Arithmetic Siegel–Weil formula on X0(N)

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 constructed some generalized Delta function for Γ 0 (N ) and proved some explicit Kronecker limit formula for Eisenstein series on X 0 (N ).

Twisted arithmetic Siegel Weil formula on X0(N)

“…Combining the above theorem with the Kronecker limit formula for Γ 0 (N) [DY,Theorem 1.5], we obtain the following result.…”

“…Case 1: We first assume that D is not a square. This case follows directly from the [DY,Theorem 5.1].…”

“…Taking residue of both sides of the equation (3.2) at s = 1, we have the following result. Recall the modular form of weight k = 12ϕ(N) for Γ 0 (N) defined in [DY,(1.6)…”

“…Let P ℓ be the cusp associated to an isotropic line ℓ ∈ Iso(V ). We have, near the cusp P ℓ , by [DY,Theorem 5.1] and (4.3) (4.14) Ξ ∆,r (n, µ, v)(z) = −g ∆,r (n, µ, v, P ℓ ) log | q ℓ | 2 −2ψ ∆ (n, µ, v; q ℓ ), where ψ ∆ (n, µ, v; q ℓ ) is a smooth function of q ℓ (as two real variables q ℓ andq ℓ ) and…”

“…Here the modular form ∆ N (z) of weight k = 12ϕ(N) is given by (3.4). We need the following two lemmas in [DY,Section 6].…”