Zeros of Eisenstein series for congruence groupsMPIM

by Gunther Cornelissen (Universiteit Utrecht/MPIM)

Wednesday Apr 30, 2025, 2:30 PM → 3:30 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)

Number theory lunch seminar

We study the distribution of zeros of Eisenstein series on an arbitrary congruence groups in the standard fundamental domain F for SL(2,Z). The main results are an upper bound on the imaginary part of such a zero (defined in terms of the (non-)vanishing of a generalisation of Ramanujan/Kloosterman sums),  a description of a limiting configuration of compact segments of geodesics to which all zeros tend as the weight increases (based on describing the limit set of a family of polynomials of increasing degree in terms of a boolean combination of inequalities), and a proof that there are only finitely many possible algebraic zeros in F (using CM theory). All results can be made explicit, e.g., principal and Hecke congruence groups, including a trichotomy for the convergence speed to the limiting configuration. Joint work with Sebastian Carrillo (Utrecht) and Berend Ringeling (Montreal).