A FINITE FIELD HYPERGEOMETRIC FUNCTION ASSOCIATED TO EIGENVALUES OF A SIEGEL EIGENFORM. DERMOT McCARTHY AND MATTHEW A. PAPANIKOLAS

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1 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 hyergeometric functions and eigenvalues of ellitic modular forms are well known, we establish in this aer that there are also connections to eigenvalues of Siegel modular forms of higher degree. Secifically, we relate the eigenvalue of the Hecke oerator of index of a Siegel eigenform of degree and level 8 to a secial value of a 4 F 3 -hyergeometric function. 1. Introduction and Statement of Results One of the more interesting alications of hyergeometric functions over finite fields is their links to ellitic modular forms and in articular Hecke eigenforms [1, 3, 8, 11, 1, 13, 1,, 4, 7, 8]. It is anticiated that these links reresent a deeer connection that also encomasses Siegel modular forms of higher degree, and the urose of this aer is to rovide new evidence in this direction. Secifically, we relate the eigenvalue of the Hecke oerator of index of a certain Siegel eigenform of degree and level 8 to a secial value of a 4 F 3 -hyergeometric function over F. We believe this is the first result connecting hyergeometric functions over finite fields to Siegel modular forms of degree > 1. Hyergeometric functions over finite fields were originally defined by Greene [15], who first established these functions as analogues of classical hyergeometric functions. Functions of this tye were also introduced by Katz [17] about the same time. In the resent article we use a normalized version of these functions defined by the first author [3], which is more suitable for our uroses. The reader is directed to [3, ] for the recise connections among these three classes of functions. Let F denote the finite field with elements, a rime, and let F denote the grou of multilicative characters of F. We extend the domain of χ F to F by defining χ0 := 0 including for the trivial character ε and denote χ as the inverse of χ. Let θ be a fixed non-trivial additive character of F and for χ F we have the Gauss sum gχ := x F χxθx. Then for A 0, A 1,..., A n, B 1,..., B n F and x F define A0, A 1,..., A n n+1f n x B 1,..., B n := 1 1 n χ F i=0 ga i χ ga i n j=1 gb j χ gb j gχχ 1n+1 χx. 1.1 Date: November 4, Mathematics Subject Classification. Primary: 11F46, 11T4; Secondary: 11F11, 11G0, 33E50. This roject was artially suorted by NSF Grant DMS

2 DERMOT McCARTHY AND MATTHEW A. PAPANIKOLAS One of the first connections between finite field hyergeometric functions and the coefficients of ellitic modular forms is due to Ahlgren and Ono [3], who roved the following. Throughout we let φ F denote the Legendre symbol. Theorem 1. Ahlgren, Ono [3, Thm. 6]. Consider the unique newform g S 4 Γ 0 8 and the integers dn defined by gz = dnq n = q 1 q m 4 1 q 4m 4, q := e πiz. n=1 Then for an an odd rime, m=1 4F 3 φ, φ, φ, φ, ε 1 = d +. Another result relating n+1 F n to the Fourier coefficients of an ellitic modular form is due to Mortenson [4]. Theorem 1.3 Mortenson [4, Pro. 4.]. Consider the unique newform h S 3 Γ 0 16, 4 and the integers cn defined by hz = cnq n = q 1 q 4m 6. Then for an odd rime, n=1 3F φ, φ, φ m=1 1 = c. Note that the original statements of Theorems 1. and 1.3 were exressed in terms of Greene s hyergeometric function. We have reformulated them in terms of n+1 F n. Our main result below Theorem 1.6 concerns the evaluation 4 F 3 φ, φ, φ, φ;, ε 1, which we relate to eigenvalues of a Siegel eigenform of degree. Before stating this result we recall some fundamental facts about Siegel modular forms see [9, 10, 18] for further details. Let A m n denote the set of all m n matrices with entries in the set A. For a matrix M we let t M denote its transose, and if M has entries in C, we let TrM denote its trace and ImM its imaginary art. We denote the r r identity matrix by I r. If a matrix M R n n is ositive definite, then we write M > 0, and if M is ositive semi-definite, we write M 0. The Siegel half-lane H of degree is defined by H := { Z C t Z = Z, ImZ > 0 }. Let Γ := S 4 Z = { M Z 4 4 t MJM = J }, J = 0 I I 0, be the Siegel modular grou of degree and let Γ q := { M Γ M I 4 mod q } be its rincial congruence subgrou of level q Z +. If Γ is a subgrou of Γ such that Γ q Γ for some minimal q, then we say Γ is a congruence subgrou of degree and level q. The modular grou Γ acts on H via the oeration M Z = AZ + B CZ + D 1

