GAUSSIAN HYPERGEOMETRIC FUNCTIONS AND TRACES OF HECKE OPERATORS

Transcription

1 GAUSSIAN HYPERGEOMETRIC FUNCTIONS AND TRACES OF HECKE OPERATORS SHARON FRECHETTE, KEN ONO, AND MATTHEW PAPANIKOLAS Abstract We establish a simple inductive formula for the trace (Γ 0 (8, p of the p-th Hece operator on the space S new (Γ 0 (8 of newforms of level 8 and weight in terms of the values of 3 F -hypergeometric functions over the finite field F p Using this formula when = 6, we prove a conecture of Koie relating 6 (Γ 0 (8, p to the values 6 F 5 (1 p and 4F 3 (1 p Furthermore, we find new congruences between (Γ 0 (8, p and generalized Apéry numbers 1 Introduction and statement of results Let p be an odd prime, and let F p denote the finite field with p elements For any multiplicative character χ on F p, extend χ to a character on F p by defining χ(0 := 0 For two characters A and B on F p, we define the normalized Jacobi sum ( B A by (11 ( A B := B(1 J(A, B = 1 A(xB(x 1, p p x F p where B is the complex conugate of the character B Let n be a positive integer For characters A 0, A 1,,A n and B 1, B,,B n on F p, Greene [9] defined the Gaussian hypergeometric series over F p by (1 n+1f n ( A0, A 1,, A n B 1,, B n x p := p p 1 χ ( A0 χ χ ( A1 χ B 1 χ ( An χ χ(x, B n χ where the sum is taen over all characters χ on F p Let ε denote the trivial character, and let φ denote the quadratic character modulo p (the prime p will always be clear from context Specializing to our purposes, define ( φ, φ,, φ (13 n+1f n (x := n+1 F n ε,, ε x p = p p 1 ( n+1 φχ χ(x χ To emphasize the dependence on p, we will occasionally write n+1 F n (x p := n+1 F n (x One important role of Gaussian hypergeometric functions is that they provide formulas for the Fourier coefficients of certain modular forms For example, if λ Q \ {0, 1} and p is Date: March 8, 004; April 7, 004 (final version 1991 Mathematics Subect Classification Primary 11F11; Secondary 11F7, 11T4, 33C99 The first author thans the Department of Mathematics at Brown University for its hospitality during her visit for the academic year The second author is grateful for the support of a grant from the National Science Foundation, and the generous support of the Alfred P Sloan, David and Lucile Pacard, H I Romnes, and John S Guggenheim Fellowships The third author thans the support of grants from the NSA MDA and NSF DMS χ

2 SHARON FRECHETTE, KEN ONO, AND MATTHEW PAPANIKOLAS a prime of good reduction for the Legendre normal form elliptic curve E(λ : y = x(x 1(x λ, then φ(1p F 1 (λ p is the p-th Fourier coefficient of the weight newform associated to E(λ by the Shimura-Taniyama correspondence [1], [17] Similarly, if λ { 1, 4, 1 4, 8, 1 8, 64, 1 64}, then, for all but finitely many primes p, it turns out that 3 F (λ p is essentially the p-th Fourier coefficient of an explicit weight 3 newform with complex multiplication that is associated to a certain singular K3 surface X λ (see Corollary 110 of [18] In view of these examples, it is natural to see further formulas for coefficients of modular forms in terms of Gaussian hypergeometric functions Here we address the problem of obtaining a Gaussian hypergeometric trace formula for the action of Hece operators For positive integers N and, let S (Γ 0 (N denote the space of cusp forms of weight on the congruence subgroup Γ 0 (N Let S new (Γ 0 (N denote the subspace generated by its level N newforms Furthermore, let Tr (Γ 0 (N, p denote the trace of the Hece operator T p on S (Γ 0 (N, and similarly let (Γ 0 (N, p denote the trace of T p on S new (Γ 0 (N The Eichler-Selberg trace formula [10] gives a precise description of Tr (Γ 0 (N, p; however, the formula is quite complicated (for example, see Theorem Here we give a simple formula, inductive in, for (Γ 0 (8, p in terms of the values 3 F (λ Moreover, Theorem 11 below provides a complete description of the traces of Hece operators T p for cusp forms on Γ 0 (8 To state this result, we first fix notation Let be even If p 1 (mod 4, then we can uniquely write p = a +b, where a and b are positive integers, and where a is odd Then we define { 1 (14 ε (p := (4a p (4b p 1 if p 1 (mod 4, (p 1 if p 3 (mod 4 Remar Using Theorem 43 (, it is straightforward to verify that [ ] (15 ε (p = φ(1 1 (p + p 3F ( (p p 3F (1 1 Also, for odd primes p and 4, define the function H (p by (16 H (p := p φ(λφ(1 λ 1 3F (λ 1, and set H (p := φ(1 Let [ n ] denote the trinomial coefficient defined by the expansion n [ ] n (17 (1 + x + x 1 n = x =n Theorem 11 If p is an odd prime and is even, then (Γ 0 (8, p = H (p ε (p 1 ([ =1 1 1 ] [ 1 ] p (Γ 0 (8, p

