Schneider–Siegel theorem for a family of values of a harmonic weak Maass form at Hecke orbits

Let {j(z)} be the modular j-invariant function. Let τ be an algebraic number in the complex upper half plane {\mathbb{H}}. It was proved by Schneider and Siegel that if τ is not a CM point, i.e., {[\mathbb{Q}(\tau):\mathbb{Q}]\neq 2}, then {j(\tau)} is transcendental. Let f be a harmonic weak Maass form of weight 0 on {\Gamma_{0}(N)}. In this paper, we consider an extension of the results of Schneider and Siegel to a family of values of f on Hecke orbits of τ. For a positive integer m, let {T_{m}} denote the m-th Hecke operator. Suppose that the coefficients of the principal part of f at the cusp {i\infty} are algebraic, and that f has its poles only at cusps equivalent to {i\infty}. We prove, under a mild assumption on f, that, for any fixed τ, if N is a prime such that {N\geq 23} and {N\notin\{23,29,31,41,47,59,71\}}, then {f(T_{m}.\tau)} are transcendental for infinitely many positive integers m prime to N.

On the Generalized Brauer–Siegel Theorem for Asymptotically Exact Families of Number Fields with Solvable Galois Closure

In 2002, M. A. Tsfasman and S. G. Vlăduţ formulated the generalized Brauer–Siegel conjecture for asymptotically exact families of number fields. In this article, we establish this conjecture for asymptotically good towers and asymptotically bad families of number fields with solvable Galois closure.