Explicit formulas for the spectral side of the trace formula of SL(2)

TL;DR

This paper derives explicit formulas for the spectral side of the trace formula for SL(2) over a number field, linking intertwining operators and L-functions, and provides bounds related to zeros of L-functions.

Contribution

It presents two new explicit expressions for the spectral side of the trace formula of SL(2), connecting it with the Riemann-Weil explicit formula and adelic distributions.

Findings

Expressed spectral side as sum over zeros of L-functions

Derived adelic distribution representation

Provided lower bounds for zero sums in Eisenstein series

Abstract

The continuous spectrum to the spectral side of the Arthur-Selberg trace formula is described in terms of intertwining operators, whose normalising factors involve quotients of $L$-functions. In this paper, we derive two expressions in the case of SL(2) over a number field in terms of the Riemann-Weil explicit formula: as a sum over zeroes of the associated $L$-functions, and as a sum of adelic distributions on Weil groups. As an application, we obtain an expression for a lower bound for the sums over zeroes with respect to the truncation parameter for Eisenstein series.