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We compare modular forms of characteristic $p>0$ (i.e. Drinfeld's modular forms) and automorphic forms. We prove that spaces of these modular forms (which are of characteristic $p$) can be described by function spaces of characteristic…

Let R=k[x_1,...,x_d] be the polynomial ring in d independent variables, where k is a field of characteristic p>0. Let D be the ring of k-linear differential operators of R and let f be a polynomial in R. In this work we prove that the…

Let $p$ be a prime. We discuss $p$-adic properties of various arithmetical functions related to the coefficients of modular form and generating functions. Modular forms are considered as a tool of solving arithmetical problems. Examples of…

Let G be a semisimple group over an algebraically closed field of characteristic p>0. We give a (partly conjectural) simple, closed formula for the character of many indecomposable tilting rational G-modules, assuming that p is large.

We investigate qualitative properties of the underlying scheme of Rapoport-Zink formal moduli spaces of p-divisible groups, resp. Shtukas. We single out those cases when the dimension of this underlying scheme is zero, resp. those where the…

Linear differential equations with polynomial coefficients over a field $K$ of positive characteristic $p$ with local exponents in the prime field have a basis of solutions in the differential extension $\mathcal{R}_p=K(z_1, z_2,…

In this paper we make an initial study on type D moduli spaces in positive characteristic $p\neq 2$, where we allow $p$ ramified in the definite quaternion algebra. We classify the isogeny classes of $p$-divisible groups with additional…

Let g be the Lie superalgebra p(3) of rank 2 over an algebraically closed field K of characteristic p > 3. We classify all irreducible modules of g, and give the character formulae for irreducible modules.

I will survey some results in the theory of modular representations of a reductive $p$-adic group, in positive characteristic $\ell \neq p$ and $\ell=p$.

The concept of descent algebras over a field of characteristic zero is extended to define descent algebras over a field of prime characteristic. Some basic algebraic structure of the latter, including its radical and irreducible modules, is…

Let $k$ be a field of characteristic $p>0$ not necessarily perfect. Using Berthelot's theory of arithmetic $\mathcal{D}$-modules, we construct a $p$-adic formalism of Grothendieck's six operations for realizable $k$-schemes of finite type.

Let k be a field and n > 0. There exists a DG k-module (V,d) and various approximations d + t d_1 + t^2 d_2 + ... + t^n d_n to a differential on V[[t]], one of which is a non-trivial deformation, another is obstructed, and another is…

We study natural D-modules on the moduli stack of elliptic curves over a field of characteristic zero. We use this to produce an algebro-geometric version of the algebra of higher depth mock modular forms, studied from a Conformal Field…

The paper presents two new results concerning the varieties of Leibnitz algebras. In the case of prime characteristic p of the base field constructed example not nilpotent variety of Leibnitz algebras satisfying an Engel condition order p.…

Let k be a field, let R be a ring of polynomials in a finite number of variables over k, let D be the ring of k-linear differential operators of R and let f be a non-zero element of R. It is well-known that R_f, with its natural D-module…

We introduce a concept of a fractional-derivatives series and prove that any linear partial differential equation in two independent variables has a fractional-derivatives series solution with coefficients from a differentially closed field…

The question whether non-isomorphic finite $p$-groups can have isomorphic modular group algebras was recently answered in the negative by Garc\'ia-Lucas, Margolis and del R\'io [J. Reine Angew. Math. 783 (2022), pp. 269-274]. We embed these…

In this paper we focus on pairs consisting of the affine $N$-space and multiideals with a positive exponent. We introduce a method "lifting to characteristic 0" which is a kind of the inversion of "modulo p reduction". By making use of it,…

Let $L$ be a finite-dimensional Lie algebra over a field of non-zero characteristic and let $S$ be a subalgebra. Suppose that $X$ is a finite set of finite-dimensional $L$-modules. Let $D$ be the category of all finite-dimensional…

Let $\phi(x) = x^d + c$ be an integral polynomial of degree at least 2, and consider the sequence $(\phi^n(0))_{n=0}^\infty$, which is the orbit of $0$ under iteration by $\phi$. Let $D_{d,c}$ denote the set of positive integers $n$ for…