 1 0 0 −1/2 0 −1/2 M0 = , M1 = , M∞ = , 0 −1 −2 0 2 0 each of finite order and they satisfy −1 M0 M∞ = M1−1 = M∞ M0 .

So the monodromy group is isomorphic to the triangle group (2, 2, 4), which is also isomorphic to the Dihedral group of order 8. There is an explicit correspondence between a hypergeometric differential equation HDE(a, b; c; z) and a Schwarz triangle ∆(p, q, r) with p, q, r ∈ Q due to a theorem of Schwarz, which we state below (see [65]). Each ∆(p, q, r) is the symmetry group of a tiling, by triangles with angles pπ, qπ, and rπ, of either the Euclidean plane, the unit sphere, or the hyperbolic plane, depending on whether p + q + r is equal to, greater than, or less than 1, respectively. We note that ∆(p, q, r) is not the same as the triangle group (e1 , e2 , e3 ), as e1 , e2 , e3 are the denominators of p, q, r when they are written in lowest terms as rational numbers. Theorem 10 (Schwarz). Let f, g be two independent solutions to HDE(a, b; c; λ) at a point z ∈ H, and let p = |1 − c|, q = |c − a − b|, and r = |a − b|. Then the Schwarz map D = f /g gives a bijection from H ∪ R onto a curvilinear triangle with vertices D(0), D(1), D(∞) and corresponding angles pπ, qπ, rπ. In [50], Schwarz determined a list of 15 (unordered) triples {|1−c|, |c−a−b|, |a− b|} for which the corresponding HDE(a, b; c; z) have finite monodromy groups. Generalizations of Schwarz’s work include finding general n+1 Fn with rational parameters and finite monodromy groups by Beukers and Heckman [12, Theorem a b 4.8]. It is well-known that 2 F1 ; z is an algebraic function of z when the c monodromy group for HDE(a, b; c; z) is a finite group; we give an example below. Example 2. For a classical example, let a = Ni be reduced, with N2 < i < N . Then, to compute the monodromy group of HDE(a − 21 , a; 2a; z), we see that p = |1 − 2a|, q = 12 , r = 12 . Thus the corresponding triangle group is (2, 2, 1/|1 − 2a|), which can


be identified with a Dihedral group of size 2N . This corresponds to a tiling of the unit sphere, since 12 + 12 + |1 − 2a| > 1. The corresponding algebraic expression for the hypergeometric functions is equation (2), as we have seen before. In particular, when N = 3, i = 2, we have that the monodromy group of HDE( 61 , 23 ; 43 ; z) is triangle group (2, 2, 3), which can be identified with the Dihedral group of size 6. This corresponds to the following tiling of the sphere.

In [61], Vid¯ unas gave a list of several other algebraic hypergeometric 2 F1 functions. In [7], Baldassarri and Dwork determined general second order linear differential equations with only algebraic solutions. 4. Finite field analogues 4.1. Periods in the finite field setting. 4.1.1. For any function f : Fq −→ C, by the orthogonality of characters [30] or Greene’s Lagrange inversion formula over finite fields (Theorem 9), f has a unique representation X fχ χ(x), (32) f (x) = δ(x)f (0) + c χ∈F× q

where fχ =

1 X f (y)χ(y). q−1 y∈Fq

Later, we will use the uniqueness in the statement to obtain identities. Now, if × A ∈ Fc q and x ∈ Fq , then we have an expression of A(1 − x) as follows, which is an analogue of (7). (33)

A(1 − x) = δ(x) +

1 X J(A, χ)χ(x) q−1 c χ∈F× q

 1 X   J(A, χ)χ(x),  q−1 c = χ∈F× q   1,

if x 6= 0, if x = 0.


17

4.1.2. We now define a new version of hypergeometric functions for the finite field Fq parallel to §3.1. We start with a natural analogue to (21) by letting 1 P0 [A; x; q]

(34)

:= A(1 − x),

and then inductively defining A1 A2 . . . An+1 (35) n+1 Pn ; λ; q := B1 . . . Bn X A An+1 (y)An+1 Bn (1 − y) · n Pn−1 1 y∈Fq

A2 B1

... ...

An ; λy; q Bn−1

as in (22). We note the asymmetry among the Ai (resp. Bj ) by definition. Part of the reason we start with the analogue of the period, rather than the hypergeometric function is because the periods are very related to point-counting. When there is no ambiguity in our choice of field Fq , we will leave out the q in this notation and simply write A1 A2 . . . An+1 A1 A2 . . . An+1 ; λ := n+1 Pn ; λ; q . n+1 Pn B1 . . . Bn B1 . . . Bn When n = 1 we have, X A1 A2 (36) P ; λ = A2 (y)A2 B1 (1 − y)A1 (1 − λy). 2 1 B1 y∈Fq

To be explicit,

(37)

2 P1

A1

   2 B1 ), J(A2 , A X A2 1 ;λ = J(A1 , χ)J(A2 χ, B1 A2 )χ(λ), B1   q − 1

if λ = 0 if λ 6= 0

c χ∈F× q

  1 ),  J(A2 , A2 B X Lemma 1 A (−1) 2 = J(A1 χ, χ)J(A2 χ, B1 χ)χ(λ),    q−1

if λ = 0, if λ 6= 0.

c χ∈F× q

While they are equivalent, the second expression is more symmetric when the finite field analogue of binomial coefficients defined in (18) is used. We give a general formula for the n+1 Pn period function as follows:

n+1 Pn

A1

A2 B1

(−1)n+1 = · q−1

... ... n Y i=1

An+1 ;λ Bn ! Ai+1 Bi (−1)

X A1 χA2 χ An+1 χ χ(λ) ··· χ B1 χ Bn χ

c χ∈F× q

+ δ(λ)

n Y

J(Ai+1 , Ai+1 Bi ).

i=1

This is similar to Greene’s definition [30, Def. 3.10], though we note that our binomial coefficient differs from Greene’s by a factor of −q, and we define the value


at λ = 0 differently. Our version gives a natural reference point for the later normalization, like in the classic case. 4.2. Hypergeometric varieties. It is well known that Greene’s hypergeometric functions can be used to count points on varieties over finite fields [8, 17, 18, 25, 26, 37, 40, 47, 59], and we similarly use the period functions n+1 Pn for that purpose. Consider the family of hypergeometric algebraic varieties y N = xi11 · · · xinn · (1 − x1 )j1 · · · (1 − xn )jn · (1 − λx1 · · · xn )k .

Xλ :

We can count the points on Xλ in a manner similar to that in [18], where the case N = n + 1, is = n, jt = 1 for k = 1 and all s, t was considered. Different algebraic models are used for the hypergeometric algebraic varieties in [11]. We choose our model in the way which makes the next statement straightforward. Proposition 1. Let q = pe ≡ 1 mod N be a prime power. Let ηN be a primitive × order N character and ε the trivial multiplicative character in Fc q . Then −mk N −1 min mi1 X ηN ηN ... ηN #Xλ (Fq ) = 1 + q n + P ; λ; q . n+1 n min +mjn mi1 +mj1 ηN . . . ηN m=1 Proof. We begin as in the proof of Theorem 2 in [18], to see that #Xλ (Fq ) = 1 + q n +

N −1 X

X

m i1 ηN (x1 · · · xn in (1 − x1 )j1 · · · (1 − xn )jn (1 − λx1 · · · xn )k ).

m=1 xi ∈Fq

Then, by applying (35) n times, we see that

n+1 Pn

=

−mk ηN X

min ηN

min +mjn ηN

mj1 mi1 ηN (x1 )ηN (1

... ...

mi1 ηN

mi1 +mj1 ; λ; q

ηN

mjn min mk − x1 ) · · · η N (xn )ηN (1 − xn )ηN (1 − λx1 · · · xn ).

xi ∈Fq

This gives the result.

4.3. Hypergeometric functions over finite fields. In the classical case (see §3.1), a hypergeometric function is normalized to have constant term 1, obtained from the corresponding n+1 Pn function divided by its value at 0. Here we will similarly normalize the finite field period functions. We thus define 1 A1 A2 A1 A2 · P ; λ . (38) F ; λ := 2 1 2 1 B1 B1 J(A2 , B1 A2 ) The 2 F1 function A 1) 2 F1 1 A 2) 2 F1 1

satisfies A2 ; 0 = 1; B1 A2 A ; λ = 2 F1 2 B1

A1 ; λ , if A1 , A2 6= ε, and A1 , A2 6= B1 . B1


19

Claim 1) follows from the definition and (36); 2) will be proved in Proposition 8 below. Intuitively, with the additional Jacobi sum factor one can rewrite the right hand side of (37) using the finite field analogues of rising factorials with the role of A1 and A2 being symmetric. More generally, we define (39)

n+1 Fn

A1

A2 B1

··· ···

An+1 ; λ Bn 1

:= Qn

n+1 Pn

A1

A2 B1

. . . An+1 ;λ . ... Bn

J(Ai+1 , Bi Ai+1 ) A A2 . . . An+1 Definition 3. A period function n+1 Pn 1 ; λ or the correB1 . . . Bn A1 A2 . . . An+1 sponding n+1 Fn ; λ is said to be primitive if Ai 6= ε and B1 . . . Bn Ai 6= Bj for all i, j. Similarly, we will call a classical period or hypergeometric function primitive in this article if ai , ai − bj ∈ / Z for any ai , bj . i=1

Like the n = 1 case, the value of the n+1 Fn functions at λ = 0 is 1, and the characters Ai (resp. Bj ) are commutative for primitive n+1 Fn . 4.4. Comparison with Greene and McCarthy’s versions. Alternative definitions for hypergeometric functions over finite fields have been given by Greene [30] and McCarthy [46], among others. We denote Greene’s version by n+1 Fn

G

A1 ,

A2 , . . . , An+1 ;λ B1 , . . . , Bn

? A2 , . . . , An+1 ;λ B1 , . . . , Bn

and McCarthy’s version by

n+1 Fn

A1 ,

Our period functions are closely related to Greene’s Gaussian hypergeometric functions and both can be used to count points. The relationship between the two is n+1 Pn

A1 ,

A2 , . . . , An+1 ;λ B1 , . . . , Bn ! n Y A1 , n =q Ai+1 Bi (−1) n+1 Fn i=1

G A2 , . . . , An+1 ;λ B1 , . . . , Bn + δ(λ)

n Y

J(Ai+1 , Ai+1 Bi ).

i=1

In the primitive case, the normalized n+1 Fn -hypergeometric function defined in (39) is the same as McCarthy’s hypergeometric function over finite fields (defined


using Gauss sums instead of Jacobi sums) when λ 6= 0. A1 A2 · · · An+1 ; λ n+1 Fn B1 · · · Bn 1 A1 , A 2 , = Qn n+1 Pn B1 , i=1 J(Ai+1 , Bi Ai+1 ) A1 , A 2 , = n+1 Fn B1 ,

To be precise,

. . . , An+1 ;λ ..., Bn ? . . . , An+1 ; λ + δ(λ). ..., Bn

5. Galois representation interpretation In this section, we will interpret the finite field analogues of period and hypergeometric functions using a Galois perspective. See books by Serre [51, 52] on basic representation theory and Galois representations. Readers can choose to skip this section as most of the later proofs can be obtained using the setup in the previous sections. The Galois interpretation gives us a global picture and allows us to predict the finite field analogues of classical formulas quite efficiently. The approach below is derived from using work of Weil [62] to reinterpret some results in [17] on generalized Legendre curve. We believe the general picture is covered by Katz [35]. We first set more notation. Given a number field K, we use OK to denote its ring of integers and use GK := Gal(K/K) to denote its absolute Galois group. We call a prime ideal p of OK unramified if it is coprime to the discriminant of K. 5.1. Notation for the mth symbol. Let m be natural number and use O(m) to denote the ring of integers of Q(ζm ). For each prime ideal p ∈ O(m) that is coprime to m, let q(p) := |O(m)/p| and use Fp to denote the residue field. Necessarily, q(p) ≡ 1 (mod m). Define a map from O(m) to the set of mth roots

of unity together with 0 using the following mth symbol notation xp . m Definition 4. Define xp := 0 if x ∈ p, and if x ∈ / p, let the symbol take the m value of the mth root of unity such that x ≡ x(q(p)−1)/m mod p. (40) p m

In particular, for a fixed prime ideal p, the mth symbol is an order m multiplicative character for the residue field Fp . For explicit examples of the mth symbols with m = 2, 3, 4, 6, see [34]. n Let σ ∈ Gal(Q(ζm )/Q) such that σ(ζm ) = ζm for some n ∈ (Z/mZ)× , by definition σ−1 x x (q(p)−1)/m σ −n(q(p)−1)/m (41) ≡ x mod p ≡ x mod p = . pσ m p m This gives the relation between mth symbols for conjugate prime ideals. Definition 5. Now for any fixed rational number of the form mi with i, m ∈ Z, we define a map ι· mi from the unramified prime ideals p of O(m) to characters of the corresponding residue fields by i i · (42) ιp = . m p m


21

By definition, for any integers i, j, m, i+j i i j −i (43) ιp = ιp ιp , and ιp = ιp . m m m m m Example 3. For each prime ideal with residue field of odd characteristic, ιp ( 21 ) = φ, the quadratic character. × Conversely, for any character A ∈ Fc q where q is an arbitrary prime power, we have another map κFq depending on the residue field Fq which assigns a rational number to A. It is defined as follows n , such that A(x) = xn/(q−1) for all x ∈ Fq . (44) κFq (A) := q−1

Then for any a =

i m,

for any prime ideal ℘ of O(m) coprime to m κFp (ιp (a)) = a

mod Z.

Next we will see the mth symbol notation is compatible with field extensions and thus the map ι· mi can be extended to finite extensions of Q(ζm ). Recall that × one can lift a multiplicative character A ∈ Fc to any finite extension of F by using q

q

the norm map. Let L be a finite extension of Q(ζm ) and let ℘ be a prime ideal in the ring of integers OL of L above p, both coprime to the discriminant of L. Then OL /℘ is the finite extension of O(m)/p and we assume the degree of the extension is f . To compare the role of mth symbols for x ∈ OL , we have ! f −1 q(℘)−1 q(p)f −1 x x · xq(p) · · · xq(p) (45) =x m . mod ℘ = x m mod ℘ = ℘ m p m · On the residue field level, this means the ℘ symbol on OL /℘ can be computed m

from thenorm map from F℘ := OL /℘ to Fp := O(m)/p composed with the induced map

· p

m

on Fp .

