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

Prescribing valuations of the order of a point in the reductions of abelian varieties and tori

Let G be the product of an abelian variety and a torus defined over a number field K. Let R be a K-rational point on G of infinite order. Call n_R the number of connected components of the smallest algebraic K-subgroup of G to which R belongs. We prove that n_R is the greatest positive integer which divides the order of (R mod p) for all but finitely many primes p of K. Furthermore, let m>0 be a multiple of n_R and let S be a finite set of rational primes. Then there exists a positive Dirichlet density of primes p of K such that for every l in S the l-adic valuation of the order of (R mod p) equals v_l(m).Comment: Final version. To appear on Journal of Number Theor

Bounded gaps between primes in special sequences

We use Maynard's methods to show that there are bounded gaps between primes in the sequence $\{\lfloor n\alpha\rfloor\}$, where $\alpha$ is an irrational number of finite type. In addition, given a superlinear function $f$ satisfying some properties described by Leitmann, we show that for all $m$ there are infinitely many bounded intervals containing $m$ primes and at least one integer of the form $\lfloor f(q)\rfloor$ with $q$ a positive integer.Comment: 14 page