# Arithmetic Siegel-Weil formula on $X_{0}(N)$

## TL;DR

This paper establishes an arithmetic Siegel-Weil formula on the modular curve X_0(N), demonstrating the modularity of certain arithmetic theta functions, and introduces generalized Delta functions along with explicit Kronecker limit formulas.

## Contribution

It proves an arithmetic Siegel-Weil formula on X_0(N) for square-free N and constructs generalized Delta functions with explicit Kronecker limit formulas.

## Key findings

- Proved the arithmetic Siegel-Weil formula on X_0(N).
- Established modularity of specific arithmetic theta functions.
- Constructed generalized Delta functions and derived Kronecker limit formulas.

## Abstract

In this paper, we proved an arithmetic Siegel-Weil formula and the modularity of some arithmetic theta function on the modular curve $X_0(N)$ when $N$ is square free. In the process, we also construct some generalized Delta function for $\Gamma_0(N)$ and proved some explicit Kronecker limit formula for Eisenstein series on $X_0(N)$.