3 HYPERGEOMETRIC FUNCTIONS AND HECKE OPERATORS 3 Remar As the referee has indly pointed out, Theorem 11 can also be formulated in terms of generating functions Moreover, if we let c (d denote the coefficient of x in (1 x(1 + x + x d, then Theorem 11 becomes the more compact 1 (18 ε (p + H (p + p c ( =0 1 Trnew (Γ 0 (8, p = 0 It is also interesting to consider the generating function Trnew (Γ 0 (8, px 1 Using properties of the numbers c (d (see [14, p 316], we then find that where and E(x + By setting φ(λ 1 φ(1 λp 3F (λx + A(px even (Γ 0 (8, p(xb(px 1 = 0, A(x = 1 + x 1 x 3x, B(x = 1 x 1 x 3x, x(1 + x x { 1/ 1(4a E(x = + 1/ if p 1 (mod 4, px 1(4b px 1 if p 3 (mod 4 1+px we then observe (19 (Γ 0 (8, px 1 = even R(x := x 1 + px + p x, [ ] 1 + px φ(λ E(R(x px + p x 1 φ(1 λp 3F (λr(x Thus Trnew (Γ 0 (8, px 1 is a rational function By computing the values of 3 F (λ explicity, we can compute this rational function for specific values of p For example, we find (110 even (Γ 0 (8, 3x 1 4x = 1 + 5x + 15x + 7x 3 For larger primes the rational functions become increasingly more complicated Remar It is reasonable to expect that there are generalizations of Theorem 11 for other Γ 0 (N, which will be the subect of further study However, there does not appear to be a simple way of obtaining a general result in which the choice of parameters in the relevant Gaussian hypergeometric functions are given a priori as a function of N With the proper hypergeometric functions in hand, it seems liely that proofs of such generalizations would follow from arguments similar to the ones presented here Theorem 11 has some immediate consequences Here we describe several applications As usual, let η(z denote Dedeind s eta-function (111 η(z := q 1/4 (1 q n, n=1

4 4 SHARON FRECHETTE, KEN ONO, AND MATTHEW PAPANIKOLAS where q := e πiz Let η(z 4 η(4z 4 = n=1 a(nqn be the unique newform in S new 4 (Γ 0 (8 If p is an odd prime, then Theorem 11 implies that 4 (Γ 0 (8, p = a(p = H 4 (p ε 4 (p = p p φ(λφ(1 λ 3 F (λ By Theorem 313 of [9] (see also Proposition 41 (, we have p φ(λφ(1λ 3F (λ = p 3 4F 3 (1, and so (11 4 (Γ 0 (8, p = a(p = p 3 4F 3 (1 p This formula is the conclusion of Theorem 6 of [3], and is equivalent to the assertion that the Calabi-Yau threefold given by x + 1 x + y + 1 y + z + 1 z + w + 1 w = 0 is modular As another application, we recall the following conecture of Koie [13] Conecture (Koie Let η(z 8 η(4z 4 + 8η(4z 1 = n=1 b(nqn be the unique newform in S6 new (Γ 0 (8 If p is an odd prime, then b(p = p 5 6F 5 (1 + p 4 4F 3 (1 + (1 φ(1p By combining Theorem 11 with transformation laws for Gaussian hypergeometric functions, we obtain the following Corollary 1 Koie s Conecture is true In addition to their relationship with coefficients of modular forms, Gaussian hypergeometric functions have also played important roles in the proofs of supercongruence conectures of Beuers [5], [6] and Rodriguez-Villegas [0] (see [3], [15], [16] For primes p 5, the following congruence due to Mortenson [16] is typical n=0 (6n! (n!(n!(3n! 4n 3 3n φ(1 (mod p Other wors by Ahlgren [1], Koie [1], and the second author [17] provide further examples of p-adic results for combinatorial expressions whose proofs require these functions As an additional application, we consider congruences of the type originally considered by Beuers [5], [6] If n is a positive integer, then define the Apéry number A(n by n ( ( n + n (113 A(n := =0 These numbers played an important role in Apéry s celebrated proof of the irrationality of ζ(3 In 1987, Beuers related these numbers to modular forms [6]; he proved that if p is an odd prime, then ( p 1 (114 4 (Γ 0 (8, p A (mod p

5 HYPERGEOMETRIC FUNCTIONS AND HECKE OPERATORS 5 He went on to conecture that ( p 1 4 (Γ 0 (8, p A (mod p Using (11, the Gross-Koblitz formula for the p-adic Gamma-function, some p-adic analysis, and the WZ method, Ahlgren and the second author [3] successfully proved this conecture Using Theorem 11, it is now possible to obtain generalizations of such congruences For brevity, we shall be content with congruences modulo primes p To state these results, for integers m, l, λ, and n, define the generalized Apéry number A(m, l, λ; n by n ( m ( l n + n (115 A(m, l, λ; n := λ Of course, we have that A(n = A(,, 1; n Theorem 13 Suppose that 4 is even, and that p is an odd prime (1 If (mod p 1, then ( p 1 (Γ 0 (8, p A (mod p ( If =0 3 (mod p 1, then ( (Γ 0 (8, p A, 4, 1; p 1 In general, we have the following (mod p Theorem 14 If 4 is even and p is an odd prime, then ( (1 + (1 1 (Γ 0 (8, p A 1,, 1; p 1 1 ( φ(λφ(1 λ 1 A 1,, λ; p 1 1 (mod p Remar Mortenson has indly pointed out that, as an immediate corollary to Theorem 13, one has the following As in the statement of Koie s conecture, we let b(nq n be the Fourier expansion of the unique newform in S 6 (Γ 0 (8 Then for all odd primes p, b(p n=0 ( 1 6 n (n! 6 (mod p, where as usual, (a n := a(a + 1 (a + n 1 for n > 0 and (a 0 = 1 In fact, Mortenson points out that this congruence appears to hold modulo p 5 In Section we recall a formulation of the Eichler-Selberg trace formula for the groups Γ 0 (4 and Γ 0 (8, and we state a formula, which will be proved in Section 7, for the group Γ 0 ( (see Theorem 3 In Section 3 we then interpret these trace formulas in terms of the numbers of F p -points on certain classes of varieties, and in Section 4 we recall essential facts regarding Gaussian hypergeometric functions Assuming the truth of Theorem 3, in Section 5 we combine all of these results to prove Theorem 11 and Corollary 1 In Section 6 we prove Theorems 13 and 14 In Section 7, we conclude with a proof of Theorem 3

