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A Finite Field Hypergeometric Function Associated to Eigenvalues of a Siegel Eigenform

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A Finite Field Hypergeometric Function Associated to Eigenvalues of a Siegel Eigenform A FINITE FIELD HYPERGEOMETRIC FUNCTION ASSOCIATED TO EIGENVALUES OF A SIEGEL EIGENFORM DERMOT McCARTHY AND MATTHEW A. PAPANIKOLAS Abstract. Although links between values of finite field hypergeometric functions and eigenvalues of elliptic modular forms are well known, we establish in this paper that there are also connections to eigenvalues of Siegel modular forms of higher degree. Specifically, we relate the eigenvalue of the Hecke operator of index p of a Siegel eigenform of degree 2 and level 8 to a special value of a 4F3-hypergeometric function. 1. Introduction and Statement of Results One of the more interesting applications of hypergeometric functions over finite fields is their links to elliptic modular forms and in particular Hecke eigenforms [1, 3, 8, 11, 12, 13, 21, 22, 24, 27, 28]. It is anticipated that these links represent a deeper connection that also encompasses Siegel modular forms of higher degree, and the purpose of this paper is to provide new evidence in this direction. Specifically, we relate the eigenvalue of the Hecke operator of index p of a certain Siegel eigenform of degree 2 and level 8 to a special value of a 4F3-hypergeometric function over Fp. We believe this is the first result connecting hypergeometric functions over finite fields to Siegel modular forms of degree > 1. Hypergeometric functions over finite fields were originally defined by Greene [15], who first established these functions as analogues of classical hypergeometric functions. Functions of this type were also introduced by Katz [17] about the same time. In the present article we use a normalized version of these functions defined by the first author [23], which is more suitable for our purposes. The reader is directed to [23, x2] for the precise connections among these three classes of functions. ∗ Let Fp denote the finite field with p elements, p a prime, and let Fcp denote the group ∗ ∗ of multiplicative characters of Fp. We extend the domain of χ 2 Fcp to Fp by defining χ(0) := 0 (including for the trivial character ") and denote χ as the inverse of χ. Let ∗ θ be a fixed non-trivial additive character of Fp and for χ 2 Fcp we have the Gauss sum g(χ) := P χ(x)θ(x). Then for A ;A ;:::;A ;B ;:::;B 2 ∗ and x 2 define x2Fp 0 1 n 1 n Fcp Fp A ;A ;:::;A F 0 1 n x n+1 n B ;:::;B 1 n p n n 1 X Y g(Aiχ) Y g(Bjχ) := g(χ)χ(−1)n+1χ(x): (1.1) p − 1 g(Ai) g(B ) ∗ i=0 j=1 j χ2Fcp Date: November 4, 2014. 2010 Mathematics Subject Classification. Primary: 11F46, 11T24; Secondary: 11F11, 11G20, 33E50. This project was partially supported by NSF Grant DMS-1200577. 1 2 DERMOT McCARTHY AND MATTHEW A. PAPANIKOLAS One of the first connections between finite field hypergeometric functions and the coeffi- cients of elliptic modular forms is due to Ahlgren and Ono [3], who proved the following. ∗ · Throughout we let φ 2 Fcp denote the Legendre symbol p . Theorem 1.2 (Ahlgren, Ono [3, Thm. 6]). Consider the unique newform g 2 S4(Γ0(8)) and the integers d(n) defined by 1 1 X Y g(z) = d(n)qn = q (1 − q2m)4(1 − q4m)4; q := e2πiz: n=1 m=1 Then for an an odd prime p, φ, φ, φ, φ 4F3 1 = d(p) + p: "; "; " p Another result relating n+1Fn to the Fourier coefficients of an elliptic modular form is due to Mortenson [24]. −4 Theorem 1.3 (Mortenson [24, Prop. 4.2]). Consider the unique newform h 2 S3(Γ0(16); ( · )) and the integers c(n) defined by 1 1 X Y h(z) = c(n)qn = q (1 − q4m)6: n=1 m=1 Then for p an odd prime, φ, φ, φ 3F2 1 = c(p): "; " p (Note that the original statements of Theorems 1.2 and 1.3 were expressed in terms of Greene's hypergeometric function. We have reformulated them in terms of n+1Fn.) Our main result below (Theorem 1.6) concerns the evaluation 4F3(φ, φ, φ, φ; "; "; "|−1)p, which we