Twisted arithmetic Siegel Weil formula on X0(N)

Abstract

In this paper, we study twisted arithmetic divisors on the modular curve X 0 (N ) with N square-free. For each pair (∆, r) where ∆ > 0 and ∆ ≡ r 2 mod 4N , we constructed a twisted arithmetic theta function φ ∆,r (τ ) which is a generating function of arithmetic twisted Heegner divisors. We prove the modularity of φ ∆,r (τ ), along the way, we also identify the arithmetic pairing φ ∆,r (τ ), ω N with special value of some Eisenstein series, where ω N is a normalized metric Hodge line bundle.