On a class of elliptic functions associated with the even Dirichlet characters

TL;DR

This paper introduces a new class of elliptic functions linked to even Dirichlet characters, exploring their representations, series expansions, and modular properties, expanding the understanding of elliptic functions in number theory.

Contribution

It constructs and analyzes a novel class of elliptic functions associated with even Dirichlet characters, including their series representations and modular transformations.

Findings

Derived representations in terms of q-series and partial fractions

Identified significance of coefficients in power series expansions

Established modular properties under arithmetic group actions

Abstract

We construct a class of companion elliptic functions associated with the even Dirichlet characters. Using the well-known properties of the classical Weierstrass elliptic function $\wp(z|\tau)$ as the blueprint, we will derive their representations in terms of $q$-series and partial fractions, explore the significance of the coefficients of their power series expansions and establish the modular properties under the actions of the arithmetic groups $\Gamma_0(N)$ and $\Gamma_1(N)$.