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Mathematics 2016, 4(1), 5; doi:10.3390/math4010005

Modular Forms and Weierstrass Mock Modular Forms

Department of Mathematics, Emory University, Emory, Atlanta, GA 30322, USA
Academic Editor: Alexander Berkovich
Received: 12 November 2015 / Revised: 12 January 2016 / Accepted: 19 January 2016 / Published: 2 February 2016
View Full-Text   |   Download PDF [260 KB, uploaded 2 February 2016]

Abstract

Alfes, Griffin, Ono, and Rolen have shown that the harmonic Maass forms arising from Weierstrass ζ-functions associated to modular elliptic curves “encode” the vanishing and nonvanishing for central values and derivatives of twisted Hasse-Weil L-functions for elliptic curves. Previously, Martin and Ono proved that there are exactly five weight 2 newforms with complex multiplication that are eta-quotients. In this paper, we construct a canonical harmonic Maass form for these five curves with complex multiplication. The holomorphic part of this harmonic Maass form arises from the Weierstrass ζ-function and is referred to as the Weierstrass mock modular form. We prove that the Weierstrass mock modular form for these five curves is itself an eta-quotient or a twist of one. Using this construction, we also obtain p-adic formulas for the corresponding weight 2 newform using Atkin’s U-operator. View Full-Text
Keywords: modular forms; weierstrass mock modular forms; eta-quotients modular forms; weierstrass mock modular forms; eta-quotients
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Clemm, A. Modular Forms and Weierstrass Mock Modular Forms. Mathematics 2016, 4, 5.

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