6 6 SHARON FRECHETTE, KEN ONO, AND MATTHEW PAPANIKOLAS Trace formulas Fix a prime p 3, and let be even Using the version of the Eichler-Selberg trace formula due to Hiiata [10, Thm ], we will prove formulas for Tr (Γ 0 (N, p when N =, 4, and 8 In the end, we consider simplifications of Hiiata s formula that relate Tr (Γ 0 (N, p to the number of points on certain varieties over F p We begin by fixing notation We will mae special use of two families of elliptic curves: (1 ( E 1 (λ : y = x(x 1(x λ, 3E (λ : y = (x 1(x + λ For a prime p 3 and λ F p, the traces of Frobenius A 1 (p, λ and 3 A (p, λ are (3 (4 A 1 (p, λ = p + 1 E 1 (λ(f p, λ 0, 1, 3A (p, λ = p E (λ(f p, λ 0, 1 We will rewrite the relevant trace formulas using these quantities Let (5 F (x, y := x1 y 1 x y The relations x + y = s and xy = p uniquely define a polynomial G (s, p = F (x, y Moreover, a straightforward induction gives 1 (6 G (s, p = =0 (1 ( p s The polynomial G (s, p, evaluated at certain values of s, is a ey part of Hiiata s formula for Tr (Γ 0 (N, p (eg see Theorem An important observation is that among the various pieces of Hiiata s formula, for fixed level N, the only part that varies with is G (s, p Moreover, the points s at which we evaluate G (s, p depend only on p and N and not on The following proposition appears in [, Thms 1 ], in the case of level 4, weight 6, and in [4, Thms 1 ], in the case of level 8, weight 4 In fact, the formulas below hold for all even 4 with exactly the same proofs Proposition 1 ((1 Ahlgren []; ( Ahlgren-Ono [4] If p is an odd prime, and 4 is even, then the following are true (1 Tr (Γ 0 (4, p = 3 G ( A 1 (p, λ, p p ( Tr (Γ 0 (8, p = 4 G ( A 1 (p, λ, p We set more notation For d < 0 and d 0, 1 (mod 4, let O(d be the order of discriminant d in the imaginary quadratic field Q( d Let h(o(d = h(d be the class number of O(d, and ω(o(d = ω(d = 1 O(d Finally, let h (d = h(d/ω(d Also,

7 HYPERGEOMETRIC FUNCTIONS AND HECKE OPERATORS 7 recall the definition of the Kronecer symbol for d, ( d 0 if d 0 (mod 4, (7 := 1 if d 1 (mod 8, 1 if d 5 (mod 8 Theorem below is Hiiata s version of the Eichler-Selberg trace formula at level, whose derivation from the general formula is a straightforward calculation which we omit Theorem (Hiiata [10, Thm ] Let p be an odd prime, and let be even Tr (Γ 0 (, p = ξ(p(p 1 G (s, p ( s h 4p c(s, f, f f t where 0<s< p s even 1 h (4p if p 1 (mod 4, ξ(p = 3h (p if p 3 (mod 8, h (p if p 7 (mod 8, and where if s 4p = t D, with D the discriminant of Q( D, and if f t, { 1 + ( D if ord c(s, f = (f = ord (t, if ord (f < ord (t As before, whenever p 1 (mod 4, we let a, b 0, a odd, be defined by the expression p = a + b We then define { 1 (8 δ (p := G (a, p + 1G (b, p if p 1 (mod 4, (p 1 if p 3 (mod 4 The following theorem provides the level version of Proposition 1 Theorem 3 For a prime p 3 and 4 even, p Tr (Γ 0 (, p = δ (p G ( 3 A (p, λ, p We postpone the proof of Theorem 3, which is self-contained, until Section 7 λ=1 3 Counting points on varieties over F p For 4 even, define three sequences of varieties U, V, and W, which are hypersurfaces in affine -space, by (31 (3 (33 U : y = (x i 1(x i + λ, i=1 V : y = x i (x i 1(x i λ, i=1 W : y = x i (x i 1(x i λ i=1

8 8 SHARON FRECHETTE, KEN ONO, AND MATTHEW PAPANIKOLAS One sees readily that U, V, and W are constructed from families of elliptic curves with subgroups of the form Z/Z, Z/Z Z/Z, and Z/4Z Z/Z respectively Geometrically, these varieties are essentially of Kuga-Sato type: for example, there is an easily defined surective map onto V from the ( -th power of the Legendre family, fibered over the λ-line Thus as in Birch [7] and Ihara [11], for a prime p 3 it is reasonable to expect that the numbers of points in U (F p, V (F p, and W (F p are directly related to Tr (Γ 0 (, p, Tr (Γ 0 (4, p, and Tr (Γ 0 (8, p respectively We mae these assertions exact in Propositions 31, 3, and 33 We now consider formulas for U (F p, V (F p, and W (F p for primes p 3 An exact formula for W 4 (F p was established by Ahlgren and the second author [4, Thm 1], and one for V 6 (F p was determined by Ahlgren [, Thm 1] Propositions extend these results to each of U (F p, V (F p, W (F p for arbitrary even 4 Let C(n = 1 n+1 ( n n be the nth Catalan number, and let δ (p be as in (8 Proposition 31 For a prime p 3 and 4 even, U (F p = p C ( 1 p 1 (p (( ( p 1 (Tr (Γ 0 (, p + δ (p + =0 Remar It is worth pointing out that the coefficients ( ( 1 are precisely the ones needed to express x in terms of Chebyshev polynomials of the second ind Proof First of all, we have (34 U (F p = { λ,x 1,,x F p { = p 1 + λ=0 1 + φ x=0 ( } (x i 1(x i + λ i=1 p = p A (p, λ λ=1 ( } φ (x 1(x + λ We will rewrite this expression in terms of the polynomials G ( 3 A (p, λ, p for 0 1 using a combinatorial argument involving inverse relations (see Riordan [19, Chs 3] One such inverse pair [19, Table 3] is the following: n (( n (35 a n = =0 We may rearrange (6 to give G (s, p p 1 = ( n b 1 n and b n = 1 =0 (1 ( n =0 ( s p (1 ( n a n

