Mahler measures, K-theory and values of L-functions

TL;DR

This paper explores the deep connections between Mahler measures of polynomials, K-theory, and special values of L-functions, reducing conjectures to Beilinson's conjectures using K-theoretical methods.

Contribution

It links Mahler measures to L-values and K-theory, providing a K-theoretic framework to approach conjectures relating these mathematical objects.

Findings

Reduced Boyd's conjectures to Beilinson's conjectures.

Established K-theoretical methods for analyzing Mahler measures.

Connected Mahler measures with special L-values for specific polynomials.

Abstract

The Mahler measure of a polynomial $P$ in $n$ variables is defined as the mean of $\log|P|$ over the $n$-dimensional torus. For certain polynomials with integer coefficients in two variables the Mahler measure is known to be related to special values of L-functions of arithmetic objects (e.g. Dirichlet characters and elliptic curves over $\mathbb{Q}$). Inspired by work of Deninger Boyd has investigated this relationship numerically. In this paper we reduce some conjectures of Boyd to Beilinson`s conjectures on special values of L-functions. The methods in use are widely of K-theoretical nature.