In fact, the ιp (·) (·) gives a map (or ‘functor’) which sends any hypergeometric series a1 a2 · · · an+1 ; λ n+1 Pn b1 · · · bn with rational parameters and algebraic arguments to a collection of hypergeometric functions over finite (residue) fields ιp (a1 ) ιp (a2 ) · · · ιp (an+1 ) P ; λ; q(p) n+1 n ιp (b1 ) · · · ιp (bn ) where p run through all unramified prime ideals of Q(λ, ζm ) with m being the least positive common denominators of all ai and bj . We will see explicitly for the n = 1 case below. 5.2. Jacobi sums and Grossencharacters. We now recall the result of Weil which is relevant to our discussion below. Weil computed that local zeta functions for (homogeneous) Fermat curves of the form X n + Y n = Z n or special generalized Legendre curves of the form y N = xm1 (1 − x)m2 (cyclic covers pf CP 1 only ramify at 0, 1, ∞). In both case the local zeta functions can be expressed in terms of explicit Jacobi sums see [34]. In [62], Weil explained how to consider Jacobi sums


as ‘Grossencharacters’ (or Hecke) characters, to be recalled below. In addition to the notation we introduced previously, we will mainly be using Weil’s notation below. Definition 6. [See Weil [62]] Let K be a degree d number field and OK its ring of integers. Let m be an ideal of OK and use I(m) to denote ideals of OK that are prime to m. A Grossencharacter of K with respect to m, according to Hecke, is a complex-valued function on I(m) such that (1) f (a)f (b) = f (ab) for all a, b ∈ I(m) (2) There are rational integers ei and complex numbers ci , with 1 ≤ i ≤ d, such that if a ∈ OK and a ≡ 1 mod m, then the value of f at the principal ideal (a) satisfies d Y c f ((a)) = aei i |ai | i i=1

where a1 = a, a2 , · · · , ad are the embeddings of a to C, and | · | denotes the complex absolute value. Example 4. Let K = Q and m = Z, the set I(m) contains all ideals of Z, which are all principal. Define a function T on I(m) by for any a ∈ Z, T ((a)) = a. Then T is a Grossencharacter of Q. By definition, the set of Grossencharacters is closed under multiplication and division. Two Grossencharacters are equivalent if they coincide whenever they are both defined. The classes of equivalent Grossencharacters are in a one-to-one correspondence with the homomorphisms from the group of idéle-classes of K to C× . By class field theory, they correspond to quasicharacters of the Galois group GK := Gal(K/K). For convenience, we sometime use the same notation for both the Grossencharacter and the corresponding character of GK . Let K = Q(λ, ζm ), with ζm a fixed primitive mth root of unity. Let p be a unramified prime ideal of K. We will now give Weil’s result (for his r = 2 case). Let a = ( am1 , am2 ) with ai ∈ Z. For each prime ideal p coprime to m, let a1 a +a a x 1−x 2 −1 1 2 X Ja (p) := − p p m p m x∈OK /p a1 +a2 a −1 a1 2 =− J ιp , ιp . p m m Next we extend Ja to all unramified ideals in OK by using (1) above, i.e. Ja (ab) = Ja (a)Ja (b). Theorem 11 (Weil, [62]). The map Ja is a Grossencharacter of K. Namely, Weil showed that (2) above also holds for this map. By class field theory, Ja corresponds to a linear quasi-character of GK . Here is a special class of Grossencharacters corresponding to finite order characters of GK . Given√an order m character of GK , it corresponds to a cyclic Galois extension field K( m c) of K which contains ζm by our assumption, for some c ∈√K. For a fixed a = mi and any prime ideal p coprime to the discriminant of K( m c), let i c i χ mi ,c (p) := ιp (c) = m p m


23

and extend this map to all unramified ideals of OK multiplicatively like (1) in Definition 6. Using elementary congruence properties, we can √ verify that equivalently, for any ideal a of OK coprime to the discriminant of K( m c) with finite quotient OK /a, one can define χ mi ,c (a) ≡ c(|OK /a|−1)i/m

mod a.

1 When mi = 2 and K = Q and c ∈ Q, this step means we extend the Legendre c symbol · to the Jacobi symbol which applies to all integers coprime to c, see [34]. For fixed mi and c, the map χ mi ,c also satisfies the condition (2) of Definition of 6. What we want to conclude is that finite order characters of GK give rise to Grossencharacters of K. Rather than giving full details, we present an example here.

Example 5. Let K = Q, c = −1 in which case d = 1 and the discriminant of √ Q( −1) is 4. In this case, if e1 = 1, c1 = −1 then the condition (2) holds for our map χ 21 ,−1 . With this in mind, it is no surprise that sometimes quotients of Jacobi sums could be finite order characters. Example 6. By Example 2 of [17], we have for m = 10 and p being any unramified prime ideal of OQ(ζ10 ) , then 8 2 6 /J 2 5 8 1 = χ 10 J( 10 , 10 ) ( 10 , 10 ) (p) = ,2 (p). p 10 Sometimes, one can even get quasi-characters. 2 5 /J 3 4 Example 7. It is shown by Yamamoto see [64, §20] that J( 12 , 12 ) ( 12 , 12 ) is not a character, but its square is. To be more precise, Yamamoto showed that 2 27/4 1 27 (p). 2 5 /J 3 4 = χ 12 J( 12 (p) = , 12 ) ( 12 , 12 ) , 4 p 12

5.3. Galois interpretation for 1 P0 . To illustrate our ideas, we will first discuss the Galois representation background behind the 1 P0 function defined in (34) using the ι· map. Recall that by our notation, i i P ; λ = (1 − λ)− m . 1 0 m For each ‘good’ prime √ ideal p of Q(λ, ζm ) (in the sense that it is coprime to the discriminant of Q(ζm , m 1 − λ), we have i i i ; λ; q(p) = ιp (1 − λ) = ιp − (1 − λ) = χ− mi ,1−λ (p), 1 P0 ιp m m m by definition. Thus 1 P0 [ιp (·); λ; q(p)] provides a map which sends rational numbers of the form mi to finite order characters of the Galois group GQ(λ,ζm ) which fix GQ( m√1−λ,ζm ) . Further for fixed mi and λ ∈ Q, one can use these 1 P0 functions to compute the corresponding Artin L-function, which will take the form of (46) −1 Y i i L , λ; s : “ = ” 1 − 1 P0 ιp ; λ; q(p) q(p)−s . m m good prime p∈OQ(λ,ζm )


Here “ = ” means we only include those ‘good’ prime ideals p that are coprime to the discriminant of GQ( m√1−λ,ζm ) . Example 8. For

i m

=

1 2

and λ = −1 as Example 5, we have Y

L(1/2, −1; s) =

p odd prime

−1 −1 −s 1− p . p

This is the Dirichlet L-function for the Dirichlet Character χ 12 ,−1 . 5.4. Galois interpretation for 2 P1 . When n = 1, the hypergeometric varieties that we introduced in §4.2 are the generalized Legendre curves, giving rise to generically GL(2) Galois representations. For full details, see [17]. As a motivation, we start with the Legendre curves. Classical, the Legendre curves Lλ : y 2 = x(1 − x)(1 − λx) are well-understood, see [53]. We summarize a few relevant properties of these Legendre curves here. • It is a double over of CP 1 which ramifies at 0, 1, λ1 , ∞ only. • For any λ ∈ Q, the curve Lλ is an algebraic curve defined over Q with genus being 1 unless λ = 0 or 1, in either case the genus of Lλ is 0. • When λ ∈ Q \ {0, 1}, the unique up to scalar differential on Lλ is of the dx form ωλ := √ . x(1−x)(1−λx) 1 1 1 1 R1 2 2 2 2 • A period of Lλ is 0 ωλ = 2 P1 ; λ = π · 2 F1 ; λ . The mon1 1 odromy group of HDE( 21 , 12 ; 1; λ) is isomorphic to the arithmetic triangle group (∞, ∞, ∞). • For simplicity, we assume λ ∈ Q\{0, 1}, then there is a compatible family of 2-dimensional `-adic representation ρ`,λ of GQ constructed from the Tate module of Lλ (see [53]) such that for any odd prime p not dividing the discriminant N (Lλ ) of the elliptic curve Lλ , X φ φ Tr ρ`,λ (Frobp ) = − φ(x(1 − x)(1 − λx)) = − 2 P1 ; λ ε x∈Fp

and det ρ`,λ (Frobp ) = p, where Frobp stands for the Frobenius conjugacy class of GQ at p. • For λ ∈ Q \ {0, 1}, the L-function of Lλ is (47)

L(Lλ , s) : “ = ”

Y

1 + 2 P1

p-N (Lλ )

φ

−1 φ −s 1−2s ; λ p +p . ε

Now we turn our attention to the generalized Legendre curves. Let a, b, c ∈ Q such that c > b, a, b, a − c, b − c ∈ / Z. Pick (48)

N = lcd(a, b, c),

i = N · (1 − b),

j = N · (1 + b − c),

k = N · a,


where lcd means the least positive common denominator. Then 2 P1

25

a

b ; λ is a c

period of the following generalized Legendre curve [N ;i,j,k]

Cλ

: y N = xi (1 − x)j (1 − λx)k ,

as in [17]. In this paper we discuss how to get the smooth models and recall how [N ;i,j,k] to compute all periods of first and second kind on Cλ using hypergeometric functions. Like Lλ , this curve is also cyclic cover of CP 1 ramified at four points 0, 1, λ1 , and ∞. When λ = 0 or 1, the covering map only ramifies at 3 points and is a quotient of a Fermat curve. This is the reason behind the degenerate situation [N ;i,j,k] happening at λ = 0 or 1. The curve Cλ admits a natural map ZN : (x, y) 7→ (x, ζN y) where ζN is any primitive root of unity of order N . We assume λ ∈ Q \ {0, 1} and also N - i, j, k, i + j + k, which is to assure there are 4 ramification points for generic [N ;i,j,k] λ. Then the smooth model of Cλ is an algebraic curve of genus g(N ; i, j, k) := 1 + N −

gcd(N, i + j + k) + gcd(N, i) + gcd(N, j) + gcd(N, k) , 2 [N ;i,j,k]

see [5] by Archinard. We use Jλ to denote the Jacobian of the smooth model [N ;i,j,k] of Cλ , which is an abelian variety of dimension given by g(N ; i, j, k) and is [N ;i,j,k] defined over Q(λ). For each proper divisor d of N , Jλ contains a factor which [d;i,j,k] prim is isogenous to Jλ over Q(λ, ζN ). Use Jλ to denote the primitive part of [N ;i,j,k] Jλ , such that none of its simple sub abelian varieties are isogenous to the [d;i,j,k] simple factors of Jλ for any proper divisor d of N . By work in [5], Jλprim is of dimension ϕ(N ), the Euler number of N , and is defined over Q(λ). Thus the Tate module of Jλprim gives rise to a compatible family of degree 2ϕ(N )-dimensional àdic Galois representations ρprim of the group Gal(Q/Q(λ)). Evaluate for any good prime p with q(p) = q, one has   X X m  Tr ρprim (Frobp ) = − ηN xi (1 − x)j (1 − λx)k  m∈(Z/N Z)×

x∈Fq

where ηN denotes any order N character in Fq . For a related discussion, see Proposition 1; for more details, see [17]. Now fix a prime ` such that Q` (λ, ζN ) is a degree ϕ(N ) extension of Q` (λ). Observe that the map ZN induces an order N automorphism on Jλprim and hence the representation spaces of ρprim over Q` (λ). Consequently, the restriction satisfies ρprim |GQ(ζN ,λ) ∼ = σ τ1 ⊕ σ τ2 · · · ⊕ σ τϕ(N ) , where σ is a compatible family of 2-dimensional `-adic representations of GQ(ζN ,λ) and τ1 , · · · τϕ(N ) are different embeddings of Q(λ, ζN ) in Q` depending on the choice primitive N th root. In Proposition 13 of [17], we show that each P of the m − x∈Fp ηN xi (1 − x)j (1 − λx)k agrees with Tr σ(Frobp ) up to an embedding, where p is any unramified prime ideal of OQ(ζN ,λ) . To prove Theorem 1 as stated m in the introduction, we will need the compatibility among the choices of ηN when we vary the prime ideals.


