Modular Forms and Certain ${}_2F_1(1)$ Hypergeometric Series

TL;DR

This paper constructs explicit weight 2 Hecke eigenforms with complex multiplication using hypergeometric motives, and computes their central L-values through hypergeometric series and special functions, revealing deep motivic connections.

Contribution

It introduces a new explicit family of modular forms linked to hypergeometric motives and provides formulas for their central L-values and Fourier coefficients.

Findings

Explicit family of weight 2 Hecke eigenforms with CM

Exact formulas for central L-values in terms of beta values

Fourier coefficients expressed via Jacobi sums

Abstract

Using the framework relating hypergeometric motives to modular forms, we define an explicit family of weight 2 Hecke eigenforms with complex multiplication. We use the theory of ${}_2F_1(1)$ hypergeometric series and Ramanujan's theory of alternative bases to compute the exact central $L$-value of these Hecke eigenforms in terms of special beta values. We also show the integral Fourier coefficients can be written in terms of Jacobi sums, reflecting a motivic relation between the hypergeometric series and the modular forms.