Now showing items 1-14 of 14

• #### Computing congruences of modular forms and Galois representations modulo prime powers ﻿

(2010)
Article
Open Access
This article starts a computational study of congruences of modular forms and modular Galois representations modulo prime powers. Algorithms are described that compute the maximum integer modulo which two monic coprime ...
• #### Congruences between modular forms and lowering the level mod l^n ﻿

(2009)
Article
Open Access
In this article we study the behavior of inertia groups for modular Galois mod ℓn representations and in some cases we give a generalization of Ribet’s lowering the level result.
• #### Differential galois theory and non-integrability of planar polynomial vector fields ﻿

(2018-02-26)
Article
Restricted access - publisher's policy
We study a necessary condition for the integrability of the polynomials vector fields in the plane by means of the differential Galois Theory. More concretely, by means of the variational equations around a particular ...
• #### Equations of hyperelliptic Shimura curves ﻿

(2012)
Article
Open Access
We describe a method for computing equations of hyperelliptic Shimura curves attached to indefinite quaternion algebras over Q and Atkin–Lehner quotients of them. It exploits Cerednik–Drinfeld 's non-archimedean uniformization ...
• #### From Galois to Hopf Galois: theory and practice ﻿

(2015-09)
Article
Open Access
Hopf Galois theory expands the classical Galois theory by con- sidering the Galois property in terms of the action of the group algebra k [ G ] on K/k and then replacing it by the action of a Hopf algebra. We review the ...
• #### Generic Galois extensions for SL_2(F_5) over Q ﻿

(Mathematical Publishing, 2007)
Article
Open Access
Let $G_n$ be a double cover of either the alternating group $A_n$ or the symmetric group $S_n$, and let $G_{n-1}$ be the corresponding double cover of $A_{n-1}$ or $S_{n-1}$. For every odd $n\geq 3$ and every field $k$ of ...
• #### On elliptic Galois representations and genus-zero modular units ﻿

(2006-07-03)
Article
Open Access
Given an odd prime \,$p$\, and a representation $\varrho$\, of the absolute Galois group of a number field $k$ onto $\mathrm{PGL}_2(\mathbb{F}_p)$ with cyclotomic determinant, the moduli space of elliptic curves defined ...
• #### On fields of definition of torsion points of elliptic curves with complex multiplication ﻿

(2011-06)
Article
Open Access
For any elliptic curve E defined over the rationals with complex multiplication (CM) and for every prime p, we describe the image of the mod p Galois representation attached to E. We deduce information about the field of ...
• #### On the integrability of polynomial fields in the plane by means of Picard-Vessiot theory ﻿

(2012-01-12)
External research report
Open Access
We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results ...
• #### Ordinary CM forms and local Galois representations ﻿

(Universitat Politècnica de Catalunya, 2009-07)
Master thesis
Open Access
"The p-adic Galois representation \rho_f attached to a p-ordinary newform f of weight k>1 is known to be reducible when restricted to a decomposition group D_p at p. Ralph Greenberg asked for a characterization of those f ...
• #### Polynomials in finite geometry ﻿

(Università degli studi di Napoli. Dipartimento di Matematica, 1999-11-01)
Part of book or chapter of book
Open Access
• #### Rational points on twists of X0(63) ﻿

(2006-02-07)
Article
Open Access
Let $\varrho\colon G_\mathbb{Q}\longrightarrow PGL_2(\mathbb{F}_p)$ be a Galois representation with cyclotomic determinant, and let $N>1$ be an integer that is square mod $p$. There exist two twisted modular curves ...
• #### The Hopf Galois property in subfield lattices ﻿

(2016-01-01)
Article
Open Access
Let K/k be a finite separable extension, n its degree and (K) over tilde /k its Galois closure. For n <= 5, Greither and Pareigis show that all Hopf Galois extensions are either Galois or almost classically Galois and they ...
• #### The polynomial method in Galois geometries ﻿

(Nova Science Publishers, 2012)
Part of book or chapter of book
Restricted access - publisher's policy
The polynomial method refers to the application of polynomials to combinatorial problems. The method is particularly effective for Galois geometries and a number of problems and conjectures have been solved using the ...