Rolen, Larry ORCID: 0000-0001-8671-8117 (2016). A NEW CONSTRUCTION OF EISENSTEIN'S COMPLETION OF THE WEIERSTRASS ZETA FUNCTION. Proc. Amer. Math. Soc., 144 (4). S. 1453 - 1457. PROVIDENCE: AMER MATHEMATICAL SOC. ISSN 1088-6826

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Abstract

In the theory of elliptic functions and elliptic curves, the Weier-strass. 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.

Item Type: Journal Article
Creators:
CreatorsEmailORCIDORCID Put Code
Rolen, LarryUNSPECIFIEDorcid.org/0000-0001-8671-8117UNSPECIFIED
URN: urn:nbn:de:hbz:38-280922
DOI: 10.1090/proc/12813
Journal or Publication Title: Proc. Amer. Math. Soc.
Volume: 144
Number: 4
Page Range: S. 1453 - 1457
Date: 2016
Publisher: AMER MATHEMATICAL SOC
Place of Publication: PROVIDENCE
ISSN: 1088-6826
Language: English
Faculty: Unspecified
Divisions: Unspecified
Subjects: no entry
Uncontrolled Keywords:
KeywordsLanguage
Mathematics, Applied; MathematicsMultiple languages
Refereed: Yes
URI: http://kups.ub.uni-koeln.de/id/eprint/28092

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