相关论文: On ordinary forms and ordinary Galois representati…

In this paper we present experimental ways of evaluating Ramanujan`s quantities which as someone can see are related with algebraic numbers. The good thing with algebraic numbers is that can be found in a closed form, from there…

In recent work, the authors proved a general result on lifting $G$-irreducible odd Galois representations $\mathrm{Gal}(\overline{F}/F) \to G(\overline{\mathbb{F}}_{\ell})$, with $F$ a totally real number field and $G$ a reductive group, to…

The arithmetic of Hilbert modular forms has been extensively studied under the assumption that the forms concerned are "paritious" -- all the components of the weight are congruent modulo 2. In contrast, non-paritious Hilbert modular forms…

This paper is a short introduction to orthogonal polynomials, both the general theory and some special classes. It ends with some remarks about the usage of computer algebra for this theory.

The present article is devoted to the description of further investigations of the author of this article. These investigations (in terms of various representations of real numbers) include the generalized Salem functions and…

In this work we provide a level raising theorem for $\mod \lambda^n$ modular Galois representations. It allows one to see such a Galois representation that is modular of level $N$, weight 2 and trivial Nebentypus as one that is modular of…

We construct plane models of the modular curve $X_H(\ell)$, and use their explicit equations to compute Galois representations associated to modular forms for values of $\ell$ that are significantly higher than in prior works.

Formulations of some Grassmann-valued systems of ordinary differential equations invariant under (infinitesimal) supersymmetry transformations, including $N$-superspace extended types, are reviewed and discussed, with use of superfields.…

The study of arboreal Galois representations (that is, Galois groups arising from iteration of polynomial and rational functions) originated with work of Odoni in the 1980s. Beginning in the early 2000s it underwent a period of renewed…

Differential equations have arithmetic analogues in which derivatives are replaced by Fermat quotients; these analogues are called arithmetic differential equations and the present paper is concerned with the "linear" ones. The equations…

To a hyperbolic smooth curve defined over a number-field one naturally associates an "anabelian" representation of the absolute Galois group of the base field landing in outer automorphism group of the algebraic fundamental group. In this…

We explain the use and set grounds about applicability of algebraic transformations of arithmetic hypergeometric series for proving Ramanujan's formulae for $1/\pi$ and their generalisations.

Explicit formulas involving a generalized Ramanujan sum are derived. An analogue of the prime number theorem is obtained and equivalences of the Riemann hypothesis are shown. Finally, explicit formulas of Bartz are generalized.

For many finite groups, the Inverse Galois Problem can be approached through modular/automorphic Galois representations. This is a report explaining the basic strategy, ideas and methods behind some recent results. It focusses mostly on the…

A new kind of deformed calculus was introduced recently in studying of parabosonic coordinate representation. Based on this deformed calculus, a new deformation of Legendre polynomials is proposed in this paper, some properties and…

In the present paper we generalize the Eulerian numbers (also of the second and third orders). The generalization is connected with an autonomous first-order differential equation, solutions of which are used to obtain integral…

This work is a collection of old and new aplications of Galois cohomology to the clasification of algebraic and arithmetical objects.

Building on lifting results of Ramakrishna, Khare and Ramakrishna proved a purely Galois-theoretic level-raising theorem for two-dimensional odd representations of the Galois group of Q. In this paper, we generalize these techniques from…

We prove the existence of $\mathrm{GSpin}_{2n}$-valued Galois representations corresponding to cohomological cuspidal automorphic representations of certain quasi-split forms of $\mathrm{GSO}_{2n}$ under the local hypotheses that there is a…

Modular and quasimodular solutions of specific second order differential equation in the upper-half plane which originates from a study of supersingular j-invariants are given explicitly. A characterization of the differential equation is…