Related papers: On the arithmetic Siegel--Weil formula for GSpin S…

Let $Z$ be the quotient of the Siegel modular threefold $\mathcal{A}^{{\rm sa}}(2,4,8)$ which has been studied by van Geemen and Nygaard. They gave an implication that some 6-tuple $F_Z$ of theta constants which is in turn known to be a…

The aim of this paper is twofold. First, we introduce a new method for evaluating the multiplicity of a given discrete series in the space of level $1$ automorphic forms of a split classical group $G$ over $\mathbb{Z}$, and provide…

Let F be a non-archimedian local field of characteristic 0, and O the ring of integres in F. We give an explicit formula for the Siegel series of a half-integral matrix over O. This formula expresses the Siegel series of a half-integral…

We study instances of Beilinson-Tate conjectures for automorphic representations of $\mathrm{PGSp}_6$ whose Spin $L$-function has a pole at $s=1$. We construct algebraic cycles of codimension three in the Siegel-Shimura variety of dimension…

The Witten $r$-spin class defines a non-semisimple cohomological field theory. Pandharipande, Pixton and Zvonkine studied two special shifts of the Witten class along two semisimple directions of the associated Dubrovin--Frobenius manifold…

We define a family of arithmetic zero cycles in the arithmetic Chow group of a modular curve X_0(N), for N>3 odd and squarefree, and identify the arithmetic degrees of these cycles as q-coefficients of the central derivative of a Siegel…

We discuss the local index theorem for cofinite Riemann surfaces in a pedagogical way, from a more computational perspective. Given a cofinite Riemann surface $X$, let $\Delta_n$ be the $n$-Laplacian and let $N_n$ be the Gram matrix of a…

In the computation of the intersection cohomology of Shimura varieties, or of the $L^2$ cohomology of equal rank locally symmetric spaces, combinatorial identities involving averaged discrete series characters of real reductive groups play…

Rapoport-Zink spaces, or more generally local Shimura varieties, are expected to provide geometric realization of the local Langlands correspondence via their $l$-adic cohomology. Along this line is a conjecture by Harris and Viehmann,…

We study Shimura varieties associated with special orthogonal groups over the field of rational numbers. We prove a version of Morel's formula for the Frobenius--Hecke traces on the intersection cohomology of the Baily--Borel…

Explicit formulas for Segal-Sugawara vectors associated with the simple Lie algebra $\mathfrak{g}$ of type $G_2$ are found by using computer-assisted calculations. This leads to a direct proof of the Feigin-Frenkel theorem describing the…

Let $X$ be an arithmetic scheme (i.e., separated, of finite type over $\operatorname{Spec} \mathbb{Z}$) of Krull dimension $1$. For the associated zeta function $\zeta (X,s)$, we write down a formula for the special value at $s = n < 0$ in…

Given a Lie superalgebra \g, we introduce several variants of the representation ring, built as subrings and quotients of the ring R_{\Z_2}(\g) of virtual \g-supermodules (up to even isomorphisms). In particular, we consider the ideal…

Imanishi, Jinzenji and Kuwata provided a recipe for computing Euler number of Grassmann manifold $G(k,N)$ using physical model and its path-integral [S.Imanishi, M.Jinzenji and K.Kuwata, Journal of Geometry and Physics, Volume 180, October…

In this paper, we prove the weak Siegel-Weil formula for the space M_2(Q) . We study the Hecke correspondence and representation numbers associated to Eichler orders, and give the explicit formula for degree of Hecke correspondence and…

We give a new formula for the special value at s=2 of the Hasse-Weil zeta function for smooth projective fourfolds under some assumptions (the Tate and Beilinson conjecture, finiteness of some cohomology groups, etc.). Our formula may be…

To any simple Lie algebra $\mathfrak g$ and automorphism $\sigma:\mathfrak g\to \mathfrak g$ we associate a cyclotomic Gaudin algebra. This is a large commutative subalgebra of $U(\mathfrak g)^{\otimes N}$ generated by a hierarchy of…

In this work we present a determinant expression for the domain-wall boundary condition partition function of rational (XXX) Richardson-Gaudin models which, in addition to $N-1$ spins $\frac{1}{2}$, contains one arbitrarily large spin $S$.…

W. Zhang's arithmetic fundamental lemma (AFL) is a conjectural identity between the derivative of an orbital integral on a symmetric space with an arithmetic intersection number on a unitary Rapoport-Zink space. In the minuscule case,…

Certain natural geometric approximation schemes are developed for Wiener measure on a compact Riemannian manifold. These approximations closely mimic the informal path integral formulas used in the physics literature for representing the…