9 HYPERGEOMETRIC FUNCTIONS AND HECKE OPERATORS 9 Setting n =, and using a n = ( s p n and bn = G n+(s,p p n 1 s = =0 (( in (35, we obtain ( p 1 G (s, p Substituting s = 3 A (p, λ, the formula in (34 therefore becomes 1 (( ( p U (F p = p p 1 G ( 3 A (p, λ, p =0 λ=1 (( ( = p p 1 (p (( ( ( p 1 Tr (Γ 0 (, p + δ (p +, =0 where for the second equality we apply Theorem 3, when, and use that G = 1, when = 1 Using standard facts about binomial coefficients, we see that ( ( 1 = C ( 1 Finally, we adust the sum to incorporate a = 1 term, which equals 3C ( 1 p 1, since Tr (Γ 0 (, p = 0 and δ (p = 1, and obtain the desired equality The derivations of the formulas for V (F p and W (F p in the following propositions are essentially the same as the proof of Proposition 31 The primary differences are that for V (F p we use Proposition 1 (1 instead of Theorem 3, and that for W (F p we use Proposition 1 ( We omit the remaining details Proposition 3 For a prime p 3 and 4 even, V (F p = p C ( 1 p 1 (p =0 (( Proposition 33 For a prime p 3 and 4 even, W (F p = p C ( 1 p 1 (p =0 (( ( p 1 (Tr (Γ 0 (4, p + 3 ( p 1 (Tr (Γ 0 (8, p + 4 Combining the numbers of F p -points on the varieties U, V, and W yields an amusing and useful relationship among the traces (Γ 0 (8, p Specifically, we put (36 N (p = U (F p + V (F p W (F p, and obtain the following theorem as an immediate consequence of Propositions 31 33

10 10 SHARON FRECHETTE, KEN ONO, AND MATTHEW PAPANIKOLAS Theorem 34 For a prime p 3 and 4 even, 1 N (p = 1 + =0 (( ( p ( 1 (Γ 0 (8, p + δ (p 4 Gaussian hypergeometric functions Gaussian hypergeometric functions over finite fields were defined by Greene [9] as character sum analogues of the classical hypergeometric functions The classical functions satisfy many interesting properties, such as transformation and summation formulas, and Greene showed that their finite field analogues enoyed many similar properties Koie [1] and the second author [17] further explored the arithmetic properties of Gaussian hypergeometric functions, including the number-theoretic significance of certain special values of these functions We continue this study below in Section 5, proving Theorem 11 and Koie s conecture In this section, we give several properties of Gaussian hypergeometric functions which we shall require Using properties of characters and of Jacobi sums, Greene proved an alternate formula for the F 1 function Also, Greene [9, Thm 313] showed that a Gaussian hypergeometric function can be expressed as a sum of Gaussian hypergeometric functions of lower degree Specializing these results to the case of n+1 F n (λ as defined above, we have the following proposition Proposition 41 (Greene [9] If n 1 and λ F p, then the following hold (1 F 1 (λ = ε(λφ(1 φ(xφ(1 xφ(1 xλ p x F p ( n+1 F n (λ = φ(1 φ(xφ(1 x n F n1 (xλ p x F p One of the transformation formulas proved by Greene [9, Thm 4] involves the relationship between a Gaussian hypergeometric series evaluated at λ and at 1/λ We will have need of two special cases of this theorem, as given in the following proposition Proposition 4 (Greene [9] If λ F p is nonzero, then (1 F 1 (λ = φ(λ F 1 ( 1 λ ( 3 F (λ = φ(λ 3 F ( 1 λ We will also have need of the special values F 1 (1 and 3 F (1 Both parts of the following theorem appear in [17]; part (1 is a special case of Theorem, and it is noted that part ( is a special case of a theorem of Evans Theorem 43 (Ono [17] Let p be an odd prime, and if p 1 (mod 4, then write p = a +b where a and b are positive integers, and where a is odd The following hold 0 if p 3 (mod 4, (1 F 1 (1 = a(1 a+b+1 if p 1 (mod 4 p 0 if p 3 (mod 4, ( 3 F (1 = 4a p if p 1 (mod 4 p

11 HYPERGEOMETRIC FUNCTIONS AND HECKE OPERATORS 11 The following theorem relates the values of F 1 (λ and 3 F (λ to the elliptic curves E 1 (λ and 3 E (λ as defined in Section We note that part ( is given in [3], as a slight reformulation of [17, Thm 5] Theorem 44 ((1 Koie [1]; ( Ono [17] Let p be an odd prime, and let A 1 (p, λ and 3A (p, λ be as given in (3 and (4 respectively Then the following are true (1 F 1 (λ = φ(1 A 1 (p, λ if λ 0, 1 ( p ( 3 F = φ(λ ( 3A (p, λ p if λ 0, 1 λ p As a consequence, with certain restrictions on λ, the values F 1 (λ and 3 F (λ are explicitly related to each other Corollary 45 If p is an odd prime, then the following hold ( 4λ (1 p F 1 (λ = p 3F + p if λ 0, 1, 1 (λ 1 ( p F 1 (1 = p 3F (1 + (1 + φ(1p Proof By Theorem 44 (1, if λ 0, 1, then p F 1 (λ = A 1 (p, λ If λ 1, then by the change of coordinates x = βx β and y = β y 3, E 1 (λ is isomorphic to the β-quadratic twist of 3 E (t, where β = λ+1 and t = (λ1 (See also Lemma 71 It follows that (λ+1 E 1 (λ(f p = p + 1 φ(β 3 A (p, t, hence A 1 (p, λ = ( 3 A (p, t when λ 0, 1, 1 Now applying Theorem 44 ( gives p F 1 (λ = p 4λ 3F (λ1 + p, if λ 0, 1, 1, since φ(t = 1 and = 4λ t (λ1 In the case λ = 1, if p 1 (mod 4 we write p = a + b with a odd By Theorem 43 (1, we have { p F 1 (1 0 if p 3 (mod 4, = 4a if p 1 (mod 4 Define g(λ := p 3F ( 4λ (λ1 + p By Theorem 43 (, we have g(1 = { p if p 3 (mod 4, 4a p if p 1 (mod 4, and thus p F 1 (1 = g(1 + φ(1p = p 3F (1 + (1 + φ(1p 5 Proofs of Theorem 11 and Corollary 1 Here we give the proofs of Theorem 11 and Corollary 1, the latter of which establishes the truth of Koie s conecture We require the formula for U (F p given in (34, and the analogous facts about V (F p, and W (F p as given below Their derivations are similar

