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

Abstract

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 4 F 3 -hypergeometric function.

The Mahler measure of a Calabi–Yau threefold and special $$L$$ L -values

“…For other values of k, work of McCarthy and the first author [19] shows that 4 F 3 (−1) can be expressed in terms of eigenvalues of a Siegel eigenform of degree 2, whose L-function is a tensor product L-function L(f 2 ⊗ f 3 , s), for classical newforms f 2 ∈ S 2 (Γ 0 (32)) and f 3 ∈ S 3 (Γ 0 (32), χ −4 ) (see [14]). Comparing with (4.1) this led us to investigate relations between m R 16 √ −1 ←→ L(f 2 ⊗ f 3 , 4), but after several attempts by the authors to search for such a relationship, the question of finding one remains open.…”

Generalized Legendre curves and quaternionic multiplication

“…[N ;i,j,k] λ over finite fields. There are multiple ways in which one can use Gaussian hypergeometric functions to count points on varieties, and these functions are also related to coefficients of various modular forms including Siegel modular forms [1,8,13,14,19,20,24,25,34]. For our purposes, we use a technique similar to one shown in [34].…”

Hypergeometric type identities in the $p$-adic setting and modular forms

“…The main method of relating finite field hypergeometric functions and Fourier coefficients of modular forms has been via the Eichler-Selberg trace formula [1,7,8,9,10,17,19,25]. But, apart from a number of special cases, including Ahlgren and Ono's relation above, most of these result are either restricted to primes in certain congruence classes to facilitate the existence of characters of certain orders, which appear as arguments in the finite field hypergeometric functions, or require much more complex relations than (4.1).…”

“…These functions are analogues of classical hypergeometric functions and were first developed to simplify character sum evaluations. Since then they have been applied to a number of areas of mathematics but the two areas of most interest to the authors are their transformation properties [11,23], which often mirror classical hypergeometric transformations, and their connections to modular forms [1,6,7,8,9,10,17,19,20,22,25].…”