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

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.

arXiv comment15 pages, added new examples in levels 16, 18 and 25

References in corpus (1)

• Value at 2 of the L-function of an elliptic curve

Cited by in corpus (1)

• Mahler measure of polynomials defining genus 2 and 3 curves