Mahler measure of a non-reciprocal family of elliptic curves

Abstract

In this article, we study the logarithmic Mahler measure of the one-parameter family \[Q_α=y^2+(x^2-αx)y+x,\] denoted by $m(Q_{\alpha })$. The zero loci of $Q_{\alpha }$ generically define elliptic curves $E_{\alpha }$ which are $3$-isogenous to the family of Hessian elliptic curves. We are particularly interested in the case $\alpha \in (-1,3)$, which has not been considered in the literature due to certain subtleties. For $\alpha$ in this interval, we establish a hypergeometric formula for the (modified) Mahler measure of $Q_{\alpha }$, denoted by $\tilde{n}(\alpha ).$ This formula coincides, up to a constant factor, with the known formula for $m(Q_{\alpha })$ with $|\alpha |$ sufficiently large. In addition, we verify numerically that if ${\alpha }^3$ is an integer, then $\tilde{n}(\alpha )$ is a rational multiple of $L^{\prime }(E_{\alpha },0)$. A proof of this identity for $\alpha =2$, which is corresponding to an elliptic curve of conductor $19$, is given.

Corrigendum: sign error in Theorem 1 and miscalculation in the proof of Lemma 9 fixed

References in corpus (6)

• Regulator of modular units and Mahler measures
• Regulators of Siegel units and applications
• Mahler measures as linear combinations of $L$-values of multiple modular forms
• Further explorations of Boyd's conjectures and a conductor 21 elliptic curve
• Parametrizing elliptic curves by modular units
• Mahler measures of a family of non-tempered polynomials and Boyd's conjectures