Related papers: On Some Hypergeometric Modularity Conjectures of D…

Let $X$ be a complex $K3$ surface with an effective action of a group $G$ which preserves the holomorphic symplectic form. Let $$ Z_{X,G}(q) = \sum_{n=0}^{\infty} e\left(\operatorname{Hilb}^{n}(X)^{G} \right)\, q^{n-1} $$ be the generating…

We give a brief review of the main results of the theory of elliptic hypergeometric functions -- a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler's…

This article describes results of joint work with Michael Rapoport and Tonghai Yang. First, we construct an modular form \phi(\tau) of weight 3/2 valued in the arithmetic Chow group of the arithmetic surface M attached toa Shimura curve…

The Dedekind eta functions plays important role in different branches of Mathematics and Theoretical Physics. One way to construct Dedekind Eta function to use the explicit formula (Kroncker limit formula) for the regularized determinants…

Based on our previous work on an arithmetic analogue of Christol's theorem, this paper studies in more detail the structure of the lambda-ring $E_K = K \otimes W_{O_K}^a (O_{\bar{K}})$ of algebraic Witt vectors for number fields $K$. First…

This is the first paper in a series where we study arithmetic applications of the multiple elliptic Gamma functions originated from mathematical physics. The main purpose of this paper is the introduction of a framework for applications of…

In Part I of the present series of papers, we adumbrate our idea of Riemannian geometry to higher order in the infinitesimals and derive expressions for the appropriate generalizations of parallel transport and the Riemannian curvature…

Hypergeometric functions of complex matrices were introduced by James in multivariate statistics. These special functions play many roles in random matrix theory. The main goal of this paper is to suggest a new use for them as holomorphic…

We introduce an $\ell$-adic analogue of Gauss's hypergeometric function arising from the Galois action on the fundamental torsor of the projective line minus three points. Its definition is motivated by a relation between the KZ-equation…

Hypergeometric functions over finite fields were introduced by Greene in the 1980s as a finite field analogue of classical hypergeometric series. These functions, and their generalizations, naturally lend themselves to, and have been widely…

The construction of Quasi-topological gravities in three-dimensions requires coupling a scalar field to the metric. As shown in arXiv:2104.10172, the resulting "Electromagnetic" Quasi-topological (EQT) theories admit charged black hole…

We first show that hypergeometric functions appear naturally as spectral functions when applying pseudo-differential calculus to decipher heat kernel asymptotic in the situation where the symbol algebra is noncommutative. Such observation…

We define hypergeometric functions using intersection homology valued in a local system. Topology is emphasized; analysis enters only once, via the Hodge decomposition. By a pull-back procedure we construct special subsets S_{pi}, derived…

Hypergeometric class equations are given by second order differential operators in one variable whose coefficient at the second derivative is a polynomial of degree $\leq2$, at the first derivative of degree $\leq1$ and the free term is a…

Let $\ell \geq 5$ be a prime, and let $\nu_\eta$ denote the Dedekind eta multiplier. For an odd integer $r$, and a real Dirichlet character $\psi$, recent work of Ahlgren, Andersen, and the author showed that quadratic congruences modulo…

We propose a formulation of the Equivariant Tamagawa Number Conjecture for modular motives with coefficients in universal deformation rings and Hecke algebras; something which seems to have been heretofore missing because the complexes of…

We use the unique canonically-twisted module over a certain distinguished super vertex operator algebra---the moonshine module for Conway's group---to attach a weak Jacobi form of weight zero and index one to any symplectic derived…

We consider spaces of modular forms attached to definite orthogonal groups of low even rank and nontrivial level, equipped with Hecke operators defined by Kneser neighbours. After reviewing algorithms to compute with these spaces, we…

In this paper, we consider $L$-functions of modular forms of weight 3, which are products of the Jacobi theta series, and express their special values at $s=3$, $4$ in terms of special values of Kamp\'e de F\'eriet hypergeometric functions.…

We prove new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor--Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the…