12 1 SHARON FRECHETTE, KEN ONO, AND MATTHEW PAPANIKOLAS to that of (34 (51 (5 V (F p = p A 1 (p, λ, W (F p = p A 1 (p, λ = p (1 + φ(λ A 1 (p, λ Proof of Theorem 11 We first note that the case = is trivial, since (Γ 0 (8, p = 0 and H (p = ε (p Now fix 4 We require the following expression for N (p in terms of the functions H (p, as defined in (36 and (16 respectively Proposition 51 Let p be an odd prime, and let 4 be even Then 1 ( (53 N (p + 1 = 1 p H (p =0 Proof By combining (34 (with λ 1, (51, and (5, and then applying Theorem 44, λ1 we see that 1 N (p + 1 = 3A (p, + (1 φ(λ A 1 (p, λ λ 1 = ( p φ(1 λ 3 F (λ + p 1 + (1 φ(λ ( p F 1 (λ 1 Using Corollary 45 on the second sum, we express N (p + 1 completely in terms of 3 F - functions Then since (1 φ(1 ( p 3F (1 + (1 + φ(1p 1 = 0, we have ( N (p + 1 = p φ(1 λ 3 F (λ + p 1 4µ p + (1 φ(µ µ= ( ( 4µ p 1 3F + p (µ 1 To simplify, note that = λ if and only if µ = λ± 1λ Thus in the second sum, a (µ1 λ term containing 3 F (λ appears with multiplicity 1 + φ(1 λ Therefore, ( N (p + 1 = p φ(1 λ 3 F (λ + p 1 ( + (1 φ(λ(1 + φ(1 λ p 3F (λ + p 1 Expanding the ( 1 -th powers using the binomial formula, we then see that (53 holds by applying the following lemma

13 HYPERGEOMETRIC FUNCTIONS AND HECKE OPERATORS 13 Lemma 5 If p is an odd prime, then for any integer n 0 the following are true (1 ( (1 φ(λ 3 F (λ n+1 = 0 φ(1 λ(1 φ(λ 3 F (λ n = 0 Proof We prove only part (1 (The proof of part ( is analogous We have (1 φ(λ 3 F (λ n+1 = ( 3F (λ n+1 φ(λ 3 F 1 n+1 λ, by splitting the sum and taing λ 1 λ on the second piece, we obtain in the second piece Then using Proposition 4 ( (1 φ(λ 3 F (λ n+1 = 3F (λ n+1 φ(λ n+ 3F (λ n+1 = 0 Next we invert the equation from Proposition 51, obtaining an expression for H (p in terms of N (p (hence in terms of the trace on spaces of newforms Recall the definition of δ (p in (8, and let γ (p := (p 1 (φ(1 1 Proposition 53 Let p be an odd prime, and let be even Then 1 ([ (54 H (p = γ (p 1 ] [ l 1 1 l ] p l ( l(γ 0 (8, p + δ l (p Proof Defining N (p := φ(1 1 means that (53 holds for all We mae use of another inverse pair [19, Table 1] given by (55 a n = n =0 Dividing through by p, (53 becomes ( n b n, and b n = n ( n (1 a n =0 N (p + 1 p 1 = =0 ( 1 H (p p Setting n = 1, and using a n = N n+(p+1 p n+1 1 H (p = =0 (1 ( 1 and b n = H n+(p p n+1 p (N (p + 1 in (55, we obtain

14 14 SHARON FRECHETTE, KEN ONO, AND MATTHEW PAPANIKOLAS Therefore by Theorem 34 and our definition of N (p, we obtain 1 H (p = γ (p =0 Now setting l = i + gives (1 ( 1 1 (56 H (p = γ (p =0 1 l= { p (1 ( 1 1 i=0 (( i ( i 1 p i ( (i+(γ 0 (8, p + δ (+i (p }, (( l ( l 1 p l ( l(γ 0 (8, p + δ l (p We may adust the sum on l to range over 0 l 1, since the binomial coefficients dependent on l will all be zero if l < We then obtain (54 by applying the fact that [ ] n n ( ( n n (57 = (1 m n m =0 The proof of Theorem 11 is then complete by applying the following lemma Lemma 54 Let p be an odd prime, and a positive even integer Then 1 ([ ε (p = γ (p + 1 ] [ l 1 1 l Proof If p 3 (mod 4, the proof reduces to showing that 1 ([ 1 ] [ l 1 1 l This follows from the easily proven fact that for any n 0, n ([ ] [ ] n n (1 n l n l + 1 l = If p 1 (mod 4, then we must show that ] (1 l = 1 n ] p l δ l (p (1 l [ n n l ] (58 1 ([ 1 l 1 ] [ 1 l ] p l (G l (a, p + G l (b, p = (4a p 1 + (4b p 1

15 HYPERGEOMETRIC FUNCTIONS AND HECKE OPERATORS 15 Using the definition of G l (s, p, we see that the left-hand side of (58 equals 1 l1 i=0 ( i l (1 i p i+l i ([ 1 ] [ l 1 1 ] ((a (l+i l + (b (l+i = 0 if l >, this expression becomes Setting = l + i, and noting that ( l l 1 { 1 ( ([ l (1 p (1 l l =0 1 ] [ l 1 1 ] } l ( (a + (b Expanding the right-hand-side of (58 using the binomial theorem, and comparing it with the above expression, we see that the proof of (58 reduces to showing the following equality for every with 0 1 (59 ( 1 = 1 (1 l ( l l ([ 1 ] [ l 1 1 l We prove (59 using a third inverse relation In [19, Table ] (modulo notation, we have the pair (510 a = b = (( p + ql l + q l (1 l+ ( p + q l l ( p + ql l b l 1 l, a l, where p and q are integer parameters Using (57, we may write [ ] [ ] n n n ( (( n n l = (1 n n + 1 l l l ] ( n l l 1 Choosing p = n and q = 1, and noting that ( ( nl l = 0 = nl l1 whenever l >, we n n see that this agrees with the first equation in the pair (510, with a = [ n ] [ n+1 ] and b = (1 ( n Inverting the pair then gives ( n (1 = ( ([ n l (1 l+ l n n l ] [ n n l + 1 ] Setting n = 1 and simplifying then gives (59 We now establish the truth of Koie s Conecture, using the = 6 case of Theorem 11

