A formula and a congruence for Ramanujan’s 𝜏-function

Abstract

We determine formulas for Ramanujan's τ-function and for the coefficients of modular forms on Γ 0 (2) in terms of finite field 3 F 2-hypergeometric functions. Using these formulas we obtain a new congruence of τ (p) (mod 11).

Hypergeometric functions over ${\mathbb {F}_p}$ and relations to elliptic curves and modular forms

“…Recently, Ahlgren, Frechette, Ono, and Papanikolas [1,2,3,6,14] have proved results linking these hypergeometric functions to modular forms and elliptic curves. The main results given here continue this line of inquiry.…”

“…These were given in terms of counting points on related varieties over finite fields. Papanikolas [14] used these to obtain a formula for Ramanujan's τ -function as well as a congruence for τ (p) modulo 11.…”

A finite field hypergeometric function associated to eigenvalues of a Siegel eigenform

“…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.…”

Distribution of values of Gaussian hypergeometric functions

“…These properties include transformation laws, explicit evaluations, and contiguous relations. These functions have played central roles in the study of combinatorial supercongruences [1,3,36,43,46,47,51,54,55,56,57,58], Dwork hypersurfaces [9,45], Galois representations [40,41], L-functions of elliptic curves [6,10,11,25,39,44,52,60,63], hyperelliptic curves [7,8], K3 surfaces [4,19,52], Calabi-Yau threefolds [2,3,64], the Eichler-Selberg trace formula [24,25,26,27,38,48,58,59], among other topics.…”