Proof. In the proof for Proposition 13 of [17], it is showed that for any fixed unramified prime ideal p of OK , then X 1 τu ιp (49) Tr σ (Frobp ) = − (xi (1 − x)j (1 − λx)k ), N x∈Fp

for some u ∈ [1, · · · , ϕ(N )], where ιp ( N1 ) is a character as defined in 5. Consider the twisting curve y N = cxi (1 − x)j (1 − λx)k √ where c ∈ K. This curve is isomorphic to y N = xi (1 − x)j (1 − λx)k over K( N c). Use ρprim to denote the primitive part of the compatible family of Galois representac τϕ(N ) . tions associated with the twisted curve. Then ρprim |GQ(ζN ,λ) ∼ = σcτ1 ⊕σcτ2 · · ·⊕σc c Moreover, σ and σc differ by some order N character of GQ(ζN ,λ) with kernel containing GQ(ζN ,λ, N√c) . It is due to the models we use and is in fact independent of n the irreducibility of σ. Thus σc = σ ⊗ c· N for some integer n ∈ (Z/N Z)× . By n τ u = c· N . This means τu only depends on n, not p. In equation (49), c· N conjunction with Prop. 13 in [17], our claim holds by letting π = σ τu . Remark 2. In other words, for fixed rational numbers a, b, c, the corresponding i h ι (a) ι (b) −2 P1 p ιpp (c) ; λ; q(p) functions are the traces (or characters) of 2-dimensional Galois representation of Gal(Q/Q(λ, ζN )) at the Frobenius elements. They allow one to determine the Galois representation up to semisimplification and compute the determinants of the representations at Frobenius elements, see (54) below. Consequently, one can use these period functions to compute the Euler p-factors of the L-function of the corresponding Galois representations, like equations (46) and (47). Remark 3. From the representation point of view, the products (resp. sums) of 2 P1 functions over the same finite field correspond to tensor products (resp. direct sums) of the corresponding representations. See [51]. The Galois perspective is quite helpful for understanding results on hypergeometric function over finite fields. For instance, the following analogue of Kummer’s evaluation (see (28)) expresses the value of the 2 P1 function at −1 in terms of two × not one Jacobi sums. To be more precise, let B, D, φ ∈ Fc where φ is of order 2. q

Then (50)

2 P1

B

C ; −1 = J(D, B) + J(Dφ, B), CB

where C = D2 . This is proved by Greene in [30, (4.11)] and is also a natural byproduct of the quadratic transformation stated in Theorem 14. When λ = 0 or 1, the 2 P1 function corresponds to a dimension 1 (instead of 2) compatible family of Galois representations. When λ = 1, the following analogue of Gauss evaluation formula (27) follows from the definition X A B P ; 1; q = B(y)ABC(1 − y) = J(B, ABC). 2 1 C y∈Fq


27

Now we relate the Theorem 1 to some results in the twisted exponential sums, which deals with the finite field period functions 2 P1 ‘vertically’. Given A, B, C ∈ × Fc q , by our notation X A B B(y)BC(1 − y)A(1 − λy). P ; λ; q = 2 1 C y∈Fq

As mentioned earlier, any character A on F× q can be extended to a multiplicative F

r

character Ar on the finite extension Fqr using the norm map NFqq , i.e. for x ∈ Fqr , F

r

Ar (x) = A(NFqq (x)). For instance, if x ∈ Fq , then r−1

r

Ar (x) = A(x · xq · · · xq ) = A(x · x · · · x) = A(xr ) = (A(x)) . r r Thus we can define 2 P1 Ar B accordingly. Using the perspective of twisted Cr ; λ; q Jacobi sums [1, 2], the generating function   r X T  Ar B r (52) Z[A, B; C; λ; q; T ] := exp  ; λ; q r · 2 P1 Cr r

(51)

r≥1

is a rational function, which is originally due to Dwork. In view of the Theorem 1 and the Weil conjectures (proved by Deligne) for local × zeta functions of smooth curves (cf. [34, §11.3]), we know for A, B, C ∈ Fc q such A B that 2 F1 ; λ is primitive with λ \ {0, 1}, the function Z[A, B; C; λ; q; T ] is C √ a degree-2 polynomial over Q with two roots of the same absolute value 1/ q. To give an example, we first recall another Hasse-Davenport relation (see [9]) which says that the Gauss sum of g(A) over Fq and the Gauss sum g(Ar ) over Fqr are related by (53)

g(Ar ) = (−1)r−1 g(A)r .

Example 9. By (53) and the finite field analogue of Kummer’s evaluation (50), one has when C = D2 , and B 2 6= C, that Z(B, C; CB; −1; q; T ) = (1 + J(D, B)T )(1 + J(Dφ, B)T ). In comparison, the finite field analogue of Gauss evaluation formula implies Z[A, B; C; 1; q; T ] = 1 + J(B, ABC)T ,

if A 6= C.

As a corollary, we can describe the determinants of the 2-dimensional Galois representation π of Theorem 1 at Frobenius elements as well. Corollary 1. Notation as Theorem 1. We have (54) h i2 h 1 ιp (a) ιp (b) (ιp (a))2 det π(Frobp ) = ; λ; q(p) + P 2 P1 2 1 ιp (c) 2

(ιp (b))2 (ιp (c))2

2

; λ; q(p)

i

Together, we are able to compute the L-function L(π, s) of π if we obmit the ramified prime factors. It will take the form of −1 Y ιp (a) ιp (b) 1 + 2 P1 ; λ; q(p) q(p)−s + det π(Frobp ) · q(p)−2s . ιp (c) p good prime


5.5. Some special cases of 2 P1 -functions. When λ ∈ Fq \ {0, 1}, and A, B 6= ε or C, the Galois representation associated with 2 P1 [ A B C ; λ] is 2-dimensional and pure in the sense that Z[A, B; C; λ; q; T ] = 1 − aq T + bq T 2 √ has two roots of absolute value 1/ q. On the opposite side, we have the impure a b cases below for the im-primitive cases, which correspond to 2 F1 ; λ with c either a, b, a − c, b − c ∈ Z. Proposition ε 2 P1 A 2 P1 A P 2 1 A P 2 1

2. Suppose λ 6= 0. Then, B ; λ = J(B, BC) − C(λ)BC(λ − 1); C B ; λ = B(λ)J(B, A) − A(1 − λ); B B ; λ = B(λ − 1)J(B, A) − B(−1)A(λ) + (q − 1)δ(1 − λ)δ(B); A ε ; λ = C(−λ)AC(1 − λ)J(C, A) − 1 + (q − 1)δ(1 − λ)δ(AC). C

Proof. When λ 6= 0, X X ε B B(y)BC(1 − y) P ; λ = B(y)BC(1 − y)ε(1 − λy) = 2 1 C y y6=1/λ X = B(y)BC(1 − y) − C(λ)BC(λ − 1) = J(B, BC) − C(λ)BC(λ − 1); y

2 P1

A

X X B B(y)A(1 − λy) ;λ = B(y)ε(1 − y)A(1 − λy) = B y

y6=1

= B(λ)J(B, A) − A(1 − λ). Due to the relation J(A, B)J(C, A) = B(−1)J(C, B)J(CB, AB) − δ(A)(q − 1) + δ(BC)(q − 1), A B the function 2 P1 ; λ can be written as A B(−1) X A B J(Aχ, χ)J(B χ, Aχ)χ(λ) ;λ = 2 P1 A q−1 χ =

X B(−1) J(B, A) J(B χ, χ)χ(−λ) − B(−1)A(λ) + δ(B)δ(1 − λ)(q − 1) q−1 χ = B(−1)J(B, A)B(1 − λ) − B(−1)A(λ) + δ(B)δ(1 − λ)(q − 1) = J(B, A)B(λ − 1) − B(−1)A(λ) + δ(B)δ(1 − λ)(q − 1).

Using the same type of argument, we get the last claim.


29

5.6. Galois interpretation for n+1 Fn . As mentioned earlier, when the parameter λ ∈ Q, we have that the period function n+1 Pn corresponds to the trace of a Galois representation of degree at most n. The normalized period, which is obtained by dividing from n+1 Pn a few Jacobi sum factors, corresponds to the tensor product of the Galois representation associated with n+1 Pn and a linear character associated with the Grossencharacter arising from the Jacobi sums as in §5.2. When n = 1 and λ ∈ Q \ {0, 1}, by Theorem 1 we have under the same assumptions, there is a 2-dimensional Galois representation π õver Q` (λ) such that ι (a) ιp (b) for any good prime ideal of OQ(λ,ζN ) , Tr π ˜ (Frobp ) = −2 F1 p ; λ; q(p) . ιp (c) Moreover, the characteristic polynomial of π ˜ (Frobp ) is degree 2 with two roots of absolute value 1. This will be particularly convenient when taking tensor products, which will be seen below. 5.7. A finite field version of Clausen formula by Evans and Greene. One version of the Clausen formula [6] states that 2 2c − 2s − 1 2s c − 12 c − s − 21 s ; λ = F ; λ . (55) F 3 2 2 1 c 2c − 1 c In terms of differential equations, this means that the symmetric square of the 2dimensional solution space of HDE(c − s − 21 , s; c; λ) is the 3-dimensional solution space of the hypergeometric differential equation satisfied by the 3 F2 ocurring on the right side of (55). In [19], Evans and Greene obtained the following analogue of (55). × Theorem 12 ([19], Theorem 1.5). Let C, S ∈ Fc q . Assume that C 6= φ, and S 2 6∈ ε, C, C 2 . Then for λ 6= 1, # " 2 2 CSφ S S 2 Cφ C 2S ; λ ; λ = 3 F2 2 F1 C C2 C ! 2 J(S , C 2 ) + φ(1 − λ)C(λ) + δ(C)(q − 1) . J(C, φ)

This theorem says the tensor product of the ‘normalized’ 2-dimensional representation on the left (when λ 6= 0, 1) equals its symmetric square, which is a 3-dimensional representation associated with the 3 F2 on the right, plus an additional linear representation from the exterior power. This expression is closer to the complex setting than the statement given in Theorem 1.5 of [19] which is in terms of period functions. We remark here that when λ = 1, by Theorem 4.38-(i) in [30] we have # " 2 X φ(−1)J(D, CS 2 )J(CS 2 , φD) C 2S S 2 Cφ ; 1 = F 3 2 2 C2 C J(φ, φC)J(S 2 , C 2 S ) c × D∈Fq D 2 =S 2

=

X J(D, CS 2 )J(CS 2 , φD) , J(φS, C)J(S, C) c

D∈F× q D 2 =S 2


which corresponds to a 2-dimensional Galois representation, while Gauss’ evaluation theorem says 2 CSφ S J(φ, S)2 F . ; 1 = 2 1 C J(S, C)2 One nice example which realizes the Clausen formula geometrically is given by Ahlgren, Ono and Penniston [3]. In their work, they consider the K3 surfaces defined by Xλ : s2 = xy(1 + x)(1 + y)(x + λy), λ 6= 0, −1, in relation to the elliptic curves Eλ : y 2 = (x − 1)(x2 − 1/(1 + λ)),

λ 6= 0, −1.

In particular, the point counting on Xλ over Fq is related to X φ, φ, φ(xy(1 + x)(1 + y)(x + λy)) = 3 P2 ε, x,y∈Fq

φ ; −λ . ε

When q ≡ 1 (mod 4), the point counting on Eλ over Fq (or Fq2 if needed), which is given by X a(λ, q) := − φ(x − 1)φ(x2 − 1/(1 + λ)), x∈Fq

η4

η4 ; −λ where η4 is an order 4 character. This follows from ε × the quadratic formula in Theorem 14 with B = C = φ ∈ Fc and x = (b + 1)/(b − 1) is essentially 2 P1

q

with b2 = 1 + λ, for some b ∈ F. Thus Theorem 1.1 of [3] is equivalent to the Clausen formula over the finite field Fq with S = η4 and C = ε. For special choices of λ ∈ Q such as 1, 8, 1/8, 4, 1/4, the corresponding elliptic curve Eλ has complex multiplication (CM). For these λ values, the period functions 2 P1 can be written in terms of Jacobi sums. Example 10. When λ = 1/8, the elliptic curve E1/8 is Q-isogenous to the curve y 2 = x3 − 9x which has complex multiplication and is of conductor 288 [42]. Hence, φ φ −a(1/8, q) =φ(3) · 2 P1 ; −1 ε ( 0, if q ≡ −1 (mod 4), =φ(3) · J(φ, η4 ) + J(φ, η 4 ), if q ≡ 1 (mod 4). By the discussion in §5.2, a corresponding Grossencharacter √ √ (Hecke character) ψ is given by χ 12 ,−3 · J( 21 , 41 ) . Explicitly, for any a + b −1 ∈ Z[ −1], √ √ √ ψ(a + b −1) = (−1)b (a + b −1)χ1 (a + b −1), where

 a+b−1 √  −1, (−1) 2 √ a+b−1 χ1 (a + b −1) = (−1) 2 ,   0,

if a ≡ 0 (mod 2), b ≡ 1 (mod 2), if a ≡ 1 (mod 2), b ≡ 0 (mod 2), otherwise.

This is computed using the fact that E1/8 is a twist of an elliptic curve with complex multiplication of conductor 64 [43].


31

Example 11. When λ = −1/4, the elliptic curve E−1/4 is Q-isgenous to the curve y 2 = x3 − 1, which has complex multiplication and is of conductor 144 [41]. Thus a(−1/4, q) can be expressed in terms of the Hecke character attached to a CMelliptic curve of conductor 144. It follows that for q ≡ 1 (mod 3), ( 0, if q ≡ −1 (mod 3) −a(−1/4, q) = φ(−1) · J(φ, η3 ) + J(φ, η 3 ), if q ≡ 1 (mod 3), where η3 is a multiplicative character of order 3. 5.8. Analogues of Ramanjuan type formulas for 1/π. In 1914, Ramanujan [48] gave a list of infinite series formulas for 1/π, where the series are values of hypergeometric functions. One example is 1 3 k ∞ X ( 2 )k 1 4 (56) (6k + 1) = . k! 4 π k=0

Later in the 1980’s, Borwein-Borwein [14] and Chudnovsky-Chudnovsky [16] proved these formulas using essentially the theory of elliptic curves with complex multiplication. The idea of their proof was recast in [15] and we will give some related discussion here. Roughly speaking, the family of elliptic curves Eλ defined in Section 5.7 have unique holomorphic differentials ωλ , up to a scalar, similar to the case for the Legendre curve Lλ that we mentioned earlier. Integrating ωλ along a suitable chosen path on Eλ leads to a period of first kind on Eλ given by 1 1 1 3 1/4 4 4 p(λ) = Γ Γ (1 − λ) ; λ . 2 F1 4 4 1 √ By the reflection formula in Theorem 5, we have Γ( 41 )Γ( 34 ) = 2π. One can d compute periods of the second kind on Eλ from dλ p(λ). Together, differentaials of the first and second kind for Eλ form a 2-dimensional vector space V (λ) and their λ corresponding period space coincides with the solution space of HDE( 14 , 34 ; 1; λ−1 ). When Eλ has CM, its endomorphism ring R := R(λ) acts on V (λ). In particular, ωλ is an eigenfunction of R. Let ηλ be another eigenfunction of R, that is linearly independent from ωλ . Then Chowla-Selberg says that for any cycle γλ of Eλ , Z Z ωλ · ηλ ∼ π, γλ

