Special $L$-values of certain CM weight three Hecke eigenforms

TL;DR

This paper explores the relationship between special values of L-functions of certain CM weight three Hecke eigenforms and Ramanujan's alternative bases, utilizing hypergeometric Galois representations and providing a classification of these representations.

Contribution

It establishes a connection between L-values of CM Hecke eigenforms and hypergeometric series, and classifies related hypergeometric Galois representations.

Findings

Relation between L-values and hypergeometric series established

Complete classification of hypergeometric Galois representations provided

Connections to modularity of hypergeometric Galois representations elucidated

Abstract

Ramanujan's theory of elliptic functions to alternative bases connects modular forms with hypergeometric series and has led to applications such as the modularity of certain hypergeometric Galois representations. In this paper, we relate special values of $L$-functions of certain CM Hecke eigenforms to Ramanujan's alternative bases via the modularity of hypergeometric Galois representations associated with hypergeometric series ${}_{3}F_{2}\!\left[ \genfrac{}{}{0pt}{}{\frac{1}{2} \ \frac{1}{d} \ \frac{d-1}{d}}{\ 1 \ \ \ \ 1} ;\ t \right]$, $d=2$, $3$, $4$, and $6$, arising from tensor products of CM elliptic curves over real quadratic fields. We also give a complete classification of these type of hypergeometric Galois representations.