Mahler measure, motivic regulators and Dirichlet $L$-values
Wei He, Jungwon Lee
TL;DR
The paper links the Mahler measure of a two-variable cyclotomic polynomial to Beilinson regulators in motivic cohomology and to derivatives of Dirichlet L-functions via a carefully constructed regulator framework. It proves a regulator formula for $m(f_N^*)-m(f_N)$ as a linear combination $\sum_{\chi} r_{N,\chi} L^{(N_\chi)'}(-\epsilon,\chi)$ with explicit coefficients and exhibits a canonical $G$-module structure on the relevant motivic cohomology, $^{i=-1}$, compatible with regulator maps. Under a linear-independence conjecture for partial L-value derivatives, it obtains a refined identity for a single $L$-value by extracting the $\chi$-part of a built cohomology class and describes a canonical splitting of the relevant sequence, linking $\chi$-parts to $L^{(N_\chi)'}(-\epsilon,\chi)$. The work also discusses applications to Chinburg’s conjecture and to the entropy of dynamical systems, and it outlines future directions toward Beilinson conjecture refinements and Euler-system-like structures in abelian extensions.
Abstract
Inspired by the work of Deninger, we present a formula that relates the Mahler measure of a two-variable variant of cyclotomic polynomial to regulator of class in motivic cohomology associated to cyclotomic fields and linear combination of special values of the derivative of Dirichlet $L$-functions. The formula is derived by studying the Beilinson regulator map applied to systematically constructed elements in the motivic cohomology group. Under linear independence hypothesis on the derivative of partial Dirichlet $L$-values at $s=0$ and $-1$, we study a Galois module structure of the relevant motivic cohomology and obtain the refined identity for a single $L$-value.