Correspondences in Arakelov geometry and applications to the case of Hecke operators on modular curves

TL;DR

This paper develops a framework for analyzing Hecke operators on modular curves within Arakelov geometry, introducing new invariants and proving self-adjointness properties using advanced intersection theory and L-series computations.

Contribution

It extends Arakelov geometry by defining Hecke actions on arithmetic Chow groups and introduces refined invariants linked to L-series, with applications to modular curves.

Findings

Hecke operators act self-adjointly on the arithmetic Chow groups of modular curves.

New numerical invariants are defined and computed using L-series and the Gross-Zagier formula.

Hecke correspondences on Jacobians are self-adjoint with respect to the Néron-Tate pairing.

Abstract

In the context of arithmetic surfaces, Bost defined a generalized Arithmetic Chow Group (ACG) using the Sobolev space L^2_1. We study the behavior of these groups under pull-back and push-forward and we prove a projection formula. We use these results to define an action of the Hecke operators on the ACG of modular curves and to show that they are self-adjoint with respect to the arithmetic intersection product. The decomposition of the ACG in eigencomponents which follows allows us to define new numerical invariants, which are refined versions of the self-intersection of the dualizing sheaf. Using the Gross-Zagier formula and a calculation due independently to Bost and Kuehn we compute these invariants in terms of special values of L series. On the other hand, we obtain a proof of the fact that Hecke correspondences acting on the Jacobian of the modular curves are self-adjoint with respect to the N\'eron-Tate height pairing.