16 16 SHARON FRECHETTE, KEN ONO, AND MATTHEW PAPANIKOLAS Proof of Corollary 1 Setting = 6 in Theorem 11 gives 6 (Γ 0 (8, p = H 6 (p ε 6 (p p 4 (Γ 0 (8, p = p 4 φ(λ 3 F (λ + p 4 4F 3 (1 + (1 φ(1p, λ=1 where in the second equality, we apply (15 and (11 The proof therefore reduces to establishing the following formula: (511 p 5 6F 5 (1 = p 4 φ(λ 3 F (λ λ=1 Applying Proposition 41 ( twice to 6 F 5 (1 gives p 5 6F 5 (1 = p 3 φ(xφ(1 xφ(λφ(1 λ 4 F 3 (xλ λ=1 x=1 λ=1 Applying the change of variables λ λ, followed by x xλ then yields x [ ] p 5 6F 5 (1 = p 3 φ(1 φ(λ 4 F 3 (λ p φ(xφ(1 xφ(1 xλ p = p 4 φ(λ 4 F 3 (λ F 1 (λ, λ=1 where the second equality follows by Proposition 41 (1 Now applying Proposition 41 ( to 4 F 3 (λ, we see that p 5 6F 5 (1 = φ(1p 3 λ 1 =1 λ =1 x=1 φ(λ 1 λ φ(1 λ 3 F (λ 1 λ F 1 (λ 1 Maing the change of variables λ 1 λ λ and using the inversion for F 1 ( 1 λ 1 given in Proposition 4 (1, we obtain p 5 6F 5 (1 = φ(1p 3 Now putting λ 1 1 λ 1, we see that p 5 6F 5 (1 = φ(1p 3 λ=1 λ=1 3F (λ 3F (λ λ 1 =1 [ λ 1 =1 φ(λφ(λ 1 λ F 1 ( 1 λ 1 φ(λλ 1 φ(1 λλ 1 F 1 (λ 1 ( By Proposition 41, the inner sum equals φ(1p 3 F 1 ( λ Finally, using the inversion for 1 3F λ 1 given in Proposition 4, we obtain (511, thus completing the proof ]

17 HYPERGEOMETRIC FUNCTIONS AND HECKE OPERATORS 17 6 Congruences for (Γ 0 (8, p modulo p Here we prove Theorems 13 and 14 using Theorem 11, as well as nown facts concerning the values of Gaussian hypergeometric functions modulo p We state some facts that we require (for example, see [1] or [17, Sect 5] Proposition 61 Suppose that p is an odd prime (1 If 1 λ p 1, then p 3F (λ A ( 1,, λ; p 1 (mod p ( We have ( p 5 6F 5 (1 A, 4, 1; p 1 (mod p Proof of Theorem 13 By Theorem 11 and the definition of H (p and ε (p, if (mod p 1, then Theorem 13 (1 follows from (114 Similarly, if 3 (mod p 1, then (Γ 0 (8, p H (p ε (p (mod p H 4 (p ε 4 (p (mod p = 4 (Γ 0 (8, p (Γ 0 (8, p H (p ε (p H 6 (p ε 6 (p (mod p 6 (Γ 0 (8, p (mod p By Corollary 1, it then follows that (Γ 0 (8, p p 5 6F 5 (1 (mod p Theorem 13 ( now follows immediately from Proposition 61 ( Proof of Theorem 14 By Theorem 11, it follows that (Γ 0 (8, p H (p ε (p (mod p In view of Proposition 61 (1, it suffices to show that (61 ε (p (1 + (1 1 Using (15, it follows that (p 3F (1 1 (mod p ε (p 1φ(1 ( 1 + (1 1( p 3F (1 1 (mod p If p 1 (mod 4, this proves (61 If p 3 (mod 4, then p 3F (1 = 0 by Theorem 43 ( Therefore, (61 is also true in these cases This completes the proof

18 18 SHARON FRECHETTE, KEN ONO, AND MATTHEW PAPANIKOLAS 7 The family 3 E (λ and the level trace formula We devote this section to the proof of Theorem 3 The proof follows similar lines to the ones in [], [4], of the formulas in Proposition 1 However, there are several differences which require explanation Before woring out the proof, we go over some facts and lemmas about the family of elliptic curves 3 E (λ : y = (x 1(x + λ Let K be any field of characteristic, and consider the family 3 E (λ to be defined over K Its -invariant is 64(3λ 13 ( 3 E (λ = λ(λ + 1 Thus if 0 or 178, then there are precisely three values of λ K so that ( 3 E (λ = Moreover, only ( 3 E ( 1 3 = 0, and only ( 3E ( 1 9 = ( 3E ( = 178 (so if char(k = 3, ( 3 E (λ is never 0 (= 178 for λ K If E/K is an elliptic curve, a K-quadratic twist of E is a quadratic twist of E by some D K Lemma 71 Let E/K be an elliptic curve with a K-rational point of order in Weierstrass form E : y = x 3 + βx + γx, with β, γ K and β 0 Then there is a λ K so that E is isomorphic over K to a K-quadratic twist of 3 E (λ Proof The change of coordinates x = βx β, y = β 3 y, gives the curve 3 E (λ, where λ = γβ Thus E is isomorphic over K to the β-quadratic twist of β 3 E (λ Remar Lemma 71 covers all isomorphism classes of elliptic curves E/K, except when (E = 178 Lemma 7 Suppose that char(k 3 Let λ 1 K \ {0, 1, 1 9 } and λ K \ {0, 1} If 3E (λ 1 = 3 E (λ over K, then λ K( λ 1 Proof Any isomorphism 3 E (λ = 3 E (λ 1 over K is given by a change of Weierstrass coordinates, x 1 = u x + r, y 1 = u 3 y Let δ ± = 1±3 λ 1 1+9λ 1 Then brute force computation yields the following possibilities: λ = λ 1, u = ±1, r = 0; λ = 5 + 3λ 1 8δ ± + 4λ 1 δ ±, u = ± δ 3(1 + 9λ 1 ±, r = δ± In the second line, there are four possibilities: two choices of δ ± and two possible signs on u In any case, the lemma follows immediately We also appeal to the following theorem of Schoof, specialized to our purposes For a prime p, let I p denote the set of all isomorphism classes of elliptic curves over F p, and define (71 (7 I(s, p := {C I p E C, E(F p = p + 1 ± s}, I (s, p := {C I(s, p E C, E(F p [] = Z/Z Z/Z} If E/F p is an elliptic curve with E(F p = p+1±s, then we write [E] for its class in I(s, p Also, if C I p, we let C tw I p be the class of quadratic twists of curves in C by non-squares

