Date of Degree
Carlos J. Moreno
Harmonic Analysis and Representation | Number Theory
Weil explicit formula, Selberg trace formula, zeroes of L-functions
In this thesis, motivated by an observation of D. Hejhal, we show that the explicit formulae of A. Weil for sums over zeroes of Hecke L-functions, via the Maass-Selberg relation, occur in the continuous spectral terms in the Selberg trace formula over various number fields. In Part I, we discuss the relevant parts of the trace formulae classically and adelically, developing the necessary representation theoretic background. In Part II, we show how show the explicit formulae intervene, using the classical formulation of Weil; then we recast this in terms of Weil distributions and the adelic formulation of Weil. As an application, we prove a lower bound for these explicit formulae using properties of the trace formula, in the spirit of Weil's criterion for the Riemann hypothesis.
Wong, Tian An, "Explicit Formulae and Trace Formulae" (2016). CUNY Academic Works.