On the Mahler measure of hyperelliptic families. (English. French summary) Zbl 1434.11211

Bertin, Marie José; Zudilin, Wadim

Ann. Math. Qué. 41, No. 1, 199-211 (2017).

Summary: We prove Boyd’s “unexpected coincidence” of the Mahler measures for two families of two-variate polynomials defining curves of genus 2. We further equate the same measures to the Mahler measures of polynomials \(y^3-y+x^3-x+kxy\) whose zero loci define elliptic curves for \(k\neq 0,\pm 3\).

MSC:

11R06	PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11G30	Curves of arbitrary genus or genus \(\ne 1\) over global fields
14H52	Elliptic curves

Keywords:

Mahler measure; \(L\)-value; elliptic curve; hyperelliptic curve; elliptic integral

Software:

LMFDB

References:

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