γλ

where ∼ means equality up to a multiple in Q. Picking γλ in the right way, one obtains a Ramanujan type formula for 1/π corresponding to the CM value λ. In this way, we obtain Ramanujan type formulas for 1/π using products of two distinct periods that are determined by R. In the spirit of the Clausen formula, the product of two periods lies in the symmetric square of the period space for Eλ ; more explicitly, it is a special element in the solution sapce of the 3 F2 . While periods of the second kind which are obtained from derivation do not have exact finite field or Galois analogues, there are analogues of ‘eigenfunctions’ of the endomorphism ring on the Galois side. Let Kλ denote the CM field of Eλ . Fix a prime ` and let ρ(λ) denote the family of 2-dimensional `-adic Galois representations of Gal(Q/Q) constructed from the elliptic curve Eλ . As Eλ has CM, ρ(λ)|Gal(Q/Kλ ) ' σ1,λ ⊕ σ2,λ


where σ1,λ and σ2,λ are 1-dimensional representations that are invariant under R. Each of them corresponds to a Grossencharacter of Kλ , as we described explicitly for two cases above (Examples 10 and 11). The product σ1,λ σ2,λ lies in the the symmetric square of ρ(λ)|Gal(Q/Kλ ) . The Clausen formula implies it is also a linear factor of the 3-dimensional Galois representation corresponding to 3 P2 . For the special cases of λ (say λ = 14 , − 18 ), one can describe the values of the 3 P2 functions explicitly. Using the above computation for a(λ, q) and the Clausen formula, one has the following proposition based on Examples 10 and 11. Proposition 3. 1. For any prime power q that is coprime to 3, we have ( q, if q ≡ −1 mod 3, φ, φ, φ ; 1/4 = φ(−1) 3 P2 2 2 ε, ε J(φ, η3 ) + J(φ, η 3 ) + q, if q ≡ 1 mod 3, where η3 is a character of order 3. 2. For any prime power q that is coprime to 4, we have ( q, φ, φ, φ ; −1/8 = φ(−2) 3 P2 ε, ε J(φ, η4 )2 + J(φ, η 4 )2 + q,

if q ≡ −1 mod 4, if q ≡ 1 mod 4,

where η4 is a character of order 4. Corollary 2. When λ = 1/4 (resp. λ = −1/8), σ1,λ σ2,λ extends to a 1-dimensional representation of GQ . It corresponds to the following Grossencharacter of Q: χ 21 ,−1 · T

(resp. χ 12 ,−2 · T ),

where T is defined in Example 4. This is compatible with the p-adic analogues of Ramanujan-type formulas for 1/π. For instance, the following supercongruence related to (56) was conjectured by van Hamme [58] and proved by Long [44]. For any prime p > 3, p−1 2 X

k=0

(6k + 1)

( 12 )k k!

3 k 1 −1 ≡ 4 p

(mod p4 ).

See [15] for a general result, and [45, 55] for some recent developments on supercongruences related to truncated hypergeometric series. 5.9. Summary. To us, the Galois representation interpretation, i.e. that the n+1 Pn or n+1 Fn functions are, up to sign, the character values of Galois representations at corresponding Frobenius conjugacy classes, provides a useful guideline when one translates classical results to the finite field setting. A few principles are summarized as follows. First, let λ ∈ Q, and N the least common multiple of the orders of the Ai and Bj . Assume q ≡ 1 (mod N ) and that λ can be embedded into Fq . Then: A1 A2 · · · An+1 • The function n+1 Fn ; λ; q corresponds, up to a sign, B1 · · · Bn to the trace of a degree at most n + 1 Galois representation of GQ(ζN ,λ) at the Frobenius conjugacy class at a prime ideal with residue of size q. When λ = 0, 1, the degree is usually less than n + 1 and the representation is often


33

impure. • When λ 6= 0, 1, the primitive 2 P1 or 2 F1 function corresponds to a 2dimensional Galois representation, which is pure. • Products (resp. sums) of n+1 Pn or n+1 Fn functions correspond to tensor products (resp. direct sums) of the corresponding representations. Remark 4. In relation to the first item above, see [1, 2] and the notes of hypergeometric motives by Rodriguez-Villegas [49] for a more general discussion. See [18] for some explicit examples.

6. Translation of Some Classical Results 6.1. Kummer’s 24 Solutions. Here, we address the relations between 2 P1 hypergeometric functions corresponding to independent solutions of the hypergeometric differential equation, and Kummer’s 24 relations. Statements below without proof or further references are due to Greene, especially see Theorem 4.4 in [30]. See also [32] by Greene for interpretation in terms of representations of SL(2, q). The following proposition corresponds to the discussion in §3.2.2 for the classical setting. Proposition A 2 P1 A P 2 1 A P 2 1

4. For B ;λ C B ;λ C B ;λ C

× any characters A, B, C ∈ Fc q , and λ ∈ Fq , we have CB CA ; λ + δ(λ)J(B, CB), = ABC(−1)C(λ) 2 P1 C A CA ; 1/λ + δ(λ)J(B, CB), = ABC(−1)A(λ) 2 P1 BA A B = B(−1) 2 P1 ;1 − λ . ABC

Remark 5. In each of the first two equalities, the extra delta term on the right is necessary as when λ = 0, the left hand side is a non-zero Jacobi sum by definition while the first term on the right hand side has value 0 by our convention. Remark 6. Let λ ∈ Q \ {0, 1}, a, b, c ∈ Q such that a, b, a − c,b − c, a + b − c∈ / Q. Use π1 (resp. π2 ) to denote a 2-dimensional `-adic Galois representation a b a b corresponding to 2 P1 ; λ (resp. 2 P1 ; 1 − λ ) via Theorem 1. c a+b−c The last equality in the previous Proposition implies that if we use the ι· notation, then ιp (a) ιp (b) ι (a) ιp (b) ; λ; q(p) = ιp (b)(−1) 2 P1 p ; 1 − λ; q(p) . 2 P1 ιp (c) ιp (c) This means π1 is isomorphic to χb,−1 ⊗ π2 up to semisimplification. Many other equalities in this section have similar Galois interpretations. The next proposition corresponds to the Pfaff and Euler transforms in §3.2.3.


Proposition A P 2 1 A P 2 1 A P 2 1

× 5. For any characters A, B, C ∈ Fc q , and λ ∈ Fq , we have B A BC λ ; ; λ = A(1 − λ) 2 P1 + δ(1 − λ)J(B, CAB), C C λ−1 B CA B λ ; λ = B(1 − λ) 2 P1 ; + δ(1 − λ)J(B, CAB), C C λ−1 B AC BC ; λ = ABC(1 − λ) 2 P1 ; λ + δ(1 − λ)J(B, CAB). C C

The relations between the normalized 2 F1 -hypergeometric functions follow immediately from Propositions 4 and 5. × Proposition 6. For any characters A, B, C ∈ Fc q , and λ ∈ Fq , we have # " CB CA A B J(CA, A) ; λ + δ(λ), ; λ = ABC(−1)C(λ) 2 F1 2 F1 C C J(B, CB) " # A B A CA J(CA, BC) ; λ = ABC(−1)A(λ) ; 1/λ + δ(λ), 2 F1 2 F1 C BA J(B, CB) A B A B J(B, CAB) ; 1−λ , ; λ = 2 F1 2 F1 C ABC J(B, CB) A B A BC λ J(B, CAB) F , ; λ = A(1 − λ) F ; + δ(1 − λ) 2 1 2 1 λ−1 C C J(B, CB) A B AC B λ J(B, CAB) , ; λ = B(1 − λ) 2 F1 ; + δ(1 − λ) 2 F1 λ − 1 C C J(B, CB) A B AC BC J(B, CAB) F ; λ = ABC(1 − λ) F ; λ + δ(1 − λ) . 2 1 2 1 C C J(B, CB)

The imprimative 2 F1 -functions can be evaluated using Proposition 2 and Lemma 1, as follows. Proposition 7. Suppose λ 6= 0, and A, B, C are nontrivial. ε B C(λ)BC(λ − 1) ; λ = 1− , 2 F1 C J(B, BC) A B F ; λ = A(1 − λ) − B(λ)J(B, A), 2 1 B A B A(λ) ; λ = B(1 − λ) − , 2 F1 A J(A, B) A ε F ; λ = 1 − C(−λ)AC(1 − λ)J(C, A) − δ(1 − λ)δ(AC)(q − 1). 2 1 C Next, we will consider the primitive cases. × Proposition 8. If A, B, C ∈ Fc q , A, B 6= ε and A, B 6= C, then we have the following.


35

(1) In general, we have

A

J(A, AC) · 2 P1 A 2 F1

B B ; λ = J(B, BC) · 2 P1 C B B A ; λ = 2 F1 ; λ ; C C

A ;λ , C

(2) For λ 6= 0, 1, we have

2 P1

A

A 2 F1

J(B, CB) A B B ;λ , ; λ = C(λ)CAB(λ − 1) 2 P1 C C J(A, CA) # " B A B J(B, CB) ; λ = C(λ)CAB(λ − 1) ; λ . 2 F1 C C J(A, CA)

Proof. Since A, B are not equal to ε, C, the first part follows from the definition and the relations J(A, CA) = A(−1)J(A, C), and g(CA)J(A, C) = g(A)g(C). To prove the second part, we use the first part, along with Propositions 4 and 5. More precisely, for λ 6= 0, 1, we have A B A B ; λ = B(−1)2 P1 ;1 − λ 2 P1 C ABC BC AC = B(−1)C(λ)2 P1 ;1 − λ ABC BC AC = C(λ)ABC(−1)2 P1 ;λ C B A = C(λ)ABC(λ − 1)2 P1 ;λ . C By the first part, one has J(A, CA) A B A ; λ = C(λ)ABC(λ − 1) 2 P1 2 P1 C J(B, CB)

B ;λ . C

Remark 7. One can interpret the above claims using Galois perspective similar to Remark 6. For a, b, c ∈ Q, a, b, a − c, b − c ∈ / Z, for each prime `, then by part (2) of the above proposition, the 2-dimensional `-adic Galois representation π of GK a b associated with 2 P1 ; λ via Theorem 1 satisfies the following property: up c to semisimplification, π∼ = ψ ⊗ π, where π is the complex conjugation of π and ψ is the linear representation of GK associated with the Grossencharacter χ−c,λ · χc−a−b,λ−1 Jb,c−b /Ja,c−a of K. The next result corresponds to the classical equation (26) in §3.2.4.


Corollary 3. Suppose A, B 6= ε, and A, B 6= C. If λ 6= 0, we have A B A AC 1 2 · 2 P1 ; λ = ABC(−1)A(λ)2 P1 ; C AB λ J(B, BC) B + ABC(−1)B(λ) 2 P1 J(A, AC)

2 · 2 F1

A

BC 1 ; , BA λ

" # A AC 1 B J(CA, BC) ; ; λ = ABC(−1)A(λ) 2 F1 C AB λ J(B, CB) # " B BC 1 J(CA, CB) . + ABC(−1)B(λ) ; 2 F1 BA λ J(A, AC)

Note that the appearance of a factor of 2 on the left hand side corresponds to the fact that the right hand side is the traces of 4-dimensional Galois representations. Proof. Using Propositions 4 and 8, we obtain that if λ 6= 0, 2 P1

A

J(B, BC) B B ;λ = P 2 1 C J(A, CA)

A ;λ C

J(B, BC) B = ABC(−1)B(λ) 2 P1 J(A, CA)

CB 1 ; . AB λ

Adding this to the result obtained by using the second part of Proposition 4 directly, we get the desired relations. 6.2. Pfaff-Saalsch¨ utz evaluation formula. Greene obtained the following analogue of (29) in [30] by comparing coefficients of the third identity in Prop. 5: A, B, C ; 1 = B(−1)J(C, AD)J(B, CD) − BD(−1)J(DB, A). 3 P2 D ABCD This means the Galois representation corresponding to the 3 P2 function evaluated at 1 is a direct sum of two Hecke characters. In terms of Jacobi sums, (57)

1 X C χ(−1)J(Aχ, χ)J(B χ, Dχ)J(C χ, DABC χ) q−1 c χ∈F× q

= J(C, AD)J(B, CD) − D(−1)J(DB, A). Remark 8. We would like to make two comments: Firstly, from the Galois perspective, the 3 P2 corresponds to a 2-dimensional Galois representation that can be described using two Grossencharacters. Secondly, the additional term −BD(−1)J(DB, A) is due to the additional term involving δ(1 − λ) in the third identity in Prop. 5. If we would ‘give up’ the point 1 in Fq , then we could have restated the Pfaff-Saalsch¨ utz evaluation formula without the additional term, which would make it an exact analogue of the classical formula. See [20, 30, 33, 46] for more evaluation formulas over finite fields.


37

6.3. A few analogues of algebraic hypergeometric formulas. In the complex setting, to equate two formal power series, it suffices to compare the coefficients of each. For a proof of the identity [54, (1.5.19)] (58)

2 F1

a

a+ 1 2

1 2

√ √ 1 ; z = (1 + z)−2a + (1 − z)−2a , 2

n we note that both sides are functions of z. The coefficient of z on the right hand −2a side is 2n , while on the left it is

(a)n (a + 1/2)n (1/2)n (1)n

Thm. 6

=

(a)2n = (1)2n

−2a . 2n

The evaluation identity (2) mentioned in the introduction can be proved similarly. In the finite field setting, one can obtain identities in a similar way using (32) which we recall says that any function f (x) : Fq → C can be written uniquely as the form f (x) = δ(x)f (0) +

X

fχ · χ(x).

c χ∈F× q

We will now give finite field analogues of (58) and (2), which generalize a few recent results of Tu-Yang in [56], proved using quotients of Fermat curves. Theorem 13. Let q be an odd prime power, z ∈ F× q , φ be the quadratic character, c × and A ∈ Fq of order larger than 2. Then A

Aφ ; z φ

A

Aφ ; z A2

2 F1

2 F1

= =

1 + φ(z) 2

2

A (1 +

1 + φ(1 − z) 2

√

2

z) + A (1 −

√ z) ,

√ √ 1+ 1−z 1− 1−z 2 2 +A . A 2 2

Remark 9. Note that the above formulas are well-defined. When z 6= 0 does not 1+φ(z) have a square root in Fq , then = 1−1 2 2 = 0. When z 6= 0 has a square 1+φ(z) root, we have = 1 and the right hand side is a sum of two characters. 2 √ √ Assume z ∈ Q. The first statement above says if z ∈ / Q(z, ζN ) (resp. 1 − z ∈ / Q(z, ζN )), then the `-adic representation corresponding to the left hand side of the first (resp. second) equation is induced from a finite order character. In general, Galois representations associated with HDE(a, b; c; z) having finite monodromy groups are either reducible over Q` or are induced from linear characters. Proof. By the first part of Proposition 4, A 2 F1

Aφ ; z φ

=

φ(z) 2 F1

A

Aφ ; z + δ(z). φ

Hence, if z is not a square in F× q , then the evaluation is equal to zero.


