Twisted arithmetic Siegel Weil formula on X0(N)

Abstract

In this paper, we study twisted arithmetic divisors on the modular curve X0(N) with N square-free. For each pair (Δ,r), where Δ≡r2mod4N and Δ is a fundamental discriminant, we construct a twisted arithmetic theta function ϕˆΔ,r(τ) which is a generating function of arithmetic twisted Heegner divisors. We prove that the arithmetic pairing 〈ϕˆΔ,r(τ),ωˆN〉 is equal to the special value, rather than the derivative, of some Eisenstein series, thanks to some cancellation, where ωˆN is a normalized metric Hodge line bundle. We also prove the modularity of ϕˆΔ,r(τ).