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 ).
Abstract. In this paper, we prove some interesting identities, among average representation numbers (associated to definite quaternion algebras) and degrees of Hecke correspondences on Shimura curves (associated to indefinite quaternion algebras).
Abstract. In this paper, I use Siegel-Weil formula and Kudla matching principle to prove some interesting identities between representation number (of ternary quadratic space ) and the degree of Heegner divisors.
In this paper, I use Siegel-Weil formula and Kudla matching principle to prove some interesting identities between representation number (of ternary quadratic space ) and the degree of Heegner divisors.
We prove a weak version of the Siegel-Weil formula on
SL
2
\operatorname {SL}_2
for the dual pair
(
SL
2
,
O
2
,
2
)
(\operatorname {SL}_2, O_{2, 2})
, where
O
2
,
2
O_{2, 2}
is the split orthogonal group. By this formula and the Siegel-Weil formula, we give explicit formulas for Hecke correspondence’s degree and average representation numbers over genus associated to Eichler orders. At last, we give explicit formulas for representations of a number as sums of three squares and four squares by local Whittaker functions, and it turns out that these functions are exactly the local factors of Hardy’s singular series.
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