We now suppose z = a2 for some a ∈ F× q . Then one has X A Aφ Aφ(−1) ; z = J(Aχ, χ)J(Aφχ, φχ)χ(a2 ) 2 F1 φ (q − 1)J(Aφ, A) χ =

X g(Aχ)g(χ) g(Aφχ)g(φχ) φA(−1) χ(a2 ) g(A) g(A) (q − 1)J(Aφ, A) χ

φ(−1) X Aχ(4)g(A2 χ2 )g(φ)2 g(χ2 )χ(4) χ(a2 ) q−1 χ qA(4)g(A2 ) 1 X = J(A2 χ2 , χ2 ) − (q − 1)δ(A2 ) χ(a2 ) q−1 χ =

2

2

=A (1 + a) + A (1 − a). We can use Proposition 6 and the duplication formula (15) to deduce A Aφ A Aφ J(φ, φA) F F ; 1 − z ; z = 2 1 2 1 J(φA, φA) φ A2 A Aφ = A(4) 2 F1 ; 1−z . φ This leads to the desired second statement.

Now we recast some of the previous discussion in terms of representations. Recall equation (2) mentioned in the introduction, which says √ 1−2a a − 21 a 1+ 1−z . ; z = F 2 1 2 2a Writing the rational number 1 − 2a in reduced form, let N be its denominator. We first notice that whenz is viewed as an indeterminant, the Galois group G √ 1−2a 1+ 1−z of the Galois closure of Q over Q(z) is a Dihedral group. This 2 is compatible with the corresponding monodromy group, which can be computed from the method explained in §3.2.7. As a consequence of the second part of Theorem 13, we have the following corollary. × Corollary 4. Let A, B ∈ Fc q such that A, B, AB, AB have orders larger than 2. Then A φA B Bφ ; z 2 F1 ; z 2 F1 A2 B2 " # AB φAB AB φAB 2 z = 2 F1 ; z +B ; z − δ(1 − z)AB(4), 2 F1 4 (AB)2 (AB)2

A 2 F1

φA ; z φ

B 2 F1

= 2 F1

AB

Bφ ; z φ

φAB AB 2 ; z + B (1 − z) 2 F1 φ

φAB ; z − δ(z). φ


39

Geometrically, this means the tensor product of two 2-dimensional representations for a Dihedral group is isomorphic to the direct sum of the other two. In other words, it describes a fusion rule for 2-dimensional representations of Dihedral groups. Proof. Note that

√ √ 1− 1−z 1+ 1−z 2 2

1 + φ(1 − z) 2

2

=

z 4

and

δ(1 − z) 1 + φ(1 − z) = 1− . 2 2

For any element z 6= 1 ∈ Fq , we have A

φA B φB ; z 2 F1 ; z = 2 F1 A2 B2 √ √ 1+ 1−z 1− 1−z 1 + φ(1 − z) 2 2 · A +A 2 2 2 √ √ 1+ 1−z 1− 1−z 2 2 · B +B 2 2 √ √ 1+ 1−z 1− 1−z 1 + φ(1 − z) 2 2 2 2 A B +A B = 2 2 2 √ √ 1+ 1−z 1− 1−z 1 + φ(1 − z) 2 z 2 2 2 2 +B A B +A B 4 2 2 2 " # AB φAB AB φAB 2 z ; z . = 2 F1 ; z +B 2 F1 4 (AB)2 (AB)2 When z = 1, A 2 F1

φA ; 1 A2

1 = 2

B

φB ; 1 = AB(4) B2 " AB AB φAB 1 2 2 F1 2 F1 2 ; 1 +B 4 (AB)

2 F1

The corollary follows.

φAB ; 1 (AB)2

#! .

Additionally, consider Slater’s equation (1.5.21) in [54], 2 F1

2a

a+1 1+z ; z = . (1 − z)2a+1 a

We have the following finite field analogue: for A 6= ε, 2 A A 2 2 ; z = A (1 − z) − A(z)J(A, A ). 2 F1 A This follows from the second part of Proposition 7.


7. Quadratic or higher transformation formulas In §6.1, we discussed the finite field analogue of Kummer 24 relations, which are relations between 2 P1 (resp. 2 F1 ) functions linked by linear fractional transformations. These are the cases that can be deduced from the finite field version of Lagrange theorem by Greene. But his theorem does not apply to higher degree transformation formulas. In work of Borwein and Borwein [13], the following cubic formula is given for real z ∈ (0, 1), 1 2 F1

3

2 3

1

; 1−z

3

3 = 2 F1 1 + 2z

"

1 3

2 3

1

;

1−z 1 + 2z

3 # .

Geometrically, this corresponds to the fact that the two elliptic curves y 2 + xy +

1 − z3 y = x3 , 27

y 2 + xy +

1 (1 − z)3 y = x3 27 (1 + 2z)3

are 3-isogenous over Q(z, ζ3 ). Hence, for q ≡ 1 (mod 3) and z that can be embedded in Fq , the local zeta functions of these curves over Fq are the same, i.e. if η3 is a × primitive cubic character in Fc , then q

2 F1

η3

" η32 η3 3 ; 1 − z = 2 F1 1

η32 ; 1

1−z 1 + 2z

3 # .

We wish to keep in mind that geometrically, transformation formulas describe ‘correspondences’ like isogenies though in general the underlying generalized Legendre curves are of higher genus.

7.1. A quadratic transformation formula. Recall the following equation, which was mentioned in §3.2.6

(59)

2 F1

c

c b ; x = (1 − x)−c 2 F1 2 c−b+1

1+c 2

−b −4x ; . c − b + 1 (1 − x)2

We now outline a proof using the multiplication and reflection formulas, along with the Pfaff-Saalsch¨ utz formula. Observe that our proof is different from the one in [4, pp. 125-126].

Proof. To begin, note that (60)

−c − 2k (c)n (c + n)k (−n)k (8) (c)n (c + n)k (−n)k = (−1)n = (−1)n k c . (c)2k n! n−k 4 ( 2 )k ( c+1 2 )k n!


41

The right hand side of (59) can be expanded as X ( c )k ( 1+c − b)k 2 2 (−4x)k (1 − x)−c−2k k!(c − b + 1)k

k≥0

=

X −c − 2k X ( c )k ( 1+c − b)k 2 2 (−4x)k (−x)i k!(c − b + 1)k i i≥0 k≥0 c 1+c X ( 2 )k ( 2 − b)k k −c − 2k n n=k+i = 4 x k!(c − b + 1)k n−k k,n≥0

X ( c )k ( 1+c − b)k (c)n (c + n)k (−n)k 2 2 xn = 4k k c c+1 k!(c − b + 1)k ) ( ) n! 4 ( k k 2 2 k,n≥0 1+c X (c)n −n 2 −b c+n = ; 1 xn . 3 F2 c+1 n! c − b + 1 2 n≥0 By the Pfaff-Saalsch¨ utz formula (29), the above equals X (c)n (b)n ( 1−c − n)n 2 xn 1+c n!( ) (b − c − n) n n 2 n≥0

reflection

=

X n≥0

(c)n (b)n xn . n!(c − b + 1)n

Now we obtain the corresponding finite field analogue of (59), starting with an analogue of (60). × 2 Lemma 2. Let D, K, χ, φ ∈ Fc p and φ of order 2, C = D , then

(61) J(CK 2 , χK) = χDφ(−1)

g(χK)g(CK χ)g(D)g(Dφ) g(DKφ) K(4)g(DK)g(C) q (q − 1)2 + δ(Dχφ)δ(DKφ) + (q − 1)δ(DK). q

Proof. J(CK 2 , χK) = χK(−1)J(χK, C χK) g(χK)g(C χK) g(D2 ) · + (q − 1)δ(CK 2 ) g(CK 2 ) g(C) g(χK)g(C χK)g(D)g(Dφ) duplication χK(−1) = + (q − 1)δ(CK 2 ) K(4)g(DK)g(DKφ)g(C) g(χK)g(C χK)g(D)g(Dφ) = χK(−1) K(4)g(DK)g(C) DKφ(−1) q−1 · g(DKφ) − δ(DKφ) + (q − 1)δ(CK 2 ). q q = χK(−1)


Breaking this into three terms, the middle term is g(χK)g(C χK)g(D)g(Dφ) (q − 1) δ(DKφ) K(4)g(DK)g(C) q (q − 1) duplication g(χDφ)g(χDφ)δ(DφK) = −χK(−1) q (q − 1)2 = −(q − 1)δ(DKφ) + δ(DφK)δ(Dφχ). q Recombining the three terms gives the Lemma. − χK(−1)

Remark 10. Among the three terms on the right hand side of (61), the first term is the major term corresponding to the term predicted by the classical case, and the other two terms deal with the possibilities of CK 2 and/or χK being the trivial character. Theorem 14. Assume C = D2 6= ε, B 2 6= C and φ is a quadratic character. Then " # DφB D −4x C(1 − x) 2 F1 ; CB (1 − x)2 2 B C J(C, B ) J(B, Dφ) ; x − δ(1 − x) − δ(1 + x) . = 2 F1 CB J(C, B) J(C, B) In order to prove Theorem 14, we will use the following identities. When B 6= C, φDB 6= ε, 2

(62)

D(4)

J(φB, D) J(C, B ) = ; J(D, DB) J(C, B)

J(B, Dφ) J(CB, Dφ) = . J(C, B) J(C, DBφ)

Proof. When x = 0, both sides are equal to 1 by definition. We now assume x 6= 0 and separate into two cases: C = B and C 6= B. First, we consider when B 6= C. By definition, we have " # DφB D −4x C(1 − x) 2 F1 ; CB (1 − x)2 X 1 = D(−1)J(DφBK, K)J(DK, BCK)K(−4x)CK 2 (1 − x) (q − 1)J(D, DB) K X 1 = D(−1)J(DφBK, K)J(DK, BCK) 2 (q − 1) J(D, DB) K X · K(−4x) J(CK 2 , ϕ)ϕ(x). ϕ

Letting χ := Kϕ, the above equals (63) X 1 D(−1)J(DφBK, K)J(DK, BCK)K(−4)J(CK 2 , χK)χ(x). (q − 1)2 J(D, DB) K,χ By Lemma 2 we can write it as 2

S1 + S2 − D(4)

J(φB, D) J(C, B ) δ(1 − x) = S1 + S2 − D(4) δ(1 − x), J(D, DB) J(C, B)


43

where the Si are obtained by applying (61). Explicitly, S1 is given by

S1 =

X 1 D(−1)J(DφBK, K)J(DK, BCK)K(−4) (q − 1)2 J(D, DB) K,χ g(χK)g(CK χ)g(D)g(Dφ) g(DKφ) χ(x) K(4)g(DK)g(C) q X g(CB) g(DK)g(BCK) χφ(−1)J(DφBK, K) 2 (q − 1) g(D)g(DB) K,χ g(BD)

· χDφ(−1) 2nd Jacobi sum

=

g(χK)g(CK χ)g(D)g(Dφ) g(DKφ) χ(x) K(4)g(DK)g(C) q X combine Gauss sums BD(−1)g(CB)g(Dφ) = g(BC χ)g(Dφχ) 2 2 (q − 1) q g(C) χ · K(−4)

·

X K

Letting Ω :=

S1 =

g(BCK)g(χK) g(CK χ)g(DKφ) K χφ(−1)J(DφBK, K) g(Dφχ) g(BC χ)

BD(−1)g(CB)g(Dφ) , q 2 g(C)

! χ(x).

we have

X Ω g(BC χ)g(Dφχ) (q − 1)2 χ

! g(BCK)g(χK) g(CK χ)g(DKφ) χ(x) · K χφ(−1)J(DφBK, K) g(Dφχ) g(BC χ) K X X Ω Jacobi sums = φ(−1)g(BC χ)g(Dφχ)χ(x) K χ(−1)J(DφBK, K) 2 (q − 1) χ K · J(BCK, χK) − (q − 1)BCK(−1)δ(BC χ) · J(CK χ, DKφ) − (q − 1)DKφ(−1)δ(Dφχ) . X

To continue we separate the delta terms,

S1 =

Ω X φ(−1)g(BC χ)g(Dφχ)χ(x)· (q − 1) χ X K χ(−1) K

q−1

J(DφBK, K)J(BCK, χK)J(CK χ, DKφ) + E1 + E2

(57)

=

Ω X φ(−1)g(BC χ)g(Dφχ)χ(x) (q − 1) χ

(J(C χ, Dφ)J(χ, B χ) − B(−1)J(CB χ, DφB)) + E1 + E2 ,


where E1 := −

X Ωφ(−1) (q − 1)

χ

g(ε)g(BDφ)

X

J(DφBK, K)J(BK, DKφ) · δ(BC χ)χ(x)

K

Gauss eval.

=

X

Ωφ(−1)g(BDφ)J(B, B)δ(BC χ)χ(x)

χ

= −φB(−1)Ωg(BDφ)BC(x),

E2 := −

X Ωφ(−1) (q − 1)

χ

Gauss eval.

=

g(ε)g(BDφ)

X

J(DφBK, K)J(BCK, DφK)δ(Dφχ)χ(x)

K

X g(CB)g(Dφ) q 2 g(C)

χ

φ(−1)g(BDφ)J(Dφ, Dφ)δ(Dφχ)χ(x) = −D(−1)Ωg(BDφ)Dφ(x).

