Modular Forms and Weierstrass Mock Modular Forms
AbstractAlfes, 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
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Clemm, A. Modular Forms and Weierstrass Mock Modular Forms. Mathematics 2016, 4, 5.
Clemm A. Modular Forms and Weierstrass Mock Modular Forms. Mathematics. 2016; 4(1):5.Chicago/Turabian Style
Clemm, Amanda. 2016. "Modular Forms and Weierstrass Mock Modular Forms." Mathematics 4, no. 1: 5.
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