相关论文: On ordinary forms and ordinary Galois representati…
In this paper, we present methods to simplify reducible linear differential systems before solving. Classical integrals appear naturally as solutions of such systems. We will illustrate the methods developed in a previous paper on several…
We present a shortcut to the familiar characterizations of finite Galois extensions, based on an idea from an earlier Monthly note by Sonn and Zassenhaus.
The article is devoted to introduce and investigation of new numeral systems. These representations are generalizations of classical representations of real numbers. The main properties of investigating representations are described. The…
Let K be an arbitrary number field, and let rho: Gal(Kbar/K) -> GL_2(E) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of rho. When K is totally real and rho is…
We compute modular Galois representations associated with a newform $f$, and study the related problem of computing the coefficients of $f$ modulo a small prime $\ell$. To this end, we design a practical variant of the complex…
In this article, I discuss material which is related to the recent proof of Fermat's Last Theorem: elliptic curves, modular forms, Galois representations and their deformations, Frey's construction, and the conjectures of Serre and of…
Paper has been accepted for publication in the Journal f\"ur die reine und Angewandte Mathematik. This version contains the corrections and additional references made in the Galley proofs.
We prove some existence results on parameterized strongly normal extensions for logarithmic equations. We generalize a result in [Wibmer, Existence of d-parameterized Picard-Vessiot extensions over fields with algebraically closed…
We show how the output of the algorithm to compute modular Galois representations described in our previous article can be certified. We have used this process to compute certified tables of such Galois representations obtained thanks to an…
Let $K$ be an imaginary quadratic field. For an order $\mathcal{O}$ in $K$ and a positive integer $N$, let $K_{\mathcal{O},\,N}$ be the ray class field of $\mathcal{O}$ modulo $N\mathcal{O}$. We deal with various subjects related to…
In their recent letter, Chaichian et.~al. present a Lagrangian for a massive ($m$) point particle on the plane, with which they claim to realize anyon statistics. However, we find that there are some inaccuracies in their formulation and,…
We extend our method to compute division polynomials of Jacobians of curves over Q to curves over Q(t), in view of computing mod ell Galois representations occurring in the \'etale cohomology of surfaces over Q. Although the division…
This note provides an insight to the diophantine properties of abelian surfaces with quaternionic multiplication over number fields. We study the fields of definition of the endomorphisms on these abelian varieties and the images of the…
The development of high-degree interpolation polynomials which use the values of the function and its subsequent derivatives is reformulated. Also, we present a variant of new formula in barycentric form.
The aim of this paper is to introduce the degenerate generalized Laguerre polynomials as the degenerate version of the generalized Laguerre polynomials and to derive some properties related to those polynomials and Lah numbers, including an…
These notes concern linear transformations on R^n and C^n, exponentials of linear transformations, and some related geometric questions.
This is a book about computational aspects of modular forms and the Galois representations attached to them. The main result is the following: Galois representations over finite fields attached to modular forms of level one can, in almost…
We construct infinitely ramified Galois representations $\rho$ such that the $a_l (\rho)$'s have distributions in contrast to the statements of Sato-Tate, Lang-Trotter and others. Using similar methods we deform a residual Galois…
In this paper we explicitly compute mod-l Galois representations associated to modular forms. To be precise, we look at cases with l<=23 and the modular forms considered will be cusp forms of level 1 and weight up to 22. We present the…
In this paper I classify, up to Cremona transformations, the Galois cover of the plane with Galois group of the form $\mathbb Z_2^r$.