3 A HYPERGEOMETRIC FUNCTION ASSOCIATED TO A SIEGEL EIGENFORM 3 where M = A B C D Γ, Z H. Let Γ be a congruence subgrou of degree and level q. A holomorhic function F : H C is called a Siegel modular form of degree, weight k Z + and level q on Γ if F k MZ := detcz + D k F M Z = F Z for all M = A B C D Γ. We note that the desired boundedness of F k MZ, for any M Γ, when ImZ ci 0, with fixed c > 0, is automatically satisfied by the Koecher rincile. The set of all such modular forms is a finite dimensional vector sace over C, which we denote Mk Γ. Every F Mk Γ has a Fourier exansion of the form F Z = πi TrN Z q where Z H and N R an ex R = { N = N ij Q t N = N 0, N ii, N ij Z }. We call F Mk Γ a cus form if an = 0 for all N 0 and denote the sace of such forms Sk Γ. The Igusa theta constant of degree with characteristic m = m, m C 1 4, m, m C 1 is defined by Θ m Z = { πi n + m Z t n + m + n + m t m }. n Z 1 ex If m = 1a, b, c, d then we will write Θ[ a b c d] for Θm. In [14], van Geemen and van Straten exhibited several Siegel cus forms of degree and level 8 which are roducts of theta constants. The rincial form of interest to us is F 7 Z := Θ [ ] [ Z Θ 0 0 ] [ 0 0 Z Θ 1 0 ] [ 0 0 Z Θ 0 1 ] [ 0 0 Z Θ 0 0 ] [ 0 1 Z Θ ] Z, 1.4 which lies in S 3Γ 4, 8, where Γ 8 Γ 4, 8 := { A B C D Γ 4 diagb diagc 0 mod 8 } Γ 4. Furthermore, F 7 Z is an eigenform [14] in the sense that it is a simultaneous eigenfunction for the all Hecke oerators acting on M 3 Γ 8. See [9] for details on Hecke oerators for Siegel forms of degree with level. In articular, for an odd rime we can define the eigenvalue λ C of F 7 by T F 7 = λ F 7, where T is the Hecke oerator of index. Our main result Theorem 1.6 relies on a connection between the Andrianov L-function L a s, F 7 of F 7 and the tensor roduct L-function Ls, f 1 f of two ellitic newforms, f 1 S Γ 0 3 and f S 3 Γ 0 3, 4, where f 1 z = anq n = q 1 q 4n 1 q 8n n=1 and taking i = 1, f z = bnq n = q + 4iq 3 + q 5 8iq 7 7q 9 4iq 11 14q iq q n=1 m=1

4 4 DERMOT McCARTHY AND MATTHEW A. PAPANIKOLAS Van Geemen and van Straten [14, 8.7] conjectured that L a s, F 7 and Ls, f 1 f have essentially the same Euler factors u to a twist of F 7 see the beginning of the roof of Theorem 1.6 for a recise statement. Our understanding is that this conjecture has been resolved recently by Okazaki [5] see also [6, 1] for additional discussion on this roblem. As a consequence, for any odd rime, we find see Section 5 for details that ab if 1 mod 8, λ = ab if 5 mod 8, if 3 mod 4. The main result of this aer relates the eigenvalues λ to a secial value of a 4 F 3 - hyergeometric function over F, conditional on the conjecture of van Geemen and van Straten. Theorem 1.6. Fix the Dirichlet character ξ = modulo 8. Assuming the conjecture of van Geemen and van Straten for F 7 [14, 8.7], for any odd rime, 4F 3 φ, φ, φ, φ, ε Moreover, when 3 mod 4, λ = 0. 1 = ξλ. Now the Fourier coefficients a and b can themselves be related to hyergeometric functions over F. The first such connection is due to Ono, and the second is established in the resent aer. Theorem 1.7 Ono [7, Thm. ]. For an odd rime, φ, φ F 1 1 = a. ε Theorem 1.8. If 1 mod 4 is rime and χ 4 F has order 4, then χ4, φ, φ 3F 1 = b. The roof of Theorem 1.8 follows lines of inquiry similar to those of Ahlgren and Ono [4], Ahlgren [], and Frechette, Ono, and the second author [11]. We use the Eichler-Selberg trace formula for Hecke oerators to isolate the Fourier coefficients of the form and connect these traces to hyergeometric values by counting isomorhism classes of members of the Legendre family of ellitic curves with rescribed torsion. The final ste in roving Theorem 1.6 is to aeal to a result of the first author [3, Thm. 1.5] see Theorem 5.13, which rovides a finite field version of a well-oised 4 F 3 - hyergeometric identity of While. From this we deduce that 4 F 3 φ, φ, φ, φ;, ε 1 = 0 for 3 mod 4 and that for 1 mod 4, 4F 3 φ, φ, φ, φ, ε 1 = F 1 φ, φ ε, χ4, φ, φ 1 3F Combining 1.5, Theorems 1.7 and 1.8, and 1.9 yields the desired result. The remainder of this aer is organized as follows. In Section we recall some roerties of class numbers of orders of imaginary quadratic fields and outline their relationshi to

