Modular equations and lattice sums

TL;DR

This paper explores modular equations from Somos and Ramanujan to establish new links between lattice sums, hypergeometric functions, and elliptic curve L-values, advancing understanding in number theory.

Contribution

It introduces novel relations between lattice sums and hypergeometric functions using modular equations and proposes a new conjecture for elliptic curve L-values.

Findings

New relations between lattice sums and hypergeometric functions

Progress towards Boyd's Mahler measure conjectures

A conjectured formula for L(E,2) for conductor 17 elliptic curves

Abstract

We highlight modular equations discovered by Somos and Ramanujan, and use them to prove new relations between lattice sums and hypergeometric functions. We also discuss progress towards solving Boyd's Mahler measure conjectures, and we conjecture a new formula for $L(E,2)$ of conductor 17 elliptic curves.