Now we continue our analysis of S1 . We separate the term before E1 + E2 into two terms. The first term is Ω X φ(−1)g(BC χ)g(Dφχ)J(C χ, Dφ)J(χ, B χ)χ(x) (q − 1) χ = ·

Ω X φ(−1)g(BC χ)g(Dφχ) (q − 1) χ g(C χ)g(Dφ) + (q − 1)Dφ(−1)δ(Dφχ) J(χ, B χ)χ(x). g(Dφχ)

By definition, it equals X B C Ω g(BDφ)D(−1)J(Dφ, BDφ)δ(Dφχ)χ(x) ; x − 2 F1 CB c χ∈F× q

= 2 F1

B

C ; x − Ω g(BDφ)φ(−1)J(Dφ, B)Dφ(x). CB

Now let E3 := Ω g(BDφ)φ(−1)J(Dφ, B)Dφ(x). The next term is Ω X − φ(−1)g(BC χ)g(Dφχ)B(−1)J(CB χ, DφB)χ(x) (q − 1) χ =−

Ω X φ(−1)g(BC χ)g(Dφχ)B(−1)CB χ(−1)J(CB χ, Dχφ)χ(x) (q − 1) χ =−

Ω X g(CB χ)g(Dχφ) χ(x) φχ(−1)g(BC χ)g(Dφχ) (q − 1) χ g(DBφ) =−

Ω X φχ(−1) g(BC χ)g(CB χ)g(Dφχ)g(Dχφ)χ(x). (q − 1) χ g(DBφ)


45

Applying the finite field reflection formula (11), the above becomes −

Ω X φχ(−1) BC χ(−1)q − (q − 1)δ(BC χ) (q − 1) χ g(DBφ) · (Dφχ(−1)q − (q − 1)δ(Dφχ))χ(x) X q2 Ω χ(−1)χ(x) − E1 − E2 =− DB(−1) (q − 1)g(DBφ) χ =−

q2 Ω DB(−1)δ(1 + x) − E1 − E2 g(DBφ) =−

J(B, Dφ) δ(1 + x) − E1 − E2 . J(C, B)

Meanwhile,

S2 =

D(4)J(φ, φBD) φ(−1)J(B, φD)φD(x) = E3 qJ(D, DB)

When B = C, in similar way, we can obtain C(1 − x) 2 F1

D −4x ; ε (1 − x)2 2 C C J(C, Dφ) J(C, C ) = 2 F1 − δ(1 + x) . ; x − δ(1 − x) ε J(C, C) J(C, C)

φD

Remark 11. This strategy of the proof can be generalized to other cases. We identify a couple of transformations below which can be proved by the reflection/multiplication formulas and other secondary results such as the Pfaff-Saalsch¨ utz formula, i.e. satisfying the (∗) condition. Interested readers should be able to obtain their finite field analogues following our strategy above. The first one is (3.1.15) of [4], 3 F2

a

c ; x = a−c+1 a−b−c+1 −a (1 − x) 3 F2

b a−b+1

a/2 a−b+1

(a + 1)/2 −4x ; . a − c + 1 (1 − x)2

See [30, Theorem 4.28] for its finite field analogue. The second one is one of Bailey’s cubic transformations [4, page 185] (64)

a 2b − a − 1 3 F2 b

a + 2 − 2b x ; = a + 3/2 − b 4 a (1 − x)−a 3 F2 3

a+2 3

a+2 3

b

a − b + 3/2

;

−27x . 4(1 − x)3


7.2. An immediate corollary of the quadratic transformation formula. From a transformation formula, one can obtain evaluation formulas either by specifying values or by comparing coefficients on both sides. Here we will show how to obtain the finite field analogue of Kummer evaluation theorem (28), which has been obtained by Greene in [30] using a different approach. Corollary 5. Assume C = D2 , then B C J(D, B) J(B, Dφ) F ; −1 = + . 2 1 CB J(C, B) J(C, B) Proof. By Theorem 14, if C = D2 6= ε, B 2 6= C, x = −1, one has # " B C DφB D J(B, Dφ) ; 1 = 2 F1 ; −1 − . C(2) 2 F1 CB CB J(C, B) Hence, B F 2 1

" C DφB ; −1 = C(2) 2 F1 CB

# D J(B, Dφ) ; 1 + CB J(C, B) = D(4)

J(φ, D) J(B, Dφ) + , J(D, DB) J(C, B)

where if C 6= B, D(4)

D(4)g(φ)g(D)g(CB) J(φ, D) J(B, D) = = , J(D, DB) g(φD)g(D)g(DB) J(C, B)

and if C = B, J(φ, D) D(4)J(φ, D) J(D, D) J(C, D) = = = , J(D, DB) J(D, D) J(D, D) J(C, C) since J(φ, χ) = χ(4)J(χ, χ) if χ 6= ε. When C = ε, the claim follows Proposition 2. D(4)

7.3. The proof of Theorem 2. In the previous proof of the quadratic formula, the delta terms produce two kinds of byproducts: the extra terms like δ(1 ± x), which are due to the degeneracy of the corresponding Galois representations at the corresponding x values, and the extra characters like the Ei terms above. While the first kind of extra terms are unavoidable, but can be omitted by excluding a few x values from the equation, the second kind of extra terms might be eliminated. In particular, some of them are ‘path dependent’. For instance, in the current case, if we swap the upper characters of the left hand side, then the delta terms do not involve BC(x). Now we generalize the above proof. Recall that, we say a classical transformation formula satisfies the condition (∗) if it can be proved using only the binomial theorem (7), the reflection and multiplication formulas, Theorems 5 and 7, or their corollaries (such as the Pfaff-Saalsch¨ utz formula). Proof of Theorem 2. As we assume the classical formula satisfying the (∗) condition, and we will similarly mimic the proof of the classical formula to get the finite field analogues. For simplicity, we also assume the finite field 2 F1 analogues are


47

primitive, which is not quite necessarily if a more general formula is desired. We expand one side (typically the more complicated side) of the formula as a double summation involving two characters K and χ, like (63). Then we reduce the double sum to the another side of the formula. The dictionary described in §2.4 predicts that the major terms for the finite field analogues, i.e. the 2 F1 terms predicted on both sides, agree. There is one subtly we should be aware of in the finite field translation: we might need to extend the current finite field Fq by adding some roots of unity in order to apply the multiplication formulas. From the representation point of view, this means we might need to restrict the Galois representations associated with 2 F1 to some finite index subgroups H so that we can translate the steps using finite field multiplication formulas. To determining the original representation before restriction, it provokes a character of the original Galois group with kernel containing H. Using the Gross-Koblitz formula and moduloing corresponding prime ideal, one can reduce the finite field 2 F1 to the Hasse-invariants of the corresponding Galois representations, appearing as the corresponding truncated hypergeometric series as of [18]. This allows us to see the finite order character is trivial. (We shall see an explicit example in the next cubic formula.) Now we consider the roles of the extra delta terms. To proceed in the way parallel to the classical proof, we might need to use either finite field reflection formula or the formula relating Jacobi sums and Gauss sums which will produce delta terms. Each delta term specializes one of the characters, i.e. K or χ, in the double summation. Thus a product of two or more independent delta terms will either give rise to an extra character or correspond to a non existing combination that has no effect on the final equation. As in the previous proof, the number of extra characters are bounded by the statement of the formula. Thus we can turn our attention to the single delta terms in the double sum. Some of the single delta terms, as pointed in Remark 10, are corresponding to the degenerate cases before we apply the multiplication formula(s), like the case described in Lemma 2. These single delta terms contribute to Ca δ(x − a) with a = 1/c3 . There might be other single delta terms arise from the extra term in the finite field Pfaff-Saalsch¨ utz formula. As we point out in Remark 8, if we give up those points a such that either c1 a or c2 an1 (1 − c3 a)n2 equals 0 or 1, then the role of the extra term in the Pfaff-Saalsch¨ utz formula can be omitted. Thus they also contribute to terms like Ca · δ(x − a) which vanish if we choose to omit a few points a in Fq . (Regardless whether the Pfaff-Saalsch¨ utz formula is used, these delta terms take care of the cases when the argument in either side becomes 0 or 1.) It is helpful to recall that Theorem 1 says each primitive 2 F1 with rational parameters at x 6= 0, 1 corresponds to a compatible family of 2-dimensional Galois representations. This often forces the extra characters of the second kind to vanish. For instance, if for one x ∈ Q \ {0, 1} and one prime ` the 2-dimensional Galois representation over Q` corresponding to either 2 F1 in the classical formula via Theorem 1 is strongly irreducible (i.e. its restriction to any finite index subgroup remains irreducible), then one can eliminate the existence of the extra characters. 7.4. Using 2 F1 symmetries to eliminate extra characters. A different way to eliminate the extra characters of the second kind is as follows. Applying a Pfaff transformation (second part of Proposition 6) to the right hand side of the above


quadratic formula, we can get # " DφB D −4x (65) C(1 − x) 2 F1 ; CB (1 − x)2 # " 2 J(B, Dφ) x C CB = C(1 − x) 2 F1 − δ(1 + x) , ; J(C, B) CB x − 1 by letting z = (66)

x x−1

1 and hence 1 − z = 1−x . We can get when z 6= 12 , 1, # " " # 2 DφB D CB C ; 4z(1 − z) = 2 F1 ; z , 2 F1 CB CB

or equivalently when A2 , B 2 6= ε, then for z 6= 21 , 1 2 A B A (67) F ; 4z(1 − z) = F 2 1 2 1 ABφ

B2 ; z . ABφ

For this one, it is easy to see that they do not differ by extra characters as both sides are invariant under the involution z 7→ 1 − z. 7.5. A cubic 2 F1 formula and a corollary. Using Theorem 2, we are going to derive the finite field analogue of the following cubic formula by Gessel and Stanton [27, (5.18)], a −a 27x(1 − x)2 3a −3a 3x (68) F ; = F ; , 2 1 2 1 1 1 4 4 2 2 which satisfies the (∗) condition and assumptions of Theorem 2. Note that (a)n (a)n−i = (−1)i , (1 − a − n)i 1−2n 2−2n 2n − 2i (−1)i (−2n)3i 27i ( −2n 3 )i ( 3 )i ( 3 )i = = (−1)i i . i i! (−2n)2i 4 · i! (−n)i (−n + 21 )i We now outline a proof using the reflection and multiplication formulas, Theorems 5 and 7. Observe, X k X a −a 27x(1 − x)2 2k (a)k (−a)k 27x (−x)i = 2 F1 1 ; 1 4 4 i (1) ( ) k k 2 2 i≥0 k≥0 n−i X (a)n−i (−a)n−i 27 2n − 2i n=k+i = (−1)i xn 1 4 i (1) ( ) n−i n−i 2 n,i≥0 −i ! X 27 n (a)n (−a)n X (−n)i ( 21 − n)i 27 2n − 2i = − xn 1 4 (1 − a − n) (1 + a − n) 4 i (1) ( ) i i n n 2 i n≥0 −2n n 1−2n 2−2n X (a)n (−a)n 27x 3 3 3 = F ; 1 . 3 2 4 1+a−n 1−a−n (1)n ( 12 )n n≥1

−2n 1−2n 2−2n 3 , 3 , 3

is a negative integer, the 3 F2 can be evaluated by the Since one of Pfaff-Saalsch¨ utz formula. It can be further related to the desired right hand side again by Theorems 5 and 7. Now we will proceed with the finite field analogue.


49

Proof of Theorem 3. To translate the proof for the complex case, note that if Aχ1 6= χ1 χ2 ) (A) ε, and Aχ1 χ2 6= ε, then (A)χ1 χ2 = g(Ag(A) = χ χ1 . Also, there is no issue in (A

1 ) χ2

applying the finite field analogue of Theorem 6 to (−2n)2i . To consider the analogue of applying Theorem 7 to (−2n)3i , note each character ϕ on F× q , when lifted to a character on F× , has cubic roots. Thus the finite field analogue of 3 q −i X (−n)i ( 21 − n)i 27 2n − 2i − (1 − a − n)i (1 + a − n)i 4 i i is an exponential sum corresponding to a reducible mixed motive which can be computed using Pfaff-Saalsch¨ utz (29) over Fq3 . The only potential ambiguity, arising from the choice of cubic root associated with Fq3 /Fq , can be eliminated by applying the Gross-Koblitz formula to 2 F1 ’s over finite fields, and reduces the discussion to truncated hypergeometric series, see [18]. By Theorem 2, the 2 F1 ’s on both sides differ by a finite number of characters and a few terms of the form cA,a δ(a − x), where a equals other values such that 3x 4 or

27x(1−x)2 4

= 0, 1, i.e. a = 0, 1, 13 , 43 . By Proposition 4, A A A 27x(1 − x)2 = A(−1) 2 F1 ; 1− 2 F1 4 φ

A 27x(1 − x)2 ; 4 φ

and " A3

# " 3x A3 A3 = A(−1) 2 F1 ; 1− 4 φ

2 F1

# A3 3x . ; 4 φ

It follows that A 2 F1

" A A3 ; 1 = 2 F1 φ

# 3 A ; 1 = A(−1), φ

and 2 F1

" A3

# " 3 1 A A3 ; = A(−1) 2 F1 φ 4

# 3 3 A ; . φ 4

To eliminate the existence of the extra characters, notice that the map x 7→ 3x 4

2

2

27x(1−x) 4

3x 4

27x(1−x) 4

4 3

−x

7→ 1 − and 7→ 1 − . This will require the correspondsends ing finite number of characters to be invariant up to at most a sign when x 7→ 43 −x, which is impossible when the combination is not zero. Next we compute cA,a for a = 0, 1, 31 , 43 . When x = 0 (resp. x = 43 ), the 2 F1 values on both sides are 1 (resp. A(−1)). Thus cA,0 = 0 (resp. aA, 43 = 0). Next by letting x = 31 , 1 we have " A(−1) = 2 F1 " 1 = 2 F1

A3

# 3 3 A ; + cA,1 φ 4

Thus cA,1 = A(−1)cA, 31 .

# 3 1 A ; + cA, 13 , φ 4 " # 3 1 A3 A = A(−1) 2 F1 ; + cA,1 . φ 4

A3


Now from Proposition 6 and Theorem 13, we know that " # 3 3 A A3 φ 1 3 3 A3 A F = A ; F ; −3 2 1 2 1 4 φ φ 4 √ √ 1 + φ(−3) 1 6 6 6 = A A (1 + −3) + A (1 − −3) 2 2 1 + φ(−3) = A(1) + A(1) = 1 + φ(−3), 2 and hence CA,1 = −φ(−3). This gives us the delta terms at x = 1 and 31 .