5 A HYPERGEOMETRIC FUNCTION ASSOCIATED TO A SIEGEL EIGENFORM 5 isomorhism classes of ellitic curves over F. Section 3 outlines our use of the Eichler- Selberg trace formula. Proerties of the trace of Frobenius of the Legendre family of ellitic curves are develoed in Section 4. The roofs of Theorems 1.6 and 1.8 are contained in Section 5. Finally, we make some closing remarks in Section 6. Acknowledgements. The authors are extremely grateful to F. Rodriguez Villegas for ointing out to them the otential connection between the Siegel eigenforms and hyergeometric function values and in articular for alerting them to the examle in [14] as a source for our identities. The authors esecially thank T. Okazaki for his generosity in sharing his results and answering questions on his forthcoming aer. The authors also thank R. Osburn for helful advice and suggestions.. Class Numbers and Isomorhism Classes of Ellitic Curves In this section we recall some roerties of class numbers of orders of imaginary quadratic fields. In articular we will note their relationshi to isomorhism classes of ellitic curves over F. See [7, 9] for further details. We first recall some notation. For D < 0, D 0, 1 mod 4, let OD denote the unique imaginary quadratic order of discriminant D. Let hd = hod denote the class number of OD, i.e., the order of the ideal class grou of OD. Let ωd = ωod := 1 OD where OD is the grou of units of OD. For brevity, we let h D := hd/ωd. We also define HD := ho and H D := h O.1 OD O O max OD O O max where the sums are over all orders O between OD and the maximal order O max. We note that H D = HD unless O max = Z[ [ ] 1] or Z. In these excetional cases, 1+ 3 HD is greater by 1 and resectively, as only the term corresonding to O 3 max in each sum differs. If O has discriminant D and O O is an order such that [O : O ] = f, then the discriminant of O is f D. We will need the following lemma which relates class numbers of certain orders. Lemma. [7, Cor. 7.8]. Let O be an order of discriminant D in an imaginary quadratic field, and let O O be an order with [O : O ] = f. Then h O = h O f D 1 1, l l l f where D l is the Kronecker symbol. l rime Finally we resent a result of Schoof which relates these class numbers to the number of isomorhism classes of ellitic curves over F. Theorem.3 Schoof [9, ]. Let 5 be rime. Suose n Z +, s Z satisfy s 4, s, n 1 and n + 1 s. Then the number of isomorhism class of ellitic curves over F whose grou of F -rational oints has order + 1 s and contains s Z/nZ Z/nZ is H 4. n

6 6 DERMOT McCARTHY AND MATTHEW A. PAPANIKOLAS 3. Eichler-Selberg Trace Formula Let Tr k Γ 0 N, χ, denote the trace of the classical Hecke oerator T acting on S k Γ 0 N, χ. Similarly let Tr new k Γ 0 N, χ, denote the trace of T on the new subsace Sk new Γ 0 N, χ. The Eichler-Selberg trace formula gives a recise descrition of Tr k Γ 0 N, χ,. In this section we use a version of the trace formula due to Hijikata [16, Thm..] to evaluate Tr 3 Γ 0 16, 4, and Tr 3 Γ 0 3, 4,. We can simlify the trace formula as given in [16, Thm..] using the articulars of our saces and [16, Lem..5]. This is a long and tedious rocess but not articularly difficult so we omit the details here. However we note that we have used some elementary facts which we resent in the following roosition. For an odd rime and s an integer such that s < 4 we may write s 4 = t D, where D is a fundamental discriminant of some imaginary quadratic field. For a given and s, let a := ord t. Proosition 3.1. If s + 1 mod 8 then if 1 mod 8 and D is odd, then a > ; if 5 mod 8 and D is odd, then a = ; if 5 mod 8 and D is even, then a <. We have also used [4, Lem. 4.]. These simlifications yield the following two Theorems. Theorem 3.. If is an odd rime, then 0 if 3 mod 4, Tr 3 Γ 0 16, 4, = 6 s h s 4 c f 1 s, f if 1 mod 4, 0< s < s +1mod 16 s 4=t D where D is a fundamental discriminant and c 1 s, f is described in the table below with a := ord t and b := ord f. c 1 s, f D even D 1 mod 8 D 5 mod 8 a b = a b = a b = a b Theorem 3.3. If is an odd rime, then 0 if 3 mod 4, Tr 3 Γ 0 3, 4, = 8 s h s 4 c f s, f if 1 mod 4, 0< s < s +1mod 16 s 4=t D where D is a fundamental discriminant and c s, f is described in the table below with a := ord t and b := ord f. c s, f D even D 1 mod 8 D 5 mod 8 a b = a b = a b = a b

