Universität zu Köln

Alfes, Claudia, Griffin, Michael, Ono, Ken and Rolen, Larry ORCID: 0000-0001-8671-8117 (2015). Weierstrass mock modular forms and elliptic curves. Res. Number Theory, 1 (1). CHAM: SPRINGER INTERNATIONAL PUBLISHING AG. ISSN 2363-9555

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Abstract

Mock modular forms, which give the theoretical framework for Ramanujan's enigmatic mock theta functions, play many roles in mathematics. We study their role in the context of modular parameterizations of elliptic curves E/Q. We show that mock modular forms which arise from Weierstrass zeta-functions encode the central L-values and L-derivatives which occur in the Birch and Swinnerton-Dyer Conjecture. By defining a theta lift using a kernel recently studied by Hovel, we obtain canonical weight 1/2 harmonic Maass forms whose Fourier coefficients encode the vanishing of these values for the quadratic twists of E. We employ results of Bruinier and the third author, which builds on seminal work of Gross, Kohnen, Shimura, Waldspurger, and Zagier. We also obtain p-adic formulas for the corresponding weight 2 newform using the action of the Hecke algebra on the Weierstrass mock modular form.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Alfes, Claudia UNSPECIFIED UNSPECIFIED UNSPECIFIED Griffin, Michael UNSPECIFIED UNSPECIFIED UNSPECIFIED Ono, Ken UNSPECIFIED UNSPECIFIED UNSPECIFIED Rolen, Larry UNSPECIFIED orcid.org/0000-0001-8671-8117 UNSPECIFIED
URN:	urn:nbn:de:hbz:38-386426
DOI:	10.1007/s40993-015-0026-2
Journal or Publication Title:	Res. Number Theory
Volume:	1
Number:	1
Date:	2015
Publisher:	SPRINGER INTERNATIONAL PUBLISHING AG
Place of Publication:	CHAM
ISSN:	2363-9555
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language FOURIER COEFFICIENTS; HEEGNER POINTS; QUADRATIC TWISTS; CM VALUES; TRACES; SERIES; DERIVATIVES; OPERATORS; FORMULAS; JACOBI Multiple languages Mathematics Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/38642