A genus two arithmetic Siegel-Weil formula on X_0(N)

TL;DR

This paper establishes a genus two arithmetic Siegel-Weil formula on modular curves, linking arithmetic zero cycles to derivatives of Siegel Eisenstein series, extending Kudla-Rapoport-Yang's work to a new setting.

Contribution

It introduces a new family of arithmetic zero cycles on X_0(N) and relates their degrees to derivatives of genus two Siegel Eisenstein series.

Findings

Arithmetic degrees match q-coefficients of the derivative of a genus two Siegel Eisenstein series.

Extends Kudla-Rapoport-Yang's results from Shimura curves to modular curves.

Provides new tools for understanding arithmetic intersections on modular curves.

Abstract

We define a family of arithmetic zero cycles in the arithmetic Chow group of a modular curve X_0(N), for N>3 odd and squarefree, and identify the arithmetic degrees of these cycles as q-coefficients of the central derivative of a Siegel Eisenstein series of genus two. This parallels work of Kudla-Rapoport-Yang for Shimura curves.