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 $2$-dimensional $2$-adic Galois representations $\rho$ of the absolute Galois group of a number field $K$, unramified outside a known finite set of primes $S$ of $K$, 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 $K$ and $S$, we show how to determine the determinant $\det\rho$, whether or not $\rho$ is residually reducible, and further information about the size of the isogeny graph of $\rho$ whose vertices are homothety classes of stable lattices. The methods are illustrated with examples for $K=\mathbb{Q}$, and for $K$ imaginary quadratic, $\rho$ 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 pp-rationality conjecture

In this paper we make a series of numerical experiments to support Greenberg's $p$-rationality conjecture, we present a family of $p$-rational biquadratic fields and we find new examples of $p$-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 $p$-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 $D \le 2000$ be a positive discriminant such that $F = \mathbf{Q}(\sqrt{D})$ has narrow class one, and $A/F$ an abelian surface of ${\rm GL}_2$-type with everywhere good reduction. Assuming that $A$ is modular, we show that $A$ is either an $F$-surface or is a base change from $\mathbf{Q}$ of an abelian surface $B$ such that ${\rm End}_{\mathbf{Q}}(B) = \mathbf{Z}$, except for $D = 353, 421, 1321, 1597$ and $1997$. In the latter case, we show that there are indeed abelian surfaces with everywhere good reduction over $F$ for $D = 353, 421$ and $1597$, 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