Progress on Chinburg's conjectures in Mahler measure theory

I present new evidence for Chinburg’s conjectures in Mahler measure theory. These conjectures predict that for each odd quadratic Dirichlet character $\chi_{-f}$ of conductor $f$, there exists a bivariate polynomial (or a rational function, in the weak form) whose Mahler measure is a rational multiple of $L'(\chi_{-f}, -1)$. Before our work, the conjecture was verified for only 18 conductors.I will highlight results from two collaborations. In joint work with M.J. Bertin, we study a family of polynomials $P_d(x, y)$ with remarkable properties. These produce new types of solutions for known conductors and, notably, relate Mahler measures to $L'(\chi, -1)$ for odd non-real primitive characters, prompting a natural generalization of Chinburg’s conjecture beyond quadratic characters. We obtain examples for $f = 5, 7, 9$ in this broader context. In ongoing work with D. Hokken and B. Ringeling, we systematically explore new polynomial families and study the properties of the related Bloch groups, leading to solutions for 11 previously unresolved conductors. Notably, the cases $f = 43, 132, 228$ provide evidence for the strong form of Chinburg’s conjecture.