Mahler measures and $L$-values of elliptic curves over real quadratic fields

TL;DR

This paper establishes new identities linking Mahler measures of specific polynomials to L-values of elliptic curves over real quadratic fields, extending previous work and guided by Beilinson's conjecture.

Contribution

It proves bounds on the degree of parameters in Mahler measure formulas and derives 28 new identities connecting Mahler measures to L-values, also expressing certain L-values via determinants of Mahler measures.

Findings

Bound on the degree of k in Villegas' formula using class numbers.

28 new identities linking Mahler measures to L-values of cusp forms.

5 formulas expressing L-values of CM elliptic curves as determinants of Mahler measures.

Abstract

A famous formula of Rodriguez Villegas shows that the Mahler measures $m(k)$ of $P_k(x,y)=x+1/x+y+1/y+k$ can be written as a Kronecker-Eisenstein series. We prove that the degree of $k$ in Villegas' formula can be bounded by the class numbers of CM points. This fact allows us to systematically derive $28$ new identities linking $m(k)$ to $L$-values of cusp forms. Guided by Beilinson's conjecture, we also prove $5$ formulas that express $L$-values of CM elliptic curves over real quadratic fields to some $2\times 2$ determinants of $m(k)$. This extends a recent work of Guo (the second author of this paper), Ji, Liu, and Qin, in which they dealt with the cases when $k=4\pm 4\sqrt{2}$.