## Explicit formulas for the spectral side of the trace formula of ${\rm SL(2)}$

### Volume 195 / 2020

Acta Arithmetica 195 (2020), 149-175
MSC: Primary 11M36; Secondary 11M26, 11F72.
DOI: 10.4064/aa190115-9-10
Published online: 29 April 2020

#### 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.