Shifted Convolution $L$-Series Values for Elliptic Curves

TL;DR

This paper presents explicit formulas for shifted convolution $L$-values of elliptic curves using mock modular forms, revealing unexpected connections between newforms and $L$-values, enabling precise computations.

Contribution

It introduces a closed-form expression for shifted convolution $L$-values of elliptic curves via mock modular forms, a novel explicit construction.

Findings

Derived explicit formulas for shifted convolution $L$-values

Established a surprising relation between weight 2 newforms and $L$-values

Enabled arbitrary precision computations of these $L$-values

Abstract

Using explicit constructions of the Weierstrass mock modular form, we offer a closed formula for generating the values of shifted convolution $L$-values for certain elliptic curves that can be computed to arbitrary precision. These identities provide a surprising relation between weight $2$ newforms and shifted convolution $L$-values.