Twisted arithmetic Siegel Weil formula on X0(N)

TL;DR

This paper develops twisted arithmetic divisors on modular curves and proves their modularity, linking their pairings with special values of Eisenstein series, advancing the understanding of arithmetic geometry on modular forms.

Contribution

It constructs twisted arithmetic theta functions on X_0(N), proves their modularity, and relates their pairings to Eisenstein series values, providing new insights into arithmetic divisors.

Findings

Construction of twisted arithmetic theta functions ( au)

Proof of their modularity property

Identification of pairings with Eisenstein series values

Abstract

In this paper, we study twisted arithmetic divisors on the modular curve X_0(N) with N square-free. For each pair (\Delta, r) where \Delta >0 and \Delta \equiv r^2 \mod 4N, we constructed a twisted arithmetic theta function \phi_{\Delta, r}(\tau) which is a generating function of arithmetic twisted Heegner divisors. We prove the modularity of \phi_{\Delta, r}(\tau), along the way, we also identify the arithmetic pairing \langle \phi_{\Delta, r}(\tau),\widehat{\omega}_N \rangle with special value of some Eisenstein series, where \widehat{\omega}_N is a normalized metric Hodge line bundle.