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Modular Forms and Weierstrass Mock Modular Forms

Author

Listed:
  • Amanda Clemm

    (Department of Mathematics, Emory University, Emory, Atlanta, GA 30322, USA)

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.

Suggested Citation

  • Amanda Clemm, 2016. "Modular Forms and Weierstrass Mock Modular Forms," Mathematics, MDPI, vol. 4(1), pages 1-8, February.
    • Handle: RePEc:gam:jmathe:v:4:y:2016:i:1:p:5-:d:63303
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