Year: 2026
Type: article
Abstract: This article provides partial solutions to Chinburg’s conjectures by studying a sequence of multivariate polynomials. These conjectures assert that for every odd quadratic Dirichlet character of conductor f , $$\chi _{-f}=\left( \frac{-f}{.}\right) $$ χ - f = - f . , there exists a bivariate polynomial (or a rational function in the weak version) whose Mahler measure is a rational multiple of $$L'(\chi _{-f},-1)$$ L ′ ( χ - f , - 1 ) . We prove that the Mahler measure of a polynomial family, denoted by $$P_d$$ P d , can be expressed as a linear combination of the derivatives of Dirichlet L -functions. Specifically, this family provides solutions to the conjectures for conductors $$f=3,4,8,15,20$$ f = 3 , 4 , 8 , 15 , 20 , and 24. We further generalize Chinburg’s conjectures from real primitive odd Dirichlet characters to all primitive odd characters. For this generalized version, the polynomials $$P_d$$ P d provide solutions for conductors 5, 7, and 9.
Source: Annales mathématiques du Québec
Authors: Marie-José Bertin, Mahya Mehrabdollahei
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W7140032638