Remark 12. One can derive a different proof by using Proposition 6 and Theorem 13 to write both sides as explicit characters. In the next few pages we will obtain a finite field analogue of an evaluation formula by Gessel and Stanton. Our main point for the discussion is as follows. In addition to what can be done for the cases satisfying (∗), there are classical results obtained by methods that have no direct translations to finite fields. However, sometimes the Galois perspective allows us to make predictions which can be verified numerically. To prove them, we need different methods that might appear to be ad hoc when compared with what we used so far. As a corollary of (68), Gessel and Stanton showed that for n ∈ N, a ∈ C 1 + 3a 1 − 3a −n 3 (1 + a)n (1 − a)n F ; = 3 2 3 4 −1 − 3n ( 23 )n ( 43 )n 2 =

B(1 + a + n, 23 )B(1 − a + n, 34 ) . B(1 + a, n + 23 )B(1 − a, n + 34 )

It follows from applying Theorem 1 of [27] to (68) with an additional factor of x−n−2 (1 − x)−2n−3 (1 − 3x), instead of x−n−2 (1 − x)−3n−2 (1 − 3x) mentioned in page 305 of [27]. As Theorem 1 of [27] implies the Lagrange inversion formula, it has no analogue over finite fields. Greene gave a Lagrange inversion formula over finite fields F, see Theorem 9 for a special case. However, as Greene pointed out it cannot be used to determine coefficients when the change of variable function is not one-to-one. To look for a finite field analogue of this formula, we notice that the corresponding Galois representations are expected to be 3-dimensional. Also, if 3a corresponds to a character A3 , then there are 3 candidates for the character analogue of a, i.e. A, Aη3 , Aη32 . That was how we first phrased the statement of Theorem 4. Now we give a proof here. Proof of Theorem 4. Let η := η3 and F := Fq for convenience. Multiplying χ(x(1 − x)2 ) by the right hand side of Theorem 3 and taking the sum over all x, we see " # X 4 A3 A3 3x 2 χ(x(1 − x) ) 2 F1 ; −χ φ(−3)A(−1) 4 27 φ x∈F " # 3 4 1 χ 3 A3 A = −χ φ(−3)A(−1). 3 P2 3 ; 3 χ 4 27 φ J(A , φA3 )


We now let " I := 3 F2

A3

3

# 4 χ( 27 )φ(−3)A(−1) 3 ; − 3 4 J(χ, χ2 ) φ χ # " 4 X χ(x(1 − x)2 ) χ( 27 )φ(−3)A(−1) A3 A3 3x ; = F − . 2 1 2 4 φ J(χ, χ ) J(χ, χ2 ) x∈F

A

χ

By Theorem 3, X χ(x(1 − x)2 )

A A 27x(1 − x)2 I= ; 2 F1 4 φ J(χ, χ2 ) x∈F X χ(x(1 − x)2 ) A A 27x(1 − x)2 = ; 2 P1 φ 4 J(A, φA)J(χ, χ2 ) x∈F X χ(x(1 − x)2 ) A(−1) X 27 J(Aϕ, ϕ)J(Aϕ, φϕ)ϕ = ϕ(x(1 − x)2 ) 2 q−1 4 J(A, φA)J(χ, χ ) x∈F

=

c ϕ∈F× q

A(−1) (q − 1)J(A, φA)J(χ, χ2 )

X

J(Aϕ, ϕ)J(Aϕ, φϕ)ϕ

27 4

J(χϕ, χ2 ϕ2 ).

c ϕ∈F× q

Using Theorem 8, we observe that when χ6 6= ε, q−1 g(χ3 ) g(ϕχ)g(ϕ2 χ2 ) J(χϕ, χ2 ϕ2 ) = + δ ϕ3 χ3 2 2 3 J(χ, χ ) g(χ)g(χ ) g(ϕ3 χ ) J(χ, χ2 ) χ χ g(η χ)g(η χ) g(φϕχ)g(ϕχ) 4 3 χ3 q − 1 g(η )g(η )g(φ) χ 27 = ϕ +δ ϕ . g(φχ)g(χ) g(ϕχη)g(ϕχη) 27 q g(φχ)g(χ) 4 Thus the sum I can be divided into two parts, I1 and I2 , where I1 =

1 A(−1) X g(η χ)g(η χ)g(φ) 27 χϕ J(Aϕ, ϕ)J(Aϕ, φϕ) χ χ g(φ )g( ) 4 J(A, φA) q ϕ3 =χ3

I2 =

1 A(−1) X g(η χ)g(η χ) g(φϕχ)g(ϕχ) J(Aϕ, ϕ)J(Aϕ, φϕ) . g(φχ)g(χ) g(ϕχη)g(ϕχη) J(A, φA) q − 1 ϕ

and

Then under our assumptions, we have I1 =

A(−1) X g(ϕ)g(φϕ) 27 χ χ χ g(η )g(η )g(Aϕ)g(Aϕ) ϕ χ χ q2 g(φ )g( ) 4 ϕ3 =χ3 X J(Aϕ, Aϕ) A(−1) X . = J(Aϕ, Aϕ)J(η χ, η χ)χϕ (27) = A(−1) q J(η χ, η χ) ϕ3 =χ3 ϕ3 =χ3

51


For the sum I2 , we will divide it into 3 parts. First, we observe that I2 =

1 A(−1) g(η χ)g(η χ) J(A, φA) q − 1 g(φχ)g(χ) X · J(Aϕ, ϕ)J(Aϕ, φϕ)g(φϕχ)g(ϕχ)g(ϕχη)g(ϕχη) c ϕ∈F× q

·

ϕχ(−1) q − 1 ϕχ(−1) q − 1 + δ (χϕη) + δ (χϕη) q q q q A(−1) 1 g(η χ)g(η χ) g(η)g(φη) = q − 1 J(A, φA) g(φχ)g(χ) X · J(Aϕ, ϕ)J(Aϕ, φϕ)J(φϕχ, η χϕ)J(χϕ, χηϕ) ϕ

·

1 q−1 q−1 χ χ ϕη) + ϕη) . + δ ( δ ( q2 q2 q2

The first delta term gives us

−

A(−1) J(η, η) g(η χ)g(φη χ) g(η χ)g(η χ)g(Aχη)g(Aχη) 2 q J(φ, η) g(φχ)g(χ) A(−1) η(4)J(η, η) =− J(η χ, η χ)J(Aχη, Aχη)g(η χ2 )g(η χ2 ) q2 J(φ, η) J(Aχη, Aχη) , = −A(−1) J(η χ, η χ)

since B(4)J(B, B) = J(φ, B) for any character B. The second delta term contributes −A(−1)

J(Aη χ, Aη χ) . J(η χ, η χ)

Therefore, we have

I = A(−1)

J(Aχ, Aχ) J(η χ, η χ) +

φ(−1) g(η)g(φη) g(η χ)g(η χ) 4 P3 q 2 J(A, φA) g(χ)g(φχ)

We are now in the position to evaluate 4 P3

A A φ

A

A φ

χφ χη

χ ;1 . χη

χφ χ ; 1 . For this χη χη

purpose, we observe that for x 6= 0, 2 P1

A

A 1 ; φ x

2 P1

η

φ(−1) X η 1 A ; = 4 P3 φ x q−1 χ

A χφ φ χη

χ ; 1 χ(x), χη


53

and by Proposition 4 and Corollary 4, the left hand side of the above equals A φA η φη Aη(x)J(φA, φA)J(φη, φη) 2 F1 ; x F ; x 2 1 A2 η ·

2 F1

Aη

= Aη(x)J(φA, φA)J(φη, φη) φAη Aη φAη ; x + η(2x) 2 F1 ; x − δ(1 − x)Aη(4) . (Aη)2 (Aη)2

From the relations J(χ, χ) = χ(4)J(φ, χ) and J(φ, φχ) = φ(−1)J(χ, φ), if χ 6= ε, we obtain Aη(x)J(φA, φA)J(φη, φη) 2 F1

Aη

φAη ; x (Aη)2

J(φ, A)J(φ, η) Aη φAη P ; x 2 1 A2 η J(φ, Aη) J(φ, A)J(φ, η) A(−1) X 2 J(AηK, K)J(φAηK, A ηK)KAη(x) = q−1 J(φ, Aη) = Aη(x)φ(−1)

K

J(φ, A)J(φ, η) A(−1) X = J(χ, Aηχ)J(φχ, Aηχ)χ(x), q−1 χ J(φ, Aη) Aη

φAη Aη(x)J(φA, φA)J(φη, φη)η(2x) 2 F1 ; x (Aη)2 J(φ, A)J(φ, η) A(−1) X J(χ, Aηχ)J(φχ, Aηχ)χ(x), = q−1 χ J(φ, Aη) and Aη(x)J(φA, φA)J(φη, φη)Aη(4)δ(1 − x) =

X 1 χ(x). J(φ, A)J(φ, η) q−1 χ

Therefore, by comparing the coefficients for χ, we get A A χφ χ φ(−1)4 P3 ;1 φ χη3 χη3 J(χ, Aηχ)J(φχ, Aηχ) J(χ, Aηχ)J(φχ, Aηχ) + = A(−1)J(φ, A)J(φ, η) J(φ, Aη) J(φ, Aη) − J(φ, A)J(φ, η), and when χ2 6= ε, it is equal to J(φ, A)J(φ, η) χ(4)J(Aηχ, Aηχ) + χ(4)J(Aηχ, Aηχ) − 1 . Our claim follows from these results and the fact that φ(−3) = 1 when p ≡ 1 (mod 6). Remark 13. Gessel and Stanton obtained in [27] many other interesting evaluation formulas. Many of them seem to have finite field analogues. Interested readers are encouraged to take a look.


7.6. Another cubic 2 F1 formula and applications. Here, we consider another cubic transformation formula satisfying the (∗) condition. This formula is a special case of the Bailey cubic formula for 3 F2 that we mentioned earlier, see (64). a a+1 a 1−a −27x −a 3 3 3 (69) . 2 F1 2 F1 4a+5 ; x = (1 − 4x) 4a+5 ; (1 − 4x)3 6 6 We now outline a proof. Observe that −3k − a (−1)n (a)n (−n)k (a + n)2k (70) = n−k n! (a)3k =

(−1)n (a)n (−n)k n!

4 27

k

a+n+1 )k ( a+n 2 )k ( 2 . a+2 a+1 (a)k ( 3 )k ( 3 )k

Using (70), the right hand side of (69) can be expanded as X ( a )k ( a+1 )k X −3k − a k 3 3 (−27x) (−4x)i 4a+5 i (1) ( ) k k 6 i k≥0 k ! n X X ( a3 )k ( a+1 ) −3k − a 27 k n=k+i 3 = (−4)n xn 4a+5 4 n − k (1) ( ) k k 6 n≥0 k=0 a+n a+n+1 X 4n (a)n −n n 2 2 = 3 F2 4a+5 a+2 ; 1 x . n! 6 3 n≥0

The claim follows as in the case of (68). Corollary 6. The above cubic formula satisfies the assumptions of Theorem 2. Thus for any prime power q ≡ 1 (mod 3) and E be any character such that E 6 6= ε. Let ω be a primitive cubic character. Then for x 6= 0, 1, 14 , − 81 , there is an integer n depending on the classical formula such that X 3 n E Eω E ωE −27x 3 ci,E χi (x) ; x = E (1 − 4x) F + ; 2 F1 2 1 E 2 φω (1 − 4x)3 E 2 φω i=1 for some character χ’s depending on E and constants ci,E depending on E and χi . Now we use 2 F1 symmetries to derive the finite field analogue of the following special case of (69), which corresponds to the triangle group (2, 3, 4): 10 ! 81 √ 1 7 1+ 1−x − 24 24 −27 (71) = . 2 F1 3 ; (1 − 4x)3 210 (1 − 4x) 4 Observe that letting a = − 18 in (69), and using (2) yields (71). We now establish the finite field analogue. Theorem 15. Let q ≡ 1 mod 24 be a prime power, and let x 6= − 81 , 14 in Fq . Let η be an order 24 character on F× q . Then η η7 −27x ; = 2 F1 η 18 (1 − 4x)3 √ √ 1 + φ(1 − x) (1 + 1 − x)10 (1 − 1 − x)10 3 η3 + η . 2 210 (1 − 4x) 210 (1 − 4x)


55

Proof. It is easy to see this formula holds when x = 0. Letting E = η¯ in Corollary 6 we know for almost all x 3 X n η η9 η η7 −27x 3 F + ; x = η ; ci,η χi (x). (1 − 4x) · F 2 1 2 1 η 18 η 18 (1 − 4x)3 i=1 Similarly letting E = η¯φ, 3 η φ η9 φ ηφ 3 2 F1 18 ; x = η φ(1 − 4x) · 2 F1 η

X n η7 φ −27x + ; ci,ηφ χ0i (x). η 18 (1 − 4x)3 i=1

Comparing the major term on the right hand side, we have η7 φ −27x ; η 18 (1 − 4x)3 η −27x Euler 3 = η φ(1 − 4x)φ 1 − 2 F1 (1 − 4x)3

η 3 φ(1 − 4x) · 2 F1

ηφ

η7 −27x ; η 18 (1 − 4x)3 η η7 −27x = φ(1 − x) · η 3 (1 − 4x) 2 F1 ; . η 18 (1 − 4x)3

Similarly, comparing the left hand sides one has 3 3 η φ η9 φ η Euler = φ(1 − x) 2 F1 2 F1 18 ; x η

η9 ; x . η 18

Thus φ(1 − x)

n X

ci,η χi (x) =

i=1

n X

ci,ηφ χ0i (x)

i=1

holds for almost all x in almost all finite fields Fq with q ≡ 1 mod 24. This is possible only when both hand sides are zero, as by (33) the expansion of φ(1 − x) involves all characters. When x = 1, using Gauss evaluation formula and [9, Theorem 3.6.6], one has 7 η 2 F1

η 4 J(φ, η) 3 = η − ; 1 = = η 3 (−3) η 9 (4) 3 η 18 J(η, η 5 ) 3 η J(φ, η 9 ) 3 = η (−3) 2 F1 = η (−3) 9 9 J(η , η ) 3

η9 ; 1 . η 18

Thus for x 6= 14 , − 18 3 η φ 2 F1

η9 φ ηφ 3 ; x = η φ(1 − 4x) · F 2 1 η 18

η7 φ −27x ; . η 18 (1 − 4x)3

By Theorem 13, √ √ 3 η η9 1 + φ(1 − x) 1+ 1−x 1− 1−x 6 6 F ; x = η + η . 2 1 2 2 2 η 18


Composing this with the cubic formula, we have 3 η η7 η η9 −27x 3 ; = η (1 − 4x) 2 F1 ; x 2 F1 η 18 (1 − 4x)4 η 18 √ √ 1 + φ(1 − x) 1+ 1−x 1− 1−x 3 6 6 = η (1 − 4x) η +η 2 2 2 √ √ 2 1 + φ(1 − x) (1 + 1 − x) (1 − 1 − x)2 3 = η3 + η , 2 22 (1 − 4x) 22 (1 − 4x) which is equivalent to our claim as η has order 24.

