Equivariant Euler characteristics and sheaf resolvents

TL;DR

This paper derives explicit formulas for equivariant Euler characteristics of canonical sheaves on certain tame abelian covers of arithmetic surfaces, revealing cases where these characteristics differ and are non-trivial.

Contribution

It introduces new formulas involving quadratic forms from intersection numbers for equivariant Euler characteristics, using resolvent techniques and local Riemann-Roch methods.

Findings

Explicit formulas for Euler characteristics of sheaves on abelian covers

Examples showing differences between various Euler characteristics

Application of resolvent and local Riemann-Roch techniques

Abstract

For certain tame abelian covers of arithmetic surfaces X/Y we obtain striking formulas, involving a quadratic form derived from intersection numbers, for the equivariant Euler characteristics of both the canonical sheaf !X/Y and also its square root !1/2 X/Y . These formulas allow us us to carry out explicit calculations; in particular, we are able to exhibit examples where these two Euler characteristics and that of the the structure sheaf of X are all different and non-trivial. Our results are obtained by using resolvent techniques together with the local Riemann-Roch approach developed in [CPT].