On the Common Index Divisors of a Dihedral Field of Prime Degree

A criterion for a prime to be a common index divisor of a dihedral field of prime degree is given. This criterion is used to determine the index of families of dihedral fields of degrees 5 and 7

Quaternionic Artin representations of Q

Isomorphism classes of dihedral Artin representations of ℚ can be counted asymptotically using Siegel's asymptotic averages of class numbers of binary quadratic forms. Here we consider the analogous problem for quaternionic representations. While an asymptotic formula is out of our reach in this case, we show that the asymptotic behaviour in the two cases is quite different

On solving systems of random linear disequations

An important subcase of the hidden subgroup problem is equivalent to the shift problem over abelian groups. An efficient solution to the latter problem would serve as a building block of quantum hidden subgroup algorithms over solvable groups. The main idea of a promising approach to the shift problem is reduction to solving systems of certain random disequations in finite abelian groups. The random disequations are actually generalizations of linear functions distributed nearly uniformly over those not containing a specific group element in the kernel. In this paper we give an algorithm which finds the solutions of a system of N random linear disequations in an abelian p-group A in time polynomial in N, where N=(log|A|)^{O(q)}, and q is the exponent of A.Comment: 13 page

Computing images of Galois representations attached to elliptic curves

Let E be an elliptic curve without complex multiplication (CM) over a number field K, and let G_E(ell) be the image of the Galois representation induced by the action of the absolute Galois group of K on the ell-torsion subgroup of E. We present two probabilistic algorithms to simultaneously determine G_E(ell) up to local conjugacy for all primes ell by sampling images of Frobenius elements; one is of Las Vegas type and the other is a Monte Carlo algorithm. They determine G_E(ell) up to one of at most two isomorphic conjugacy classes of subgroups of GL_2(Z/ell Z) that have the same semisimplification, each of which occurs for an elliptic curve isogenous to E. Under the GRH, their running times are polynomial in the bit-size n of an integral Weierstrass equation for E, and for our Monte Carlo algorithm, quasi-linear in n. We have applied our algorithms to the non-CM elliptic curves in Cremona's tables and the Stein--Watkins database, some 140 million curves of conductor up to 10^10, thereby obtaining a conjecturally complete list of 63 exceptional Galois images G_E(ell) that arise for E/Q without CM. Under this conjecture we determine a complete list of 160 exceptional Galois images G_E(ell) the arise for non-CM elliptic curves over quadratic fields with rational j-invariants. We also give examples of exceptional Galois images that arise for non-CM elliptic curves over quadratic fields only when the j-invariant is irrational.Comment: minor edits, 47 pages, to appear in Forum of Mathematics, Sigm