7 A HYPERGEOMETRIC FUNCTION ASSOCIATED TO A SIEGEL EIGENFORM 7 The dimension of S 3 Γ 0 16, 4 is one, so we can combine Theorems 3. and 3.3 to evaluate Tr new 3 Γ 0 3, 4,. Corollary 3.4. If is an odd rime, then 0 if 3 mod 4, Tr new 3 Γ 0 3, 4, = 4 s h s 4 c f 3 s, f if 1 mod 4, 0< s < s +1mod 16 s 4=t D where D is a fundamental discriminant and c 3 s, f is described in the table below with a := ord t and b := ord f. c 3 s, f D even D 1 mod 8 D 5 mod 8 a b = a b = a b = a b Legendre Family of Ellitic Curves In this section we recall some roerties of ellitic curves, and in articular roerties of the Legendre family of ellitic curves over finite fields. For further details lease refer to [19, 3]. We will also develo some reliminary results which we will use in Section 5. Theorem 4.1 [19, Thm. 4.]. Suose E is an ellitic curve over a field K, chark, 3, given by y = x ax bx c, with a, b, c K all distinct. Then, given a oint P = x, y EK there exists Q EK with P = []Q if and only if x a, x b and x c are all squares in K. The Legendre family of ellitic curves E λ over F is given by Proosition 4.3. E λ : y = xx 1x λ, λ F \ {0, 1} The j-invariant of E λ is je λ = 8 λ λ λ λ 1. je λ = 178 if and only if λ {, 1, 1}. 3 je λ = 0 if and only if λ λ + 1 = 0, i.e., λ { 1± 3 }. 4 The ma λ je λ is surjective and six-to-one excet above j = 0 and j = 178. In articular, { λ, 1 λ, 1 λ, 1 1 λ, λ λ 1, λ 1 } je λ. λ If t F \ {0}, then we define the t-quadratic twist of E λ by E t λ : y = xx tx tλ. If t is a square in F then E λ is isomorhic to E t λ over F. Straightforward calculations yield the following result. Proosition 4.4. Let 5 be rime. 1 E λ is the λ quadratic twist of E 1 λ 1 quadratic twist of E 1 λ. 3 E λ is the 1 λ quadratic twist of E λ λ 1 quadratic twist of E 1 1 λ. 5 E λ is the λ quadratic twist of E λ 1. λ. E λ is the. 4 E λ is the λ 1

8 8 DERMOT McCARTHY AND MATTHEW A. PAPANIKOLAS The curve E λ has 3 oints of order namely, 0, 0, 1, 0 and λ, 0. The following lemma gives us certain information on the -ower torsion of E λ which we will require in Section 5. Please see [19, V.5], [, Lemma 3.] and [4, Pro. 3.3] for similar arguments. Lemma 4.5. Let 5 be rime and let E λ be defined by E λ F contains Z/Z Z/8Z but not Z/4Z Z/4Z if 1 is a square, λ is a fourth ower and λ 1 is not a square in F. Let E/F be an ellitic curve such that EF contains Z/Z Z/8Z but not Z/4Z Z/4Z. Then, if 1 mod 4, E is isomorhic to E λ for some λ F \ {0, 1} with λ a fourth ower and λ 1 not a square in F. Proof. 1 Using Theorem 4.1 we see that 0, 0 is a double but 1, 0 and λ, 0 are not. Therefore E λ F cannot contain Z/4Z Z/4Z. Let t F be fixed such that t = λ and let i F be fixed such that i = 1. Using the dulication formula [3, III.] we see that if 1 and λ are squares then Q 1 = t, itt 1, Q = t, itt 1, Q 3 = t, itt + 1, Q 4 = t, itt + 1, are four oints of order 4 whose double is 0, 0. If λ 1 = t 1 is not a square then either t 1 or t+1 is a square but not both. Therefore, using Theorem 4.1 we see that either Q 1 and Q are doubles or Q 3 and Q 4 are doubles, but not both airs. Therefore E λ F contains Z/Z Z/8Z. As E has full -torsion we can assume it is of the form y = xx αx β for some α, β F. Also, E contains Z/Z Z/4Z but not Z/4Z Z/4Z so one and only one of 0, 0, α, 0 and β, 0 is a double. We can assume without loss of generality that 0, 0 is the double oint. Therefore, if 1 mod 4, we see from Theorem 4.1 that α and β are squares but α β is not. Letting m = α and n = β then E has the form y = xx m x n which is isomorhic to E λ with λ = n. Make the change of m variables x m x and y m 3 y. Therefore λ is a square but λ 1 is not. We see from art 1 that as 1 and λ are both squares there are four oints of order 4 whose double is 0, 0. Two of of these oints have x coordinate of t and the other two have x coordinate of t, where t = λ. As E contains Z/Z Z/8Z exactly two of these oints of order 4 are doubles. Therefore, by Theorem 4.1, t must be a square and either t 1 or t + 1 is a square but not both. As t = λ and t 1t + 1 = λ 1 we see that λ is a fourth ower and λ 1 is not a square. For an ellitic curve E/F we define the integer a E by a E := +1 EF. A curve E/F is suersingular if and only if a E, which if 5, is equivalent to a E = 0 by the Hasse bound. As usual if E is given by y = fx then a E = x F φ fx, 4.6 where φ is the Legendre symbol modulo and the character of order of F. We will often omit the subscrit when it is clear from the context. Consequently, We now rove some identities for a λ := a E λ. Lemma 4.8. For an odd rime, a E λ = φt a E t λ φλ=1 a λ = 1 + φ 1.

