Related papers: The Regularized Siegel-Weil Formula (The Second Te…
We derive a (weak) second term identity for the regularized Siegel-Weil formula for the even orthogonal group, which is used to obtain a Rallis inner product formula in the "second term range". As an application, we show the following…
Following W.T.Gan and S.Takeda, we obtain a weak second term identity of the regularized Siegel-Weil formula for the unitary dual pair $(U(n,n),U(V))$, where $V$ is a split hermitian space of dimension $2r$ with $r+1 \leq n \leq 2r-1$. As…
Let $\pi$ be a genuine cuspidal representation of the metaplectic group of rank $n$. We consider the theta lifts to the orthogonal group associated to a quadratic space of dimension $2n+1$. We show a case of regularised Rallis inner product…
In this paper, we examine the tower property concerning the genericity of global theta lifts between various classical groups, drawing inspiration from Rallis' tower property. By exploring the relationship between the analytic properties of…
In this paper, we propose a formula relating certain residues of Eisenstein series on symplectic groups. These Eisenstein series are attached to parabolic data coming from Speh representations. The proposed formula bears a strong similarity…
The classical theta correspondence, based on the Weil representation, allows one to lift automorphic representations on symplectic groups or their double covers to automorphic representations on special orthogonal groups. It is of interest…
In a previous paper (arxiv:1409.7353), we introduced a regularized theta lift for reductive dual pairs of the form $(Sp_4,O(V))$ with $V$ a quadratic vector space over a totally real number field $F$. The lift takes values in the space of…
The main goal of this expository article is to survey recent progress on the arithmetic Siegel-Weil formula and its applications. We begin with the classical sum of two squares problem and put it in the context of the Siegel-Weil formula.…
This research provides a formal definition of the arithmetic theta lift for cusp forms of weight $3/2$ and establishes the arithmetic inner product formula, thereby completing the Kudla program on modular curves. This formula is…
We prove a compatibility theorem between the Stark conjecture and the Harris-Venkatesh conjecture for imaginary dihedral modular forms of weight $1$. The key technical input is a general two-variable $\mathrm{PGL}_2$ Siegel-Weil formula…
We explicitly give an inner product formula and a Bessel period formula for theta series on GSp(4), which was studied by Harris, Soudry and Taylor. As a consequence, we prove a non-vanishing of the theta series of small weights and we give…
We determine the occurrence and explicitly describe the theta lifts on all levels of all the irreducible generic representations for the dual pair of groups $(\mathrm{Sp}_{2n}, \mathrm{O}(V))$ defined over a local nonarchimedean field…
Previously, we proved an identity for theta functions of degree eight, and several applications of it were also discussed. This identity is a natural extension of the addition formula for the Weierstrass sigma-function. In this paper we…
We establish a Siegel-Weil formula for classical groups over a function field with odd characteristic, which asserts in many cases that the Siegel Eisenstein series is equal to an integral of a theta function. This is a function-field…
There are many Rankin-Selberg integrals representing Langlands $L$-functions, and it is not apparent what the limits of the Rankin-Selberg method are. The Dimension Equation is an equality satisfied by many such integrals that suggests a…
In this paper we generalize the famous Jacobi's triple product identity, considered as an identity for theta functions with characteristics and their derivatives, to higher genus/dimension. By applying the results and methods developed in…
Following the approach of B. Roberts, we characterize the non-vanishing of global theta lifts for symplectic-orthogonal dual pairs in terms of its local counterpart. In particular, we replace the temperedness assumption present in Robert's…
In this paper we prove a regularized product expansion for the two-variable zeta functions of number fields introduced by van der Geer and Schoof. The proof is based on a general criterion for zeta-regularizability due to Illies. For number…
Using combinations of weight-1 and weight-2 of Kronecker-Eisenstein series to construct currents in the distributional de Rham complex of a squared elliptic curve, we find a simple explicit formula for the type II $(\text{GL}_2,…
We study the Siegel-Weil formula in the second term range ($n+1\leq m\leq n+r$) for unitary groups of hermitian forms over a skew-field $D$ with involution of the second kind.