Remark 14. We obtain another algebraic hypergeometric function corresponding to the triangle group (2, 3, 3), namely, !1/4 p 1 1 − 12 4 2z(3 + z 2 ) 1 + 1 + z 2 /3 . = (72) 2 F1 1 ; (1 + z)3 2(1 + z) 2 Numerical evidence suggests that the finite field analogue of (72) is the following. Let q ≡ 1 (mod 12) and η be any order 12 character. Then for z ∈ Fq such that z 6= 0, ±1, and z 2 + 3 6= 0, η η 3 2z(3 + z 2 ) = ; (73) 2 F1 (1 + z)3 φ ! !! p p 1 + 1 + z 2 /3 1 − 1 + z 2 /3 1 + φ(1 + z 2 /3) 3 3 η +η . 2 2(1 + z) 2(1 + z) We leave this to the interested reader. 8. Acknowledgments Both Long and Tu were supported by NSF DMS #1303292. We would like to thank Microsoft Research and the Office of Research & Economic Development at Louisiana State University (LSU) for support which allowed Fuselier and Swisher to visit LSU in 2015. Tu was supported in part by the National Center for Theoretical Sciences (NCTS) in Taiwan for her visit to LSU. We are very grateful to Irene Bouw, Henri Cohen, Jerome W. Hoffman, Wen-Ching Winnie Li, Matt Papanikolas, Daqing Wan and Tonghai Yang for their interests, enlightening discussions and helpful references from some of them. Special thanks to Rodriguez-Villegas for his lecture on Hypergeometric Motives given at NCTS in 2014. Our statements are verified by Magma or Sage programs. The collaboration was carried out on Sagamath cloud. Long and Swisher would further thank Institute for Computational and Experimental Research in Mathematics (ICERM) at Brown University for hospitality as part of work was done when they were both ICERM research fellows in Fall 2015. References [1] Alan Adolphson and Steven Sperber. On twisted exponential sums. Math. Ann., 290(4):713– 726, 1991. [2] Alan Adolphson and Steven Sperber. Twisted exponential sums and Newton polyhedra. J. Reine Angew. Math., 443:151–177, 1993.


57

[3] Scott Ahlgren, Ken Ono, and David Penniston. Zeta functions of an infinite family of K3 surfaces. Amer. J. Math., 124(2):353–368, 2002. [4] George E. Andrews, Richard Askey, and Ranjan Roy. Special functions, volume 71 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1999. [5] Nat´ alia Archinard. Hypergeometric abelian varieties. Canad. J. Math., 55(5):897–932, 2003. [6] Wilfrid Norman Bailey. Generalized hypergeometric series. Cambridge Tracts in Mathematics and Mathematical Physics, No. 32. Stechert-Hafner, Inc., New York, 1964. [7] Francesco Baldassarri and Bernard Dwork. On second order linear differential equations with algebraic solutions. Amer. J. Math., 101(1):42–76, 1979. [8] Rupam Barman and Gautam Kalita. Hypergeometric functions and a family of algebraic curves. Ramanujan J., 28(2):175–185, 2012. [9] Bruce C. Berndt, Ronald J. Evans, and Kenneth S. Williams. Gauss and Jacobi sums. Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons, Inc., New York, 1998. A Wiley-Interscience Publication. [10] Frits Beukers. Notes of differential equations and hypergeometric functions. unpublished notes. [11] Frits Beukers, Henri Cohen, and Anton Mellit. Finite hypergeometric functions. arXiv: 1505.02900. [12] Frits Beukers and Gert Heckman. Monodromy for the hypergeometric function n Fn−1 . Invent. Math., 95(2):325–354, 1989. [13] Jonathan M. Borwein and Peter B. Borwein. A cubic counterpart of Jacobi’s identity and the AGM. Trans. Amer. Math. Soc., 323(2):691–701, 1991. [14] Jonathan M. Borwein and Peter B. Borwein. Pi and the AGM. Canadian Mathematical Society Series of Monographs and Advanced Texts, 4. John Wiley & Sons, Inc., New York, 1998. A study in analytic number theory and computational complexity, Reprint of the 1987 original, A Wiley-Interscience Publication. [15] Sarah Chisholm, Alyson Deines, Ling Long, Gabriele Nebe, and Holly Swisher. p−adic analogues of ramanujan type formulas for 1/π. Mathematics, 1(1):9–30, 2013. [16] D. V. Chudnovsky and G. V. Chudnovsky. Approximations and complex multiplication according to Ramanujan. In Ramanujan revisited (Urbana-Champaign, Ill., 1987), pages 375– 472. Academic Press, Boston, MA, 1988. [17] Alyson Deines, Jenny G. Fuselier, Ling Long, Holly Swisher, and Fang Ting Tu. Generalized Legendre Curves and Quaternionic Multiplication. Journal of Number Theory (in press), arXiv:1412.6906, 2014. [18] Alyson Deines, Jenny G. Fuselier, Ling Long, Holly Swisher, and Fang Ting Tu. Hypergeometric series, truncated hypergeometric series, and Gaussian hypergeometric functions. ArXiv e-prints 1501.03564, Procceedings for Women in Number Theory 3 workshop, Association for Women in Mathematics series, January 2015. [19] Ron Evans and John Greene. Clausen’s theorem and hypergeometric functions over finite fields. Finite Fields Appl., 15(1):97–109, 2009. [20] Ron Evans and John Greene. Evaluations of hypergeometric functions over finite fields. Hiroshima Math. J., 39(2):217–235, 2009. [21] Ronald J. Evans. Identities for products of Gauss sums over finite fields. Enseign. Math. (2), 27(3-4):197–209 (1982), 1981. [22] Ronald J. Evans. Character sum analogues of constant term identities for root systems. Israel J. Math., 46(3):189–196, 1983. [23] Ronald J. Evans. Hermite character sums. Pacific J. Math., 122(2):357–390, 1986. [24] Ronald J. Evans. Character sums over finite fields. In Finite fields, coding theory, and advances in communications and computing (Las Vegas, NV, 1991), volume 141 of Lecture Notes in Pure and Appl. Math., pages 57–73. Dekker, New York, 1993. [25] Sharon Frechette, Ken Ono, and Matthew Papanikolas. Gaussian hypergeometric functions and traces of Hecke operators. Int. Math. Res. Not., (60):3233–3262, 2004. [26] Jenny G. Fuselier. Hypergeometric functions over Fp and relations to elliptic curves and modular forms. Proc. Amer. Math. Soc., 138(1):109–123, 2010. [27] Ira Gessel and Dennis Stanton. Strange evaluations of hypergeometric series. SIAM J. Math. Anal., 13(2):295–308, 1982.


[28] I. J. Good. Generalizations to several variables of Lagrange’s expansion, with applications to stochastic processes. Proc. Cambridge Philos. Soc., 56:367–380, 1960. ´ [29] Edouard Goursat. Sur l’´ equation diff´ erentielle lin´ eaire, qui admet pour int´ egrale la s´ erie hy´ perg´ eom´ etrique. Ann. Sci. Ecole Norm. Sup. (2), 10:3–142, 1881. [30] John Greene. Hypergeometric functions over finite fields. Trans. Amer. Math. Soc., 301(1):77– 101, 1987. [31] John Greene. Lagrange inversion over finite fields. Pacific J. Math., 130(2):313–325, 1987. [32] John Greene. Hypergeometric functions over finite fields and representations of SL(2, q). Rocky Mountain J. Math., 23(2):547–568, 1993. [33] John Greene and Dennis Stanton. A character sum evaluation and Gaussian hypergeometric series. J. Number Theory, 23(1):136–148, 1986. [34] Kenneth Ireland and Michael Rosen. A classical introduction to modern number theory, volume 84 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1990. [35] Nicholas M. Katz. Exponential sums and differential equations, volume 124 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1990. [36] Neal Koblitz. The number of points on certain families of hypersurfaces over finite fields. Compositio Math., 48(1):3–23, 1983. [37] Masao Koike. Hypergeometric series over finite fields and Ap´ ery numbers. Hiroshima Math. J., 22(3):461–467, 1992. [38] J. L. Lagrange. Nouvelle m´ ethode pour r´ esoudre des ´ equations litt´ erales par le moyen de s´ eries. M´ em. Acad. Roy. des Sci. et Belles-Lettres de Berlin, 24, 1770. [39] Serge Lang. Cyclotomic fields I and II, volume 121 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1990. With an appendix by Karl Rubin. [40] Catherine Lennon. Gaussian hypergeometric evaluations of traces of Frobenius for elliptic curves. Proc. Amer. Math. Soc., 139(6):1931–1938, 2011. [41] The LMFDB Collaboration. The L-functions and Modular Forms Database, Home page of the Elliptic Curve 144.a3. http://www.lmfdb.org/EllipticCurve/Q/144/a/3, 2013. [Online; accessed 6 October 2015]. [42] The LMFDB Collaboration. The L-functions and Modular Forms Database, Home page of the Elliptic Curve 288.d3. http://www.lmfdb.org/EllipticCurve/Q/288/d/3, 2013. [Online; accessed 6 October 2015]. [43] The LMFDB Collaboration. The L-functions and Modular Forms Database, Home page of the Elliptic Curve 64.a4. http://www.lmfdb.org/EllipticCurve/Q/64/a/4, 2013. [Online; accessed 6 October 2015]. [44] Ling Long. Hypergeometric evaluation identities and supercongruences. Pacific J. Math., 249(2):405–418, 2011. [45] Ling Long and Ravi Ramakrishna. Some supercongruences occurring in truncated hypergeometric series. preprint. arXiv:1403.5232. [46] Dermot McCarthy. Transformations of well-poised hypergeometric functions over finite fields. Finite Fields Appl., 18(6):1133–1147, 2012. [47] Ken Ono. Values of Gaussian hypergeometric series. Trans. Amer. Math. Soc., 350(3):1205– 1223, 1998. [48] Srinivasa Ramanujan. Modular equations and approximations to π [Quart. J. Math. 45 (1914), 350–372]. In Collected papers of Srinivasa Ramanujan, pages 23–39. AMS Chelsea Publ., Providence, RI, 2000. [49] Fernando Rodriguez-Villegas. Hypergeometric motives. 2015. [50] Hermann A. Schwarz. Ueber diejenigen F¨ alle, in welchen die Gaussische hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt. J. Reine Angew. Math., 75:292–335, 1873. [51] Jean-Pierre Serre. Repr´ esentations lin´ eaires des groupes finis. Hermann, Paris, revised edition, 1978. [52] Jean-Pierre Serre. Abelian l-adic representations and elliptic curves. Advanced Book Classics. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, second edition, 1989. With the collaboration of Willem Kuyk and John Labute. [53] Joseph H. Silverman. The arithmetic of elliptic curves, volume 106 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1986. [54] Lucy Joan Slater. Generalized hypergeometric functions. Cambridge University Press, Cambridge, 1966.


59

[55] Holly Swisher. On the supercongruence conjectures of van Hamme. Research in the Mathematical Sciences, 2015. To appear. [56] Fang-Ting Tu and Yifan Yang. Evaluations of certain hypergeometric functions over finite fields. Preprint. [57] Fang-Ting Tu and Yifan Yang. Algebraic transformations of hypergeometric functions and automorphic forms on Shimura curves. Trans. Amer. Math. Soc., 365(12):6697–6729, 2013. [58] Lucien van Hamme. Some conjectures concerning partial sums of generalized hypergeometric series. Lecture Notes in Pure and Appl. Math, 192:223–236, 1997. [59] M. Valentina Vega. Hypergeometric functions over finite fields and their relations to algebraic curves. Int. J. Number Theory, 7(8):2171–2195, 2011. [60] Raimundas Vid¯ unas. Transformations of some Gauss hypergeometric functions. J. Comput. Appl. Math., 178(1-2):473–487, 2005. [61] Raimundas Vid¯ unas. Algebraic transformations of Gauss hypergeometric functions. Funkcial. Ekvac., 52(2):139–180, 2009. [62] Andr´ e Weil. Jacobi sums as “Gr¨ ossencharaktere”. Trans. Amer. Math. Soc., 73:487–495, 1952. [63] J¨ urgen Wolfart. Werte hypergeometrischer Funktionen. Invent. Math., 92(1):187–216, 1988. [64] Koichi Yamamoto. On a conjecture of Hasse concerning multiplicative relations of Gaussian sums. J. Combinatorial Theory, 1:476–489, 1966. [65] Masaaki Yoshida. Fuchsian differential equations. Aspects of Mathematics, E11. Friedr. Vieweg & Sohn, Braunschweig, 1987. With special emphasis on the Gauss-Schwarz theory. High Point University, High Point, NC 27262, USA E-mail address: [email protected] Louisiana State University, Baton Rouge, LA 70803, USA E-mail address: [email protected] Cornell University, Ithaca, NY 14853, USA E-mail address: [email protected] Oregon State University, Corvallis, OR 97331, USA E-mail address: [email protected] National Center for Theoretical Sciences, Hsinchu, Taiwan 300, R.O.C. E-mail address: [email protected]