9 A HYPERGEOMETRIC FUNCTION ASSOCIATED TO A SIEGEL EIGENFORM 9 Proof. Using 4.6 we see that 1 φλ=1 a λ = 1 a t = φ 1 t= x F \{0,1} φxφx 1 φt x. Using roerties of Jacobsthal sums see [6, 6.1] we note that, for a F \ {0}, we have 1 t=1 φt +a = 1+φa and 1 x=1 φx x = 1. Therefore, we have t= φt x = 1 + φ x + φ1 x, and 1 φλ=1 a λ = φ 1 x F \{0,1} φx x + 1 x F \{0,1} φx 1 + t= x F \{0,1} We then aly the second Jacobsthal identity, noting that x F φx = 0. Lemma 4.9. If 1 mod 4, let χ 4 F denote a character of order 4. Then 1 1 φλλ 1= 1 1 φλλ 1=1 a λχ 4 λλ 1φλ 1 = 0; a λχ 4 λλ 1φλ 1 = 1 φλ=1 a λχ 4 λφλ 1. φx. Proof. The roofs of 1 and are relatively routine alications of Proosition 4.4 together with 4.7, using the fact that 1 is a square in F if 1 mod 4. We omit the details for reasons of brevity. Lemma If 1 mod 4, let χ 4 F denote a character of order 4. Then 1 χ 4 λ=1 φλ 1= 1 a λ = 1 φλ=1,χ 4 λ= 1 φλ 1=1 a λ. Proof. The curve E λ is -isogenous to the ellitic curve W λ : y = x 3 +1+λx +1 λ x, λ F \ {0, 1}, via the mas f λ : W λ E λ and its dual f λ : E λ W λ given by y f λ x, y =, y1 λ x y, fλ x, y =, yλ x, 4x 8x x x see [3, Ex. III.4.5]. In turn, when φλ = 1 and t F is fixed such that t = λ, W λ is isomorhic over F to E ψ, where ψ = 1 t. 1+t This follows from the change of variables x, y u x, u 3 y where u = 1 + t. Noting that isogenous curves over F have the same number of F -rational oints, we see that a λ = a ψ. Also, as 1 mod 4, it is easy to check that if χ 4 λ = 1 and φλ 1 = 1 then φψ = 1, χ 4 ψ = 1 and φψ 1 = 1. We now consider the sets S λ = { λ 1 χ 4 λ = 1, φλ 1 = 1}, S ψ = { ψ 1 φψ = 1, χ 4 ψ = 1, φψ 1 = 1}.

10 10 DERMOT McCARTHY AND MATTHEW A. PAPANIKOLAS If λ S λ then 1 λ S λ also, and, as 1 mod 4, λ and 1 λ are distinct. Similarly if ψ S ψ then 1 ψ S ψ with ψ and 1 ψ distinct. We define the equivalence relation on F \ {0, 1} by x y y x or 1 x mod. [ 1 t ], with 1+t We can then consider the bijective ma g : S λ / S ψ / given by g[λ] = [ inverse ma given by g 1 [ψ] = 1 s ], where s F 1+s is fixed such that s = ψ. It is easy to check that these mas are well-defined and indeendent of choice of square roots, t and s. There are exactly two elements in each equivalence class of S λ /, namely λ and 1. Also by λ Proosition 4.41, 4.7 and the fact that φλ = 1 we see that a λ = a 1 λ. Similarly there are exactly two elements in each equivalence class of S ψ / with a the same for both. Thus 1 χ 4 λ=1 φλ 1= 1 a λ = [λ] S λ / a λ = [ψ] S ψ / a ψ = 1 ψ= φψ=1,χ 4 ψ= 1 φψ 1=1 a ψ. 5. Proofs Lemma 5.1. The sace S3 new Γ 0 3, 4 is -dimensional over C with basis {f, f }, where f = n=1 bnqn = q + 4iq 3 + q 5 8iq 7 +, i = 1, and f is obtained by alying comlex conjugation to the Fourier coefficients of f. Furthermore, for an odd rime, b is real if 1 mod 4 and urely imaginary if 3 mod 4. Proof. Using Sage Mathematics Software [33], one comutes easily the dimension of the sace as well as the first several Fourier coefficients of f and f. To see the second art of the lemma, we consider the twist f,ψ = n=1 ψnbnqn of f by the character ψ = 4, which by [30, Lem. 3.6] is a form in S 3 Γ 0 3, ψ. Likewise, the twist f,ψ of f is also in S 3 Γ 0 3, ψ. One checks via Sage that f,ψ and f,ψ are both new at level 3. Therefore, we can exress the twists in terms of our basis for S3 new Γ 0 3, ψ, and it must be that f,ψ = f, f,ψ = f. We then have f + f = n=1 Rebnqn = n 1 mod 4 bnqn, roviding the result. Proof of Theorem 1.8. Assume that 1 mod 4 throughout, and choose a character χ 4 F of order 4. By Lemma 5.1, and so by Corollary 3.4, it suffices to rove χ4, φ, φ 3F 1 = b = 1 Trnew 3 Γ 0 3, 4,, 0< s < s +1mod 16 s s 4=t D h s 4 f c 3 s, f/, 5.