19 HYPERGEOMETRIC FUNCTIONS AND HECKE OPERATORS 19 in F p If the -invariant of curves in a class C is not 178, then C C tw Also, for d the discriminant of an order O in an imaginary quadratic field, define the sum of class numbers (73 H(d := h(o O O O max Theorem 73 (Schoof [1, (45 (49] If p is an odd prime, and s is an even integer with 0 s < p, then { H(s 4p if s 0, I(s, p = H(4p if s = 0 If additionally s p + 1 (mod 4, then { H ( s 4p if s 0, I (s, p = 4 h(p if s = 0 We will need the following lemma on relations between class numbers Lemma 74 ([8, Cor 78] 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 l prime Finally, in the proof of Theorem 3 we will mae frequent use of the following easily proven lemma Lemma 75 Let D be a fundamental discriminant of an imaginary quadratic field If p is an odd prime and s 4p = t D, then the following hold (1 If s p + 1 (mod 4, then t is even ( If s p 1 (mod 4, then t is odd and D 0 (mod 4 Proof of Theorem 3 Fix s even with 0 s < p, and write s 4p = t D as in the statement of Theorem Define (74 L(s, p := {λ 1 λ p, 3 A (p, λ = ±s} We handle the case p = 3 first The only values of s to consider are 0 and, and one simply checs that L(0, 3 = 0 and L(, 3 = 1 It is then a routine matter to chec that the p = 3 case follows directly from Theorem For the remainder of the proof, we will assume that p 5 The elements of I(s, p can be paired up by quadratic twists so that We define (75 (76 I(s, p = {C 1,,C h, C tw 1,,C tw h } Ĩ(s, p := {C 1 C tw 1,,C h C tw h }, Ĩ (s, p := {C C tw C I (s, p}

20 0 SHARON FRECHETTE, KEN ONO, AND MATTHEW PAPANIKOLAS Ultimately we want to relate L(s, p, Ĩ(s, p, and Ĩ(s, p To do this we define (77 F : L(s, p Ĩ(s, p, λ [ 3 E (λ] [ 3 E (λ] tw and follow with some analysis of its properties The case to consider now is when s 0 Under this assumption we will show that (78 L(s, p = ( { s h 4p 1 if 1 L(s, p, c(s, f 9 f 0 otherwise f t When s 0, we have 1 L(s, p if and only if p 1 (mod 4 with s = a or s = b We 9 can therefore use (78 to match up terms in Theorem with those in the statement of Theorem 3: ( s h 4p c(s, f f 0<s< p s even G (s, p f t p = G ( 3 A (p, λ, p + λ=1 { 1 G (a, p + 1 G (b, p, if p 1 (mod 4, 0 if p 3 (mod 4 The main argument behind (78 is contained in the case where s 0 and also 1 3, 1 9 / L(s, p We will wor out this case first and then consider the remaining details For s even with 0 < s < p, we will show that (79 L(s, p = Ĩ(s, p + Ĩ(s, p Now by Lemma 71, the function F is surective Since 1, 3 1 / L(s, p, it follows from 9 Lemma 7 that F is 3-to-1 at λ L(s, p if and only if λ F p, which holds if and only if 3 E (λ(f p [] = Z/Z Z/Z For all other values of λ, F is 1-to-1 Thus (79 holds Moreover, if s p 1 (mod 4, then Ĩ(s, p = 0 and also, by Lemma 75, t is odd and D 0 (mod 4 Thus (710 L(s, p = Ĩ(s, p = H(s 4p = ( s h 4p, f f t where the second equality follows from Theorem 73 and the third from the assumption that 1, 3 1 / L(s, p (so h( ( s 4p 9 f = h s 4p f for all f in consideration The result then agrees with (78 On the other hand, if s p + 1 (mod 4, then t is even by Lemma 75, and so by Theorem 73, ( s (711 L(s, p = Ĩ(s, p + Ĩ(s, p = H(s 4p 4p + H 4 = ( s h 4p + f f t f t ( s h 4p, 4f

21 HYPERGEOMETRIC FUNCTIONS AND HECKE OPERATORS 1 where the last equality follows because 1, 3 1 / L(s, p From Lemma 74 it follows that 9 (71 L(s, p = ( s h 4p + ( s h 4p 1 + f f 1 ( D 1 ( s h 4p f f t f t f t 4 Now applying Lemma 74 again, we find ( ( D (713 L(s, p = 1 + f t f t ( s h 4p + f f t f t 4 ( s h 4p, f which verifies (78 The next case to consider is 1 L(s, p, where still s 0 Then p 1 (mod 3 and 3 D = 3, from which it follows that t is even Again by Lemma 71, F is surective Also 1 3 F p, and so [ 3 E ( 1] [ 3 3E ( 1 3 ]tw Ĩ(s, p Since F is only 1-to-1 at λ = 1 and not 3 3-to-1, we find (714 L(s, p = Ĩ(s, p + Ĩ(s, p The argument then follows the same lines as in (711 (713, except that when f = t, we have h ( s 4p f = h(3 = 1 = 3h (3 Thus from this fact and from (714 we modify (711 slightly: [ ( ] [ s L(s, p = + h 4p + 3 f + ( ] s h 4p + 4 4f 3 f t f t Then (78 follows precisely as in (71 and (713 Now suppose 1 L(s, p, where still s 0 Then p 1 (mod 4 and D = 4 There 9 are 4 isomorphism classes of curves over F p with -invariant 178 [, Prop X54], each quartic twists of each other As elsewhere, write p = a + b, with a, b > 0 and a odd, and then it follows that s = a or s = b We have (715 [ 3 E ( 1] [ 9 3E ( 1 9 ]tw Ĩ(a, p so if s = a the map F : L(a, p Ĩ(a, p is surective but fails to be 3-to-1 at 1 9 Therefore, L(a, p = Ĩ(a, p + Ĩ(a, p Since b p+1 (mod 4, we have Ĩ(b, p = 0 Furthermore by (715, F misses the class pair of -invariant 178 in Ĩ(b, p, and so f t L(b, p = Ĩ(b, p 1 Since h(4 = 1 = h (4, we find the present versions of (710 and (711 to be [ ( ( ] s L(a, p = + h 4p s h 4p + 1, f 4f ] [ f t [ ( ] s L(b, p = 1 + h 4p + 1 f f t The rest of (78 follows exactly as in (71 and (713, which concludes the case s 0

