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Based on the theory of rigid cohomology, we provide an explicit formula of zeta functions of certain K3 families, which we call the hypergeometric type. The central point of our argument is the comparison between the 2nd rigid cohomology of…

Elliptic Macdonald polynomials of sl(2)-type and level 2 are introduced. Suitable limits of elliptic Macdonald polynomials are the standard Macdonald polynomials and conformal blocks. Identities for elliptic Macdonald polynomials, in…

Superstring compactification on a manifold of Spin(7) holonomy gives rise to a 2d worldsheet conformal field theory with an extended supersymmetry algebra. The $\mathcal{N}=1$ superconformal algebra is extended by additional generators of…

In this article we examine the Ruelle type spectral functions $\cR(s)$,which define an overall description of the content of the work. We investigate the Gopakumar-Vafa reformulation of the string partition functions, describe the N=2…

In 2013, Lemke Oliver classified all eta-quotients which are theta functions. In this paper, we unify the eta-theta functions by constructing mock modular forms from the eta-theta functions with even characters, such that the shadows of…

We study modular invariance of normalized supercharacters of tame integrable modules over an affine Lie superalgebra, associated to an arbitrary basic Lie superalgebra $ \mathfrak{g}. $ For this we develop a several step modification…

In a series of papers we have been studying the geometric theta correspondence for non-compact arithmetic quotients of symmetric spaces associated to orthogonal groups. It is our overall goal to develop a general theory of geometric theta…

This is an introduction to the geometry of compact Riemann surfaces, largely following the books Farkas-Kra, Fay, Mumford Tata lectures. 1) Defining Riemann surfaces with atlases of charts, and as locus of solutions of algebraic equations.…

We express the logarithmic Mahler measure of two families of polynomials defining K3 hypersurfaces of generic Picard number 19 in terms of Eisenstein-Kronecker series. For some singular K3 surfaces (i.e of Picard number 20) we write this…

We study the realization spaces of $10_3$ line configurations. Answering a question posed by Sturmfels in 1991, we use elliptic surface techniques to show that realizations over $\mathbb{Q}$ are dense in those over $\mathbb{R}$ for all…

This work gives a manual for constructing superconformal field theories associated to a family of smooth K3 surfaces. A direct method is not known, but a combination of orbifold techniques with a non-classical duality turns out to yield…

These notes will give an introduction to the theory of K3 surfaces. We begin with some general results on K3 surfaces, including the construction of their moduli space and some of its properties. We then move on to focus on the theory of…

In this paper, we study modular transformation properties of a certain class of functions with indefinite quadratic forms.

We give a complete classification of finite subgroups of automorphisms of K3 surfaces up to deformation. The classification is in terms of Hodge theoretic data associated to certain conjugacy classes of finite subgroups of the orthogonal…

In this study, we construct four-dimensional F-theory models with 3 to 8 U(1) factors on products of K3 surfaces. We provide explicit Weierstrass equations of elliptic K3 surfaces with Mordell-Weil ranks of 3 to 8. We utilize the method of…

Arithmetic of K3 surfaces defined over finite fields is investigated. In particular, we show that any K3 surface of finite height over a finite field k of characteristic p > 3 has a quasi-canonical lifting to characteristic 0, and that for…

As a necessary step in constructing elliptic matrix models, which preserve the superintegrability property $<char>\sim {\rm char}$, we suggest an elliptic deformation of the peculiar loci $p_k^{\Delta_n}$, which play an important role in…

A protagonist here is a new-type invariant for type II degenerations of K3 surfaces, which is explicit PL (piecewise linear) convex function from the interval with at most 18 non-linear points. Forgetting its actual function behaviour, it…

We provide a moduli-theoretic framework for the collapsing of Ricci-flat Kahler metrics via compactification of moduli varieties of Morgan-Shalen and Satake type. In patricular, we use it to study the Gromov-Hausdorff limits of hyperKahler…

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…