Untitled

Abstract

We establish a simple inductive formula for the trace Tr new k (Γ 0 (8), p) of the p-th Hecke operator on the space S new k (Γ 0 (8)) of newforms of level 8 and weight k in terms of the values of 3 F 2-hypergeometric functions over the finite field F p. Using this formula when k = 6, we prove a conjecture of Koike relating Tr new 6 (Γ 0 (8), p) to the values 6 F 5 (1) p and 4 F 3 (1) p. Furthermore, we find new congruences between Tr new k (Γ 0 (8), p) and generalized Apéry numbers.

A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds

“…The explicit expression for g(τ ) was kindly informed to us by J. Wan who also noticed its historical cast in [9] (see the last column of the table on p. 56 there). As we learned later, the conjecture (17) was reported in [8] and attributed to E. Mortenson; it is now shown to be true modulo p 3 in the joint work [19] F k (z)ε k + O ε 6 as ε → 0 can be related, at z = 1, to the critical L-values as follows:…”

“…The explicit expression for g(τ ) was kindly informed to us by J. Wan who also noticed its historical cast in [9] (see the last column of the table on p. 56 there). As we learned later, the conjecture (17) was reported in [8] and attributed to E. Mortenson; it is now shown to be true modulo p 3 in the joint work [19] with R. Osburn and A. Straub. Numerically, the Taylor ε-expansion As pointed out to us by F. Rodriguez Villegas and D. Roberts the related hypergeometric motive is also linked to the modular form f (τ ) from the introduction, defined in (2).…”

“…Remark 3. The case (r, t) = 1 3 , 1 4 in Observation 1 corresponds to a particularly simple CM modular form of level 9, namely, to f (τ ) = η(3τ ) 8 . Its critical L-values possess closed-form evaluation…”

“…Another interesting instance corresponds to choosing z = 1 in the hypergeometric series F 0 (z) = 6 F z k related to a Calabi-Yau fivefold -a complete intersection of six degree 2 surfaces in P 12 ; the associated Hodge structure for each fiber z of the family can be conjecturally computed with a help of the hypergeometric motives[22]. Consider the newformg(τ ) = ∞ n=1 b(n)q n = q + 20q 3 − 74q 5 − 24q 7 + 157q 9 + 124q 11 + • • • = η(2τ ) 12 + 32η(2τ ) 4 η(8τ ) 8of weight 6 on Γ 0(8). Its coefficients satisfy Weil's bound |b(p)| ≤ 2p 5/2 and numerics suggest that b(p) mod p 5…”

Evaluation of Certain Hypergeometric Functions over Finite Fields

“…Key ingredients in their approach are some transformation formulas of hypergeometric functions over finite fields, such as a finite field analogue of Clausen's theorem. In recent years, Frechette, Fuselier, Goodson, Lennon, Ono, Papanikolas, and Salerno [10,11,12,13,15,22,23,26] investigated connections between hypergeometric functions over finite fields and elliptic curves, algebraic varieties, and Hecke eigenforms. Very recently, McCarthy-Papanikolas [24] linked the hypergeometric functions to Siegel modular forms.…”