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 ( Δ , r ) , where Δ ≡ r 2 mod 4 N and Δ is a fundamental discriminant, we construct a twisted arithmetic theta function ϕ ˆ Δ , r ( τ ) which is a generating function of arithmetic twisted Heegner divisors. We prove that the arithmetic pairing 〈 ϕ ˆ Δ , r ( τ ) , ω ˆ N 〉 is equal to the special value, rather than the derivative, of some Eisenstein series, thanks to some cancellation, where ω ˆ N is a normalized metric Hodge line bundle. We also prove the modularity of ϕ ˆ Δ , r ( τ ) .