11 A HYPERGEOMETRIC FUNCTION ASSOCIATED TO A SIEGEL EIGENFORM 11 where D is a fundamental discriminant and c 3 s, f is as described in the Corollary 3.4. By Greene [15, Thm. 3.13] with [3, Pro..5] we have χ4, φ, φ 1 φ, φ 3F 1 = χ 4 1 F 1 λ χ ε 4 λχ 4 1 λ. Using a result of Koike [0, Sec. 4] again with [3, Pro..5], we exress the F 1 in the sum above in terms of the quantity a λ = a E λ, which we defined in Section 4. This yields 3F χ4, φ, φ 1 Using Lemmas 4.8 to 4.10 we see that 1 a λ χ 4 λλ 1 φλ 1 = 1 = a λ χ 4 λλ 1 φλ = We now use Theorem.3 to show that 1 a λ = Define the set χ 4 λ=1 φλ 1= 1 1 φλ=1 1 φλ=1 = + = + 4 0< s < s +1mod 16 a λχ 4 λφλ 1 a λ φλ=1 φλ=1 χ 4 λφλ 1= 1 1 χ 4 λ=1 φλ 1= 1 s H a λ a λ[1 χ 4 λφλ 1] a λ. 5.4 s 4 H 4 Ls, := {λ λ 1, χ 4 λ = 1, φλ 1 = 1, a λ = s}. Then, by the Hasse bound and Lemma 4.51, we obtain 1 χ 4 λ=1 φλ 1= 1 a λ = 0< s < λ Ls, s = 0< s < s +1mod 16 s s Ls,. 5.6 Let I denote the set of all isomorhism classes of ellitic curves over F. Define { } E C, a Is, := C I E = s, EF contains. Z/Z Z/8Z but not Z/4Z Z/4Z

12 1 DERMOT McCARTHY AND MATTHEW A. PAPANIKOLAS We now consider the ma F : Ls, Is, given by λ [E λ ], where [E λ ] I is the isomorhism class containing E λ. Lemma 4.5 ensures F is well-defined and surjective. For λ Ls, we see by Proosition 4.4 and 4.7 that E λ is isomorhic to E 1, E 1 λ and E λ 1 λ λ but not to E λ nor E 1. However, if λ Ls, then 1 λ 1 Ls, but 1 λ, Ls,, λ 1 1 λ λ λ as they do not meet the conditions. By Proosition 4.3 we then see that, for a given λ Ls,, if j[eλ] 0, 178 then F is two-to-one. We note that there are no classes of curves in Is, with j = 0 or 178. Otherwise, in the case j = 0, Ls, would have to contain λ = 1± 3 which satisfies λ λ + 1 = 0, i.e, λ 1 is a square. In the case j = 178, Ls, would have to contain λ {, 1, 1 }, none of which satisfy the conditions. Also, as s 0 we note that there are [ ] no classes of suersingular curves in Is,. Hence, O or Z [ 1 ]. See [9, 3] for details. By Theorem.3 we know that Is, = mod 16. Therefore Ls, = H s 4 H 4 H s 4 4 s 4 = H 16 O s 4 H 4 s 4 16 s 4 H 4 and 5.5 holds via 5.6. So, combining 5.3, 5.4 and 5.5 we obtain χ4, φ, φ 3F 1 = + 8 s H s 4 H 4 0< s < s +1mod 16 Comaring 5. and 5.7 it now suffices to show 8 0< s < s +1mod 16 s H s 4 H 4 s 4 16 = 0< s < s +1mod 16 s s 4=t D Z 1+ 3 s 4, as s s 4, 16 s h s 4 f c 3 s, f/, 5.8 where D is a fundamental discriminant and c 3 s, f is as described in the Corollary 3.4. We now use.1 and Lemma. to show that 5.8 holds. We roceed on a case by case basis deending on the congruence class of both and D. Recall a := ord t and b := ord f. Let 1 mod 8 and D 1 mod 8. Then by Proosition 3.1 we have a >. We first examine the left-hand side of 5.8. By Lemma. h s 4 = h s 4 f f { if a b, 1 if a b = 1, and h s 4 f = h s 4 4f { 4 if a b 3, if a b =.