22 SHARON FRECHETTE, KEN ONO, AND MATTHEW PAPANIKOLAS Finally we suppose s = 0 We observe that G (0, p = (p 1, so to conclude the proof of the theorem, we need to verify that { 0 if p 1 (mod 4, (716 L(0, p = ξ(p 1 if p 3 (mod 4 The main reason for the discrepancy modulo 4 is that 1 L(0, p if and only if p 3 9 (mod 4 As before we consider the various cases If p 1 (mod 4, then 1 / L(0, p, so by Lemma 71, F is surective Since 4 (p + 1, 9 it follows from Lemma 7 that F is 1-to-1, L(0, p = Ĩ(0, p = 1 H(4p = 1 h (4p = ξ(p, where the second equality follows from Theorem 73 If p 3 (mod 4, we note that 1 L(0, p Also 1 L(0, p if and only if p 9 3 (mod 3, in which case 1 / F 3 p, and so regardless [ 3 E ( 1] / I 3 (0, p Thus the function F is surective (Lemma 71 but fails to be 3-to-1 at 1, so as in previous cases, 9 L(0, p = Ĩ(0, p + Ĩ(0, p However, now there is a slight difference from the other cases We note that in fact [ 3 E ( 1] = [ 9 3E ( 1 9 ]tw since p 3 (mod 4 [, Prop X54] Otherwise, for C I(0, p with C [ 3 E ( 1], we have C 9 Ctw For this reason, Ĩ(0, p = 1 I (0, p + 1 Then combining these equations with Theorem 73 and Lemma 74, L(0, p = Ĩ(0, p + Ĩ(0, p = 1 H(4p + h(p 1 = 1 h (4p + 1 h (p + h (p 1 { 3 = 3 if p 3 (mod 8 h (p + 1 if p 7 (mod 8 which agrees with (716 References } h (p, [1] S Ahlgren, Gaussian hypergeometric series and combinatorial congruences, Symbolic computation, number theory, special functions, physics and combinatorics (Gainesville, FL, 1999, Kluwer Acad Publ, Dordrecht, 001, pp 1 1 [] S Ahlgren, The points of a certain fivefold over finite fields and the twelfth power of the eta function, Finite Fields Appl 8 (00, [3] S Ahlgren and K Ono, A Gaussian hypergeometric series evaluation and Apéry number congruences, J Reine Angew Math 518 (000, [4] S Ahlgren and K Ono, Modularity of a certain Calabi-Yau threefold, Monatsh Math 19 (000, [5] F Beuers, Some congruences for the Apéry numbers, J Number Theory 1 (1985, [6] F Beuers, Another congruence for the Apéry numbers, J Number Theory 5 (1987, [7] B J Birch, How the number of points of an elliptic curve over a fixed prime field varies, J London Math Soc 43 (1968, [8] D A Cox, Primes of the form x + ny, John Wiley & Sons Inc, New Yor, 1989 [9] J Greene, Hypergeometric functions over finite fields, Trans Amer Math Soc 301 (1987,

23 HYPERGEOMETRIC FUNCTIONS AND HECKE OPERATORS 3 [10] H Hiiata, A K Pizer, and T R Shemanse, The basis problem for modular forms on Γ 0 (N, Mem Amer Math Soc 8 (1989, vi+159 [11] Y Ihara, Hece Polynomials as congruence ζ functions in elliptic modular case, Ann of Math ( 85 (1967, [1] M Koie, Hypergeometric series over finite fields and Apéry numbers, Hiroshima Math J (199, [13] M Koie, private communication [14] D Merlini, D G Rogers, R Sprugnoli, and M C Verri, On some alternative characterizations of Riordan arrays, Canad J Math 49 (1997, [15] E Mortenson, A supercongruence conecture of Rodriguez-Villegas for a certain truncated hypergeometric function, J Number Theory 99 (003, [16] E Mortenson, Supercongruences between F 1 hypergeometric functions and their Gaussian analogs, Trans Amer Math Soc 355 (003, [17] K Ono, Values of Gaussian hypergeometric series, Trans Amer Math Soc 350 (1998, [18] K Ono, The web of modularity: arithmetic of the coefficients of modular forms and q-series, Amer Math Soc, Providence, RI, 004 [19] J Riordan, Combinatorial identities, John Wiley & Sons Inc, New Yor, 1968 [0] F Rodriguez-Villegas, Hypergeometric families of Calabi-Yau manifolds, Calabi-Yau varieties and mirror symmetry (Toronto, ON, 001, Amer Math Soc, Providence, RI, 003, pp 3 31 [1] R Schoof, Nonsingular plane cubic curves over finite fields, J Combin Theory Ser A 46 (1987, [] J H Silverman, The arithmetic of elliptic curves, Springer-Verlag, New Yor, 1986 Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, MA address: sfrechet@mathcsholycrossedu Department of Mathematics, University of Wisconsin, Madison, WI address: ono@mathwiscedu Department of Mathematics, Texas A&M University, College Station, TX address: map@mathtamuedu