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

Author: Larry Rolen
Journal: Proc. Amer. Math. Soc. 144 (2016), 1453-1456
MSC (2010): Primary 11F03, 11F37, 11F50, 33E05
Published electronically: July 8, 2015
MathSciNet review: 3451223
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Abstract: In the theory of elliptic functions and elliptic curves, the Weierstrass $ \zeta $ 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 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.

References [Enhancements On Off] (What's this?)

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Additional Information

Larry Rolen
Affiliation: Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D-50931 Köln, Germany

Received by editor(s): April 14, 2015
Received by editor(s) in revised form: April 17, 2015
Published electronically: July 8, 2015
Additional Notes: The author thanks the University of Cologne and the DFG for their generous support via the University of Cologne postdoc grant DFG Grant D-72133-G-403-151001011. The author is also grateful to Ken Ono for his encouragement to write this note, and to Kathrin Bringmann, Michael Mertens, and the anonymous referee for useful comments. The author would also like to thank Martin Raum for pointing out the connection of this work with functions considered in [6], as well as Nikolaos Diamantis for pointing out connections to higher order modular forms as in the comment preceding the proof of Theorem 0.1.
Communicated by: Ken Ono
Article copyright: © Copyright 2015 American Mathematical Society

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