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

TL;DR

This paper introduces finite formulas for twisted central L-values of elliptic curves using locally harmonic Maass forms, simplifying calculations and extending previous frameworks to non-CM elliptic curves.

Contribution

It develops finite sum formulas for L-values of elliptic curves via locally harmonic Maass forms, broadening applicability beyond prior methods.

Findings

Finite formulas for twisted central L-values derived

Simplification of L-value computations achieved

Extension to non-CM elliptic curves demonstrated

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.