141 research outputs found
Effective Invariant Theory of Permutation Groups using Representation Theory
Using the theory of representations of the symmetric group, we propose an
algorithm to compute the invariant ring of a permutation group. Our approach
have the goal to reduce the amount of linear algebra computations and exploit a
thinner combinatorial description of the invariant ring.Comment: Draft version, the corrected full version is available at
http://www.springer.com
Black Box Galois Representations
We develop methods to study -dimensional -adic Galois representations
of the absolute Galois group of a number field , unramified outside a
known finite set of primes of , which are presented as Black Box
representations, where we only have access to the characteristic polynomials of
Frobenius automorphisms at a finite set of primes. Using suitable finite test
sets of primes, depending only on and , we show how to determine the
determinant , whether or not is residually reducible, and
further information about the size of the isogeny graph of whose
vertices are homothety classes of stable lattices. The methods are illustrated
with examples for , and for imaginary quadratic, being
the representation attached to a Bianchi modular form.
These results form part of the first author's thesis.Comment: 40 pages, 3 figures. Numerous minor revisions following two referees'
report
Numerical verification of the Cohen-Lenstra-Martinet heuristics and of Greenberg's -rationality conjecture
In this paper we make a series of numerical experiments to support
Greenberg's -rationality conjecture, we present a family of -rational
biquadratic fields and we find new examples of -rational multiquadratic
fields. In the case of multiquadratic and multicubic fields we show that the
conjecture is a consequence of the Cohen-Lenstra-Martinet heuristic and of the
conjecture of Hofmann and Zhang on the -adic regulator, and we bring new
numerical data to support the extensions of these conjectures. We compare the
known algorithmic tools and propose some improvements
Mod-2 dihedral Galois representations of prime conductor
For all odd primes N up to 500000, we compute the action of the Hecke
operator T_2 on the space S_2(Gamma_0(N), Q) and determine whether or not the
reduction mod 2 (with respect to a suitable basis) has 0 and/or 1 as
eigenvalues. We then partially explain the results in terms of class field
theory and modular mod-2 Galois representations. As a byproduct, we obtain some
nonexistence results on elliptic curves and modular forms with certain mod-2
reductions, extending prior results of Setzer, Hadano, and Kida.Comment: 16 pages; v2: final submitted versio
On the existence of abelian surfaces with everywhere good reduction
Let be a positive discriminant such that has narrow class one, and an abelian surface of -type with everywhere good reduction. Assuming that is modular, we show that is either an -surface or is a base change from of an abelian surface such that , except for and . In the latter case, we show that there are indeed abelian surfaces with everywhere good reduction over for and , which are non-isogenous to their Galois conjugates. These are the first known such examples
A table of elliptic curves over the cubic field of discriminant -23
Let F be the cubic field of discriminant -23 and O its ring of integers. Let
Gamma be the arithmetic group GL_2 (O), and for any ideal n subset O let
Gamma_0 (n) be the congruence subgroup of level n. In a previous paper, two of
us (PG and DY) computed the cohomology of various Gamma_0 (n), along with the
action of the Hecke operators. The goal of that paper was to test the
modularity of elliptic curves over F. In the present paper, we complement and
extend this prior work in two ways. First, we tabulate more elliptic curves
than were found in our prior work by using various heuristics ("old and new"
cohomology classes, dimensions of Eisenstein subspaces) to predict the
existence of elliptic curves of various conductors, and then by using more
sophisticated search techniques (for instance, torsion subgroups, twisting, and
the Cremona-Lingham algorithm) to find them. We then compute further invariants
of these curves, such as their rank and representatives of all isogeny classes.
Our enumeration includes conjecturally the first elliptic curves of ranks 1 and
2 over this field, which occur at levels of norm 719 and 9173 respectively
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