relate to eigenvalues of a Siegel eigenform of degree 2. Before stating this result we recall some fundamental facts about Siegel modular forms (see [9, 10, 18] for further details). Let Am×n denote the set of all m × n matrices with entries in the set A. For a matrix M we let tM denote its transpose, and if M has entries in C, we let Tr(M) denote its trace and n×n Im(M) its imaginary part. We denote the r × r identity matrix by Ir. If a matrix M 2 R is positive definite, then we write M > 0, and if M is positive semi-definite, we write M ≥ 0. The Siegel half-plane H2 of degree 2 is defined by 2 2×2 t H := Z 2 C j Z = Z; Im(Z) > 0 : Let Γ2 := Sp ( ) = M 2 4×4 j tMJM = J ;J = 0 I2 ; 4 Z Z −I2 0 be the Siegel modular group of degree 2 and let 2 2 Γ (q) := M 2 Γ j M ≡ I4 (mod q) be its principal congruence subgroup of level q 2 Z+. If Γ0 is a subgroup of Γ2 such that Γ2(q) ⊂ Γ0 for some minimal q, then we say Γ0 is a congruence subgroup of degree 2 and level q. The modular group Γ2 acts on H2 via the operation M · Z = (AZ + B)(CZ + D)−1 A HYPERGEOMETRIC FUNCTION ASSOCIATED TO A SIEGEL EIGENFORM 3 AB 2 2 0 where M = CD 2 Γ , Z 2 H . Let Γ be a congruence subgroup of degree 2 and level q. A holomorphic function F : H2 ! C is called a Siegel modular form of degree 2, weight k 2 Z+ and level q on Γ0 if −k F jkM(Z) := det(CZ + D) F (M · Z) = F (Z) AB 0 2 for all M = CD 2 Γ . We note that the desired boundedness of F jkM(Z), for any M 2 Γ , when Im(Z) − cI2 ≥ 0, with fixed c > 0, is automatically satisfied by the Koecher principle. The set of all such modular forms is a finite dimensional vector space over C, which we 2 0 2 0 denote Mk (Γ ). Every F 2 Mk (Γ ) has a Fourier expansion of the form X 2πi F (Z) = a(N) exp q Tr(NZ) N2R2 where Z 2 H2 and 2 2×2 t R = N = (Nij) 2 Q j N = N ≥ 0;Nii; 2Nij 2 Z : 2 0 We call F 2 Mk (Γ ) a cusp form if a(N) = 0 for all N 6> 0 and denote the space of such 2 0 forms Sk(Γ ). The Igusa theta constant of degree 2 with characteristic m = (m0; m00) 2 C1×4, m0; m00 2 C1×2 is defined by X 0 t 0 0 t 00 Θm(Z) = exp πi (n + m )Z (n + m ) + 2(n + m ) m : n2Z1×2 1 a b If m = 2 (a; b; c; d) then we will write Θ c d for Θm. In [14], van Geemen and van Straten exhibited several Siegel cusp forms of degree 2 and level 8 which are products of theta constants. The principal form of interest to us is 0 0 0 0 1 0 0 1 0 0 0 0 F7(Z) := Θ 0 0 (Z) · Θ 0 0 (Z) · Θ 0 0 (Z) · Θ 0 0 (Z) · Θ 0 1 (Z) · Θ 1 1 (Z); (1.4) 2 2 which lies in S3 (Γ (4; 8)), where 2 2 AB 2 2 Γ (8) ⊂ Γ (4; 8) := CD 2 Γ (4) j diag(B) ≡ diag(C) ≡ 0 (mod 8) ⊂ Γ (4): Furthermore, F7(Z) is an eigenform [14] in the sense that it is a simultaneous eigenfunction 2 2 for the all Hecke operators acting on M3 (Γ (8)). (See [9] for details on Hecke operators for Siegel forms of degree 2 with level.) In particular, for p an odd prime we can define the eigenvalue λ(p) 2 C of F7 by T (p) F7 = λ(p) F7; where T (p) is the Hecke operator of index p. Our main result (Theorem 1.6) relies on a connection between the Andrianov L-function a L (s; F7) of F7 and the tensor product L-function L(s; f1 ⊗ f2) of two elliptic newforms, −4 f1 2 S2(Γ0(32)) and f2 2 S3(Γ0(32); ( · )), where 1 1 X n Y 4n 2 8n 2 f1(z) = a(n)q = q (1 − q ) (1 − q ) n=1 m=1 p and taking i = −1, 1 X n 3 5 7 9 11 13 15 17 f2(z) = b(n)q = q + 4iq + 2q − 8iq − 7q − 4iq − 14q + 8iq + 18q + ··· : n=1 4 DERMOT McCARTHY AND MATTHEW A. PAPANIKOLAS a Van Geemen and van Straten [14, x8.7] conjectured that L (s; F7) and L(s; f1 ⊗ f2) have essentially the same Euler factors up to a twist of F7 (see the beginning of the proof of Theorem 1.6 for a precise statement). Our understanding is that this conjecture has been resolved recently by Okazaki [25] (see also [26, x1] for additional discussion on this problem). As a consequence, for any odd prime p, we find (see Section 5 for details) that 8 a(p)b(p) if p ≡ 1 (mod 8), <> λ(p) = −a(p)b(p) if p ≡ 5 (mod 8), (1.5) :>0 if p ≡ 3 (mod 4).
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