Related papers: The transformation laws of algebraic theta functio…
In this paper we are interested in developments of elliptic functions of Jacobi. In particular a trigonometric expansion of the classical theta functions introduced by the author (Algebraic methods and q-special functions, Editors: C.R.M.…
The paper consists of two sections. The first section provides a new definition of mirror symmetry of abelian varieties making sense also over $p$-adic fields. The second section introduces and studies quantized theta-functions with…
Let $F$ be a totally real number field and $\mathfrak{o}$ the ring of integers of $F$. We study theta functions which are Hilbert modular forms of half-integral weight for the Hilbert modular group $\mathrm{SL}_2(\mathfrak{o})$. We obtain…
We discuss various aspects of the geometry of theta characteristics including the birational geometry of the spin moduli space of curves, parametrization of moduli via special K3 surfaces, as well as the relation with classical theta…
We investigate the representation theory of the Temperley-Lieb algebra, $TL_n(\delta)$, defined over a field of positive characteristic. The principle question we seek to answer is the multiplicity of simple modules in cell modules for…
We characterize the theta functions for vectors in the imaginary wall in a cluster algebra of acyclic affine type and compute some of their structure constants. One of the structure constant computations can be interpreted as new…
We study the holomorphic projection of mixed mock modular forms involving sesquiharmonic Maass forms. As a special case, we numerically express the holomorphic projection of a function involving real quadratic class numbers multiplied by a…
This is an integrated part of our Geo-Arithmetic Program. In this paper we initiate a geometrically oriented construction of non-abelian zeta functions for curves defined over finite fields by a weighted count of semi-stable bundles. Basic…
We give a moduli-theoretic proof of the classical theorem of Gabriel, stating that a scheme can be reconstructed from the abelian category of quasi-coherent sheaves over it. The methods employed are elementary and allow us to extend the…
Algebraic deformations of modules over a ring are considered. The resulting theory closely resembles Gerstenhaber's deformation theory of associative algebras.
We study the theta map which assigns to a real quadratic form its theta series. We introduce two invariants reflecting whether the differential of the theta map vanishes or is degenerate. We provide examples of lattices where this…
We show that some $q$-series such as universal mock theta functions are linear sums of theta quotients and mock Jacobi forms of weight 1/2, which become holomorphic parts of real analytic modular forms when they are restricted to torsion…
We introduce a systematic theory of Weil bundles over \( p \)-adic analytic manifolds, forging new connections between differential calculus over non-archimedean fields and arithmetic geometry. By developing a framework for infinitesimal…
We provide necessary and sufficient conditions for when an algebraic stack admits a good moduli space and prove a semistable reduction theorem for points of algebraic stacks equipped with a $\Theta$-stratification. These results provide a…
We introduce and investigate an infinite family of functions which are shown to have generalised quantum modular properties. We realise their "companions" in the lower half plane both as double Eichler integrals and as non-holomorphic theta…
This work is devoted to the algebraic and arithmetic properties of Rankin-Cohen brackets allowing to define and study them in several natural situations of number theory. It focuses on the property of these brackets to be formal…
We suggest to compactify the universal covering of the moduli space of complex structures by non-commutative spaces. The latter are described by certain categories of sheaves with connections which are flat along foliations. In the case of…
We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function and establish tropical versions of the…
Partial zeta functions of algebraic varieties over finite fields generalize the classical zeta function by allowing each variable to be defined over a possibly different extension field of a fixed finite field. Due to this extra variation…
Ideas from deformation quantization applied to algebras with one generator lead to methods to treat a nonlinear flat connection. It provides us elements of algebras to be parallel sections. The moduli space of the parallel sections is…