Related papers: Round about Theta. Part I Prehistory
We construct Hida families of theta lifts from definite orthogonal and unitary groups. A major ingredient of the construction is the choice of test Schwartz functions at places dividing $p$. We select a special type of Schwartz functions…
The Schwartz kernel of the multiplication operation on a quantum torus is shown to be the distributional boundary value of a classical multivariate theta function. The kernel satisfies a Schr\"odinger equation in which the role of time is…
In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…
In this paper we introduce new generalizations of the zeta function, the Tricomi functions; their main properties are studied. This opens the way to a deeper, better application of these functions both in the theory of special functions,…
Any permutation has a disjoint cycle decomposition and concept generates an equivalence class on the symmetry group called the cycle-type. The main focus of this work is on permutations of restricted cycle-types, with particular emphasis on…
This note is a short survey of two topics: Archimedean zeta functions and Archimedean oscillatory integrals. We have tried to portray some of the history of the subject and some of its connections with similar devices in mathematics. We…
The original article expressed the special values of the zeta function of a variety over a finite field in terms of the $\hat{Z}$-cohomology of the variety. As the article was being completed, Lichtenbaum conjectured the existence of…
This set of lecture notes is an expanded version of a mini-course the author gave in March of 2025 for the program ``Representation Theory \& Noncommutative Geometry" at the Institut Henri Poincar\'e, Paris. The goal is to provide a survey…
Ordinary theta-functions can be considered as holomorphic sections of line bundles over tori. We show that one can define generalized theta-functions as holomorphic elements of projective modules over noncommutative tori (theta-vectors).…
Quantum mechanics and classical mechanics are two very different theories, but the correspondence principle states that quantum particles behave classically in the limit of high quantum number. In recent years much research has been done on…
Aspects of the properties, enumeration and construction of points on diagonal and Hermitian surfaces have been considered extensively in the literature and are further considered here. The zeta function of diagonal surfaces is given as a…
This paper deals with some simple results about spherical functions of type $\delta$, namely new integral formulas, new results about behavior at infinity and some facts about the related $C_\sigma$ functions.
The main objective of this paper is to give a summary of our recent work on recursion formulae for intersection numbers on moduli spaces of curves and their applications. We also present a conjectural relation between tautological rings and…
Pure type systems arise as a generalisation of simply typed lambda calculus. The contemporary development of mathematics has renewed the interest in type theories, as they are not just the object of mere historical research, but have an…
The Riemann theta function is a complex-valued function of g complex variables. It appears in the construction of many (quasi-) periodic solutions of various equations of mathematical physics. In this paper, algorithms for its computation…
We prove or conjecture several relations between the multizeta values for positive genus function fields of class number one, focusing on the zeta-like values, namely those whose ratio with the zeta value of the same weight is rational (or…
We obtain formulas for the coefficients of positive and negative powers of a partial theta function.
We study links between first-order formulas and arbitrary properties for families of theories, classes of structures and their isomorphism types. Possibilities for ranks and degrees for formulas and theories with respect to given properties…
The principal aim of this paper is to construct torsion cohomology classes in the initial terms of a spectral sequence computing the cohomology of a Kottwitz-Harris-Taylor Shimura variety. Beside we produce some global congruences between…
This article is the first article of a serie of articles on the generalization of Kato's Euler system. The main subject of this article is to construct a family of Kato's Euler systems and a family of Kato's explicit reciprocity laws over…