Related papers: Green forms and the arithmetic Siegel-Weil formula

Kudla conjectured that certain Eisenstein series contain important arithmetical and geometric information. The following note describes a certain aspect of this general picture in the special case concerning the orthogonal group…

We prove a general formula for the $p$-adic heights of Heegner points on modular abelian varieties with potentially ordinary (good or semistable) reduction at the primes above $p$. The formula is in terms of the cyclotomic derivative of a…

Let $F$ be a totally real field in which a fixed prime $p$ is inert, and let $E$ be a CM extension of $F$ in which $p$ splits. We fix two positive integers $r,s \in \mathbb N$. We investigate the Tate conjecture on the special fiber of…

There exist conjectural formulas on relations between $L$-functions of submotives of Shimura varieties and automorphic representations of the corresponding reductive groups, due to Langlands -- Arthur. In the present paper these formulas…

In the present paper we find explicit formulas for the degrees of Heegner divisors on arithmetic quotients of the orthogonal group $\Orth(2,p)$ and for the integrals of certain automorphic Green's functions associated with Heegner divisors.…

We construct a well-behaved Weil-\'etale complex for a large class of $\mathbb{Z}$-constructible sheaves on a regular irreducible scheme $U$ of finite type over $\mathbb{Z}$ and of dimension $1$. We then give a formula for the special value…

We classify Siegel modular cusp forms of weight two for the paramodular group K(p) for primes p< 600. We find that weight two Hecke eigenforms beyond the Gritsenko lifts correspond to certain abelian varieties defined over the rationals of…

We produce first examples of p-local height three TAF homology theories. The corresponding one-dimensional formal groups arise as split summands of the formal groups of certain abelian three-folds, the Shimura variety of which can be…

We prove function field versions of the Zilber-Pink conjectures for varieties supporting a variation of Hodge structures. A form of these results for Shimura varieties in the context of unlikely intersections is the following. Let $S$ be a…

Let $A$ be an abelian variety defined over a number field $k$, let $p$ be an odd prime number and let $F/k$ be a cyclic extension of $p$-power degree. Under not-too-stringent hypotheses we give an interpretation of the $p$-component of the…

We compute Hecke eigenform bases of spaces of level one, degree~three Siegel modular forms and 2-Euler factors of the eigenforms through weight 22. Our method uses the Fourier coefficients of Siegel Eisenstein series, which are fully known…

Given a Hecke eigenform $f$ of weight $2$ and square-free level $N$, by the work of Kohnen, there is a unique weight $3/2$ modular form of level $4N$ mapping to $f$ under the Shimura correspondence. Furthermore, by the work of Waldspurger…

We present an explicit and computationally actionable blueprint for constructing vector-valued Siegel modular forms associated to real multiplication (RM) abelian surfaces, leveraging the theta correspondence for the unitary dual pair…

Inspired by recent work of Aslanyan and Daw, we introduce the notion of $\Sigma$-orbits in the general framework of distinguished categories. In the setting of connected Shimura varieties, this concept contains many instances of…

In this paper, we formulate conjectural formulas for the arithmetic intersection numbers of special cycles on unitary Shimura varieties with minuscule parahoric level structure. Also, we prove that these conjectures are compatible with all…

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.…

In this paper, we reformulate conjectural formulas for the arithmetic intersection numbers of special cycles on unitary Shimura varieties with minuscule parahoric level structure in terms of weighted counting of lattices containing special…

We show that the systems of Hecke eigenvalues that appear in the coherent cohomology with coefficients in automorphic line bundles of any mod $p$ abelian type compact Shimura variety at hyperspecial level are the same as those appearing in…

In this paper, we investigate a general method to establish tame and norm relations for special cycles in Shimura varieties, using unitary cycles in odd orthogonal Shimura varieties as a guiding example.

We construct indecomposable cycles in the motivic cohomology group $H^3_{{\mathcal M}}(A,{\mathbb Q}(2))$ where $A$ is an Abelian surface over a number field or the function field of a base. When $A$ is the self product of the universal…