Mahler measures at interior CM points: proofs of two conjectures of Samart

Abstract

We prove two conjectures of Samart as identities of genuine Mahler measures. The first is the case $k=1$ of his evaluations for the family $(x+x^{-1})(y+y^{-1})(z+z^{-1})+k^{1/2}$: $m((x+x^{-1})(y+y^{-1})(z+z^{-1})+1)=4L'(g_7,0)$, where $g_7(\tau)=\eta(\tau)^3\eta(7\tau)^3$ is the unique newform of $S_3(\Gamma_0(7),\chi_{-7})$ (LMFDB label 7.3.b.a). The second is the conjugate pair of quadratic CM entries of his 2015 table: $n_2((47\pm 45\sqrt{-7})/2)=(4/7)(54L'(g_7,0)+L'(\chi_{-7},-1))$, where $n_2(s):=2m((x+x^{-1})(y+y^{-1})(z+z^{-1})+\sqrt{s})$. In both cases the parameter lies inside (or on the boundary of) the critical locus, so the passage from the holomorphic (modified) Mahler measure---where Samart computed the $L$-value side conditionally, and Fei the real-part level---to the true Mahler measure was open. For the first theorem we close the gap by a differential-comparison continuation of the Mahler differential along an explicitly certified path, followed by a continuity argument at the interior point. For the second theorem no continuation is needed: the second preimage of the parameter under the modular parametrization is the Fricke partner of Samart's CM point and lies in his proved region, and the evaluation is an exact CM lattice-sum computation. The analytic and arithmetic inputs not proved here are stated explicitly with references; the finite numerical inequalities used in the continuation argument are certified by interval arithmetic, and all identities are confirmed numerically to 41--60 digits.