Related papers: A theta operator for the group $\mathrm{GSp}_4$

The spectral eta-invariant of a self-adjoint elliptic differential operator on a closed manifold is rigid, provided that the parity of the order is opposite to the parity of dimension of the manifold. The paper deals with the calculation of…

We construct explicit differential operators on hermitian modular forms, extending methods developed for Siegel modular forms. These differential operators are closely related to the two-variable spherical pluriharmonic polynomials. We…

This note outlines an approach to defining $p$-adic Shimura classes and $p$-adic derived Hecke operators on the completed cohomology of modular curves from upcoming work by the author. After reviewing the modulo-$p$ constructions of Harris…

A generalization of Serre's $p$-adic Eisenstein series in the case of Siegel modular forms is studied and a coincidence between a $p$-adic Siegel Eisenstein series and a genus theta series associated with a quaternary quadratic form is…

We show that analytic torsion of smooth theta divisor is represented by a Siegel modular form characterizing the Andreotti-Mayer locus if its dimension is positive.

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…

We present an explicit set of matrices giving the action of the Hecke operators $T(p)$, $T_j(p^2)$ on Siegel modular forms.

We study differential operators on an elliptic curve of order higher than 2 which are algebraically integrable (i.e., finite gap). We discuss classification of such operators of order 3 with one pole, discovering exotic operators on special…

Let (S,H) be a rational algebraic surface with an ample divisor. We compute generating functions for the Hodge numbers of the moduli spaces of H-stable rank 2 sheaves on S in terms of certain theta functions for indefinite lattices that…

We give a geometric interpretation of the Siegel operators for holomorphic differential forms on Siegel modular varieties. This involves extension of the differential forms over a toroidal compactification, and we show that the Siegel…

We propose a systematic scheme for computing the variation of rearrangement operators arising in the recently developed spectral geometry on noncommutative tori and $\theta$-deformed Riemannian manifolds. It can be summarized as a category…

We introduce axioms for towers of infinite-dimensional algebras such that the corresponding Grothendieck groups of projective and finite-dimensional modules are Hopf dual to each other. This duality gives rise to an action of the Hesienberg…

Starting with work of Serre, Katz, and Swinnerton-Dyer, theta operators have played a key role in the study of $p$-adic and $\bmod p$ modular forms and Galois representations. This paper achieves two main results for theta operators on…

We derive a tensorial formula for a fourth-order conformally invariant differential operator on conformal 4-manifolds. This operator is applied to algebraic Weyl tensor densities of a certain conformal weight, and takes its values in…

We deduce one-parameter group properties for pseudo-differential operators $\operatorname{Op} (a)$, where $a$ belongs to the class $\Gamma ^{(\omega _0)}_*$ of certain Gevrey symbols. We use this to show that there are pseudo-differential…

We consider the classical theta operator $\theta$ on modular forms modulo $p^m$ and level $N$ prime to $p$ where $p$ is a prime greater than 3. Our main result is that $\theta$ mod $p^m$ will map forms of weight $k$ to forms of weight…

We give a classification of $1^{st}$ order invariant differential operators acting between sections of certain bundles associated to Cartan geometries of the so called metaplectic contact projective type. These bundles are associated via…

We prove a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a $G$-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of…

We introduce a simplified (coarse) version of pseudo-differential calculus for operators of order zero on complete Riemannian manifolds. This calculus works for the usual Hormander (1,0) class of operators, as well as for…

The main aim of this article is to establish an $L_p$-theory for elliptic operators on manifolds with singularities. The particular class of differential operators discussed herein may exhibit degenerate or singular behavior near the…