15 - Explicit Formulae as Trace Formulae

Abstract

This chapter illustrates that Weil's explicit formula can be interpreted as a trace formula on a suitable space. The simplest space that is constructed for this purpose is the semi-direct product of the ideles of norm one with the adeles, the semi-direct product of the multiplicative group of rational numbers with the additive group of rational numbers. For a suitable kernel function on this space, the conjugacy class side of the Selberg trace formula is the sum over the primes occurring in Weil's explicit formula. This implies that the sum of the eigenvalues of the self-adjoint integral operator associated to the kernel function is the sum over the critical zeroes of the Riemann zeta-function occurring on the other side of Weil's formula. However, the relation between the eigenvalues of this integral operator and the zeroes of the zeta-function appears mysterious at present. The approach of interpreting explicit formula as the trace formula leads to various new equivalences to the Riemann Hypothesis, such as, certain positivity hypotheses for the integral operators.