Modular Forms and Weierstrass Mock Modular Forms

Amanda Clemm
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Amanda Clemm: Department of Mathematics, Emory University, Emory, Atlanta, GA 30322, USA

Mathematics, 2016, vol. 4, issue 1, 1-8

Abstract: Alfes, Griffin, Ono, and Rolen have shown that the harmonic Maass forms arising from Weierstrass ? -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 ? -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.

Keywords: modular forms; weierstrass mock modular forms; eta-quotients (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2016
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