A new construction of Eisenstein's completion of the Weierstrass zeta function

TL;DR

This paper presents a new proof of Eisenstein's completion of the Weierstrass zeta function, enhancing understanding of elliptic functions and their applications in harmonic Maass forms and elliptic curve theory.

Contribution

It offers a novel proof using differential operators for Jacobi forms and classical lattice identities, enabling construction of more general non-holomorphic elliptic functions.

Findings

Provides a new proof of Eisenstein's theorem on the Weierstrass zeta function completion.

Demonstrates how to construct more general non-holomorphic elliptic functions.

Highlights applications in harmonic Maass forms and elliptic curve L-functions.

Abstract

In the theory of elliptic functions and elliptic curves, the Weierstrass $zeta$ function (which is essentially an antiderivative of the Weierstrass $\wp$ function) plays a prominent role. Although it is not an elliptic function, Eisenstein constructed a simple (non-holomorphic) completion of this form which is doubly periodic. This theorem has begun to play an important role in the theory of harmonic Maass forms, and was crucial to work of Guerzhoy as well as Alfes, Griffin, Ono, and the author. In particular, this simple completion of $\zeta$ provides a powerful method to construct harmonic Maass forms of weight zero which serve as canonical lifts under the differential operator $\xi_{0}$ of weight 2 cusp forms, and this has been shown in to have deep applications to determining vanishing criteria for central values and derivatives of twisted Hasse-Weil $L$-functions for elliptic curves. Here we offer a new and motivated proof of Eisenstein's theorem, relying on the basic theory of differential operators for Jacobi forms together with a classical identity for the first quasi-period of a lattice. A quick inspection of the proof shows that it also allows one to easily construct more general non-holomorphic elliptic functions.