L-values of certain weight 3 Modular Forms and Transformations of Hypergeometric Series

2024/12/09 by Rosen, Esme · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research

paper · pdf · doi:10.48550/arxiv.2412.07054

Abstract

Recently, Allen, Grove, Long, and Tu proposed an explicit Hypergeometric-Modularity method which gives a concrete link between certain hypergeometric objects and modular forms. The theory is exemplified by a collection of 199 weight 3 modular forms. Among other properties their process shows that the L-value of such a modular form at 1 is an explicit multiple of a ₃F₂(1) hypergeometric series. Using the framework of a finite Coxeter group governing the invariance group of normalized ₃F₂(1) series, this paper fully classifies and describes the possible Hecke eigenforms whose L-values that can be obtained using this method. In addition, we determine when these modular forms differ by twist of a finite-order character using the perspective of hypergeometric functions. As one application, we reinterpret a classical identity of hypergeometric series as a formula involving L-values of two Hecke eigenforms that differ by a twist.

Cited by

• The Explicit Hypergeometric Modularity Method III

Related

• Modular Forms and Certain ₂F₁(1) Hypergeometric Series
• The Explicit Hypergeometric-Modularity Method I
• Hypergeometric series, modular linear differential equations, and vector-valued modular forms
• The Explicit Hypergeometric-Modularity Method II
• The Explicit Hypergeometric Modularity Method III