13 A HYPERGEOMETRIC FUNCTION ASSOCIATED TO A SIEGEL EIGENFORM 13 Therefore using.1 we see that 8 H H s 4 4 s 4 16 = 8 a b 1 = a b 3 h s 4 8 h s 4 f 4f a b h s h s f f a b=1 Now the coefficient of s in the exression on the right-hand side of 5.8 in this case equals h s 4 + h s h s h s f f f f a b 3 a b= a b=1 a b=0 As a >, and a b=0 h s 4 = h s 4 = h s 4, f f f a b=1 a b=1 h s 4 = 4 h s 4 = 4 h s 4. f f f a b= a b= a b=1 Thus 5.10 equals h s h s f f a b 3 a b=1 Exressions 5.11 and 5.9 are equal and so 5.8 holds in the case 1 mod 8 and D 1 mod 8. The other cases roceed in a similar manner. We omit the details for reasons of brevity. We conclude, therefore, that Theorem 1.8 is true. Remark 5.1. Using the same methods as in the roof of Theorem 1.8, together with Theorem 3., we obtain a new roof of Theorem 1.3, but we do not resent the details here. We now recall a theorem of the first author that is a finite field version of While s 4F 3 -hyergeometric transformation for well-oised series. If A F is a square we will write A =. Theorem 5.13 McCarthy [3, Thm. 1.5]. For A, B, C, D F, A, B, C, D 4F 3 1 AB, AC, AD 0 if A, = ga gacd RB, C, D if A =, A ε, B ε, 3F 1 gac gad R, AB B A and CD A. R =A Proof of Theorem 1.6. Van Geemen and van Straten [14, 8.7] conjectured that there exists a relationshi between the Andrianov L-function of the form F 7 and the tensor roduct L- function of the forms f 1 and f. As mentioned reviously we understand that this conjecture

14 14 DERMOT McCARTHY AND MATTHEW A. PAPANIKOLAS has been resolved by Okazaki [5] and is the subject of a forthcoming aer see also [6, 1] for additional discussion on this roblem. Secifically, the conjecture imlies that there exists an eigenform F S 3Γ 4, 8 such that F = F 7 γ for some γ S 4 Z and L a s, F = Ls, f 1 f. The Andrianov L-Function for forms of degree with level is described in [9]. We see that L a s, F 7 has an Euler roduct over odd rimes with local factor L a s, F 7 = [ 1 λ s + λ λ s λ 3 3s + 6 4s] 1, 5.14 where λ is the eigenvalue associated to the Hecke oerator of index. From [6, Pro..] we then obtain L a s, F = [ 1 ζλ s + λ λ s ζλ 3 3s + 6 4s] 1 for a certain function ζt which is defined on odd t modulo 8 see also [5, Thm..]. We now examine Ls, f 1 f. Let ψ = 4 be the character associated to f. Given that f 1 and f are both Hecke eigenforms we know by the work of Shimura [31] that Ls, f 1 f has an Euler roduct with local factors at odd rimes of Ls, f 1 f = [ 1 ab s + b + ψ a ψ 3 s ψab 3 3s + 6 4s] 1. We note also that Ls, f 1 f = [1 ab s ] 1 = 1. Comaring the coefficients of s and 3s in L a s, F and Ls, f 1 f we conclude that ab = ζλ = 4 ab. Therefore, we conclude that λ = 0 when 3 mod 4. This is also aarent from the fact that f 1 is a CM-form and a = 0 for rimes 3 mod 4 e.g., see [7, Pro. 1]. For 1 mod 4 we determine ζ by examining the values of a, b and λ when = 5, 17, which are listed in [14, 7 8]. We determine therefore that ζ = 1, if 1 mod 8, and ζ = 1, if 5 mod 8, which coincides with ξ. Overall then, we find { ξ a b if 1 mod 4, λ = 0 if 3 mod 4. By Theorem 5.13 we have that, when 3 mod 4, φ, φ, φ, φ 4F 3 1, ε Thus the theorem is roved in the case 3 mod 4. We now concentrate on the case when 1 mod 4 and assume this throughout the remainder of the roof. Combining Theorems 1.7 and 1.8 it suffices to show that 4F 3 φ, φ, φ, φ, ε Now Theorem 5.13 yields 4F 3 φ, φ, φ, φ, ε 1 1 = F 1 φ, φ ε, = [ 1 χ4, φ, φ 3F χ 4, ε = 0. 3F χ4, φ, φ F χ4, φ, φ χ 4, ε ] 1.

