On the Mahler measure associated to $X_1(13)$

TL;DR

This paper establishes a precise relationship between the Mahler measure of a defining equation of the modular curve $X_1(13)$ and the derivative at zero of an associated L-function, using advanced number theory techniques.

Contribution

It provides a novel proof linking Mahler measures to L-function derivatives for the modular curve $X_1(13)$, combining Deninger's method, Beilinson's theorem, and period analysis.

Findings

Mahler measure equals the L-function derivative at zero for $X_1(13)$

Explicit connection between Mahler measures and modular forms

Examples for modular curves of levels 16, 18, and 25

Abstract

We show that the Mahler measure of a defining equation of the modular curve $X_1(13)$ is equal to the derivative at $s=0$ of the $L$-function of a cusp form of weight 2 and level 13 with integral Fourier coefficients. The proof combines Deninger's method, an explicit version of Beilinson's theorem together with an idea of Merel to express the regulator integral as a linear combination of periods. Finally, we present further examples related to the modular curves of level 16, 18 and 25.