Related papers: Enumerating Galois representations in Sage

Fix a prime $p > 2$. Let $\rho : \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\mathbb{I})$ be the Galois representation coming from a non-CM irreducible component $\mathbb{I}$ of Hida's $p$-ordinary Hecke algebra. Assume the…

Suppose that F/F+ is a CM extension of number fields in which the prime p splits completely and every other prime is unramified. Fix a place w|p of F. Suppose that rbar : Gal(F-bar/F) -> GL_3(Fp-bar) is a continuous irreducible Galois…

Let $G$ be a finite group and let $ram^{t}(G)$ denote the minimal positive integer $n$ such that $G$ can be realized as the Galois group of a tamely ramified extension of $\mathbb{Q}$ ramified only at $n$ finite primes. Let $d(G)$ denote…

Let L/K be a finite Galois extension of complete local fields with finite residue fields and let G=Gal(L/K). Let G_1 and G_2 be the first and second ramification groups. Thus L/K is tamely ramified when G_1 is trivial and we say that L/K is…

Let $K$ be a complete, discretely valued field with finite residue field and $G_K$ its absolute Galois group. The subject of this note is the study of the set of positive integers $d$ for which there exists an absolutely irreducible…

Let $m$ be an integer greater than three and $\ell$ be an odd prime. In this paper, we prove that at least one of the following groups: $\mbox{P}\Omega^\pm_{2m}(\mathbb{F}_{\ell^s})$, $\mbox{PSO}^\pm_{2m}(\mathbb{F}_{\ell^s})$,…

A probabilistic representation for initial value semilinear parabolic problems based on generalized random trees has been derived. Two different strategies have been proposed, both requiring generating suitable random trees combined with a…

We consider the Galois representation associated with a finite slope $p$-adic family of modular forms. We prove that the Lie algebra of its image contains a congruence Lie subalgebra of a non-trivial level. We describe the largest such…

In this paper we extend a conjecture of Ash and Sinnott relating niveau one Galois representation to the mod p cohomology of congruence subgroups of SL(n,Z) to include Galois representations of higher niveau. We then present computational…

We provide two hybrid numeric-symbolic optimization algorithms, computing exact sums of nonnegative circuits (SONC) and sums of arithmetic-geometric-exponentials (SAGE) decompositions. Moreover, we provide a hybrid numeric-symbolic decision…

Representations of the non-semisimple superalgebra $gl(2|2)$ in the standard basis are investigated by means of the vector coherent state method and boson-fermion realization. All finite-dimensional irreducible typical and atypical…

The computation of the characters of supercuspidal representations of a p-adic group involves some 4th roots of unity whose values are defined in terms of orbits of the Galois group of a p-field on a root system. The part of the definition…

Let $K$ be any finite extension of $Q_{p}$, $L$ any finite Galois extension of $K$ and $E$ any finite large enough coefficient field containing $L$. We classify two-dimensional, F-semistable $E$-representations of $G_{K}$, by listing the…

An elementary approach is shown which derives the values of the Gauss sums over $\mathbb F_{p^r}$, $p$ odd, of a cubic character without using Davenport-Hasse's theorem. New links between Gauss sums over different field extensions are shown…

We introduce and develop a structure theory of a new class of noncommutative rings - Galois orders, that generalize classical orders in noncommutative rings. Galois orders realized as certain subrings of invariants in skew semigroup rings.…

We describe algorithms to compute fixed fields, splitting fields and towers of radical extensions without using polynomial factorisation in towers or constructing any field containing the splitting field, instead extending Galois group…

We introduce a heuristic prediction for the distribution of the isomorphism class of the Galois group of the maximal pro-p extension of Q unramified outside a "random" set of primes. This is guided by reasoning similar to that governing the…

We prove a version of the weight part of Serre's conjecture for mod $p$ Galois representations attached to automorphic forms on rank 2 unitary groups which are non-split at $p$. More precisely, let $F/F^+$ denote a CM extension of a totally…

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 integral polynomials…

We introduce a new notation for representing labeled regular bipartite graphs of arbitrary degree. Several enumeration problems for labeled and unlabeled regular bipartite graphs have been introduced. A general algorithm for enumerating all…