Central $L$-values of elliptic curves and local polynomials

Stephan Ehlen; Pavel Guerzhoy; Ben Kane; Larry Rolen

doi:10.1112/plms.12305

Central $L$-values of elliptic curves and local polynomials

Number Theory 2019-11-13 v5

Authors: Stephan Ehlen , Pavel Guerzhoy , Ben Kane , Larry Rolen

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Abstract

Here we study the recently introduced notion of a locally harmonic Maass form and its applications to the theory of $L$ -functions. In particular, we find finite formulas for certain twisted central $L$ -values of a family of elliptic curves in terms of finite sums over canonical binary quadratic forms. This yields vastly simpler formulas related to work of Birch and Swinnerton-Dyer for such $L$ -values, and extends beyond their framework to special non-CM elliptic curves.

Keywords

elliptic curves central l-values locally harmonic maass forms binary quadratic forms l-functions birch and swinnerton-dyer

Cite

@article{arxiv.1803.10555,
  title  = {Central $L$-values of elliptic curves and local polynomials},
  author = {Stephan Ehlen and Pavel Guerzhoy and Ben Kane and Larry Rolen},
  journal= {arXiv preprint arXiv:1803.10555},
  year   = {2019}
}

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29 pages, 4 tables and 1 diagram

Central $L$-values of elliptic curves and local polynomials

Abstract

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