15 A HYPERGEOMETRIC FUNCTION ASSOCIATED TO A SIEGEL EIGENFORM 15 We use transformations of Greene [15, 4.3, 4.5] with [3, Pro..5] to relate both 3F s in this equation to the 3 F in This gives us φ, φ, φ, φ χ4, φ, φ [ gφ gχ4 4F 3 1 =, ε 3 F 1 + gφ gχ ] 4. gχ 4 gχ 4 By [3, Thm. 1.10], F 1 φ, φ ε, 1 = gφ gχ 4 gχ 4 Therefore 5.15 holds and the theorem is roved. 6. Closing Remarks gφ gχ 4. gχ 4 We first note that the definition of n+1 F n as described in 1.1 can be easily extended to finite fields with a rime ower number of elements [3]. We also note that the reresentation of the Hecke algebra in Mk Γ q is generated by T and T, the Hecke oerators of index and resectively [9]. Therefore, for a given eigenform F of degree we are also interested in λ, the eigenvalue associated to the action of T on F. Based on limited numerical evidence it aears that the eigenvalue λ associated to the eigenform F 7 is also related to a hyergeometric function but over F, as follows. Let a := λ and a := λ λ corresonding u to sign to the coefficients of s and s in 5.14 resectively. Then we have observed for rimes < 0 that φ, φ, φ, φ a a = 4 F 3 1, ε It is conceivable that this identity holds for all odd rimes, and we believe similar methods to those emloyed in the main body of this aer could be alied to roving it. References [1] S. Ahlgren, Gaussian hyergeometric series and combinatorial congruences, in: Symbolic comutation, number theory, secial functions, hysics and combinatorics Gainesville, FL, 1999, 1 1, Dev. Math., 4, Kluwer, Dordrecht, 001. [] S. Ahlgren, The oints of a certain fivefold over finite fields and the twelfth ower of the eta function, Finite Fields Al. 8 00, no. 1, [3] S. Ahlgren, K. Ono, A Gaussian hyergeometric series evaluation and Aéry number congruences, J. Reine Angew. Math , [4] S. Ahlgren, K. Ono, Modularity of a certain Calabi-Yau threefold, Monatsh. Math , no. 3, [5] A. N. Andrianov, F. A. Andrianov, Action of Hecke oerators on roducts of Igusa theta constants with rational characteristics, Acta Arith , no., [6] B. C. Berndt, R. J. Evans, K. S. Williams, Gauss and Jacobi Sums, John Wiley & Sons, Inc., New York, [7] D. A. Cox, Primes of the Form x + ny, John Wiley & Sons, Inc., New York, [8] R. Evans, Hyergeometric 3 F 1/4 evaluations over finite fields and Hecke eigenforms, Proc. Amer. Math. Soc , no., [9] S. A. Evdokimov, Euler roducts for congruence subgrous of the Siegel grou of genus, Math. USSR- Sb , no. 4, [10] S. A. Evdokimov, A basis of eigenfunctions of Hecke oerators in the theory of modular forms of genus n, Math. USSR-Sb , no. 3,

16 16 DERMOT McCARTHY AND MATTHEW A. PAPANIKOLAS [11] S. Frechette, K. Ono, M. Paanikolas, Gaussian hyergeometric functions and traces of Hecke oerators, Int. Math. Res. Not. 004, no. 60, [1] J. G. Fuselier, Hyergeometric functions over F and relations to ellitic curves and modular forms, Proc. Amer. Math. Soc , no. 1, [13] J. G. Fuselier, D. McCarthy, Hyergeometric tye identities in the -adic setting and modular forms, 014, arxiv: [14] B. van Geemen, D. van Straten, The cus forms of weight 3 on Γ, 4, 8, Math. Com , no. 04, [15] J. Greene, Hyergeometric functions over finite fields, Trans. Amer. Math. Soc , no. 1, [16] H. Hijikata, A. K. Pizer, T. R. Shemanske, The basis roblem for modular forms on Γ 0 N, Mem. Amer. Math. Soc , no. 418, vi+159. [17] N. M. Katz, Exonential Sums and Differential Equations, Princeton Univ. Press, Princeton, [18] H. Klingen, Introductory Lectures on Siegel Modular Forms, Cambridge Univ. Press, Cambridge, [19] A. W. Kna, Ellitic Curves, Princeton Univ. Press, Princeton, 199. [0] M. Koike, Orthogonal matrices obtained from hyergeometric series over finite fields and ellitic curves over finite fields, Hiroshima Math. J , no. 1, [1] C. Lennon, Trace formulas for Hecke oerators, Gaussian hyergeometric functions, and the modularity of a threefold, J. Number Theory , no. 1, [] D. McCarthy, On a suercongruence conjecture of Rodriguez-Villegas, Proc. Amer. Math. Soc , no. 7, [3] D. McCarthy, Transformations of well-oised hyergeometric functions over finite fields, Finite Fields Al , no. 6, [4] E. Mortenson, Suercongruences for truncated n+1 F n hyergeometric series with alications to certain weight three newforms, Proc. Amer. Math. Soc , no., [5] T. Okazaki, Trile geometric forms for SU,, in rearation. [6] T. Okazaki, L-functions of S 3 Γ4, 8, J. Number Theory , no. 1, [7] K. Ono, Values of Gaussian hyergeometric series, Trans. Amer. Math. Soc , no. 3, [8] M. Paanikolas, A formula and a congruence for Ramanujan s τ-function, Proc. Amer. Math. Soc , no., [9] R. Schoof, Nonsingular lane cubic curves over finite fields, J. Combin. Theory Ser. A , no., [30] G. Shimura, On modular forms of half integral weight, Ann. of Math , [31] G. Shimura, The secial values of the zeta functions associated with cus forms, Comm. Pure Al. Math , no. 6, [3] J. H. Silverman, The Arithmetic of Ellitic Curves, nd ed., Sringer-Verlag, New York, 009. [33] W. A. Stein et al., Sage Mathematics Software Version 4.8, The Sage Develoment Team, 01, htt:// Deartment of Mathematics & Statistics, Texas Tech University, Lubbock, TX , USA address: dermot.mccarthy@ttu.edu Deartment of Mathematics, Texas A&M University, College Station, TX , USA address: ma@math.tamu.edu