Nathan Jones

Almost all elliptic curves are Serre curves

Using a multidimensional large sieve inequality, we obtain a bound for the mean-square error in the Chebotarev theorem for division fields of elliptic curves that is as strong as what is implied by the Generalized Riemann Hypothesis. As an application we prove that, according to height, almost all elliptic curves are Serre curves, where a Serre curve is an elliptic curve whose torsion subgroup, roughly speaking, has as much Galois symmetry as possible.

Geometry and arithmetic of verbal dynamical systems on simple groups

We study dynamical systems arising from word maps on simple groups. We develop a geometric method based on the classical trace map for investigating periodic points of such systems. These results lead to a new approach to the search of Engel-like sequences of words in two variables which characterize finite solvable groups. They also give rise to some new phenomena and concepts in the arithmetic of dynamical systems.

A bound for the torsion conductor of a non-CM elliptic curve

Given a non-CM elliptic curve E over Q of discriminant ∆E , define the “torsion conductor” mE to be the smallest positive integer so that the Galois representation on the torsion of E has image π−1(Gal(Q(E[mE ])/Q)), where π denotes the natural projection GL2(Ẑ) → GL2(Z/mEZ). We show that, uniformly for semi-stable non-CM elliptic curves E over Q, one has

Missing class groups and class number statistics for imaginary quadratic fields

ABSTRACT The number of imaginary quadratic fields with class number h is of classical interest: Gauss’ class number problem asks for a determination of those fields counted by . The unconditional computation of for h ⩽ 100 was completed by Watkins, using ideas of Goldfeld and Gross–Zagier; Soundararajan has more recently made conjectures about the order of magnitude of as h → ∞ and determined its average order. In the present paper, we refine Soundararajan’s conjecture to a conjectural asymptotic formula for odd h by amalgamating the Cohen–Lenstra heuristic with an archimedean factor, and obtain an adelic, or global, refinement of the Cohen–Lenstra heuristic. We also consider the problem of determining the number of imaginary quadratic fields with class group isomorphic to a given finite abelian group G. Using Watkins’ tables, one can show that some abelian groups do not occur as the class group of any imaginary quadratic field (for instance, does not). This observation is explained in part by the Cohen–Lenstra heuristics, which have often been used to study the distribution of the p-part of an imaginary quadratic class group. We combine heuristics of Cohen–Lenstra together with our prediction for the asymptotic behavior of to make precise predictions about the asymptotic nature of the entire imaginary quadratic class group, in particular addressing the above-mentioned phenomenon of “missing” class groups, for the case of p-groups as p tends to infinity. Furthermore, conditionally on the Generalized Riemann Hypothesis, we extend Watkins’ data, tabulating for odd h ⩽ 106 and for G a p-group of odd order with |G| ⩽ 106. (In order to do this, we need to examine the class numbers of all negative prime fundamental discriminants − q, for q ⩽ 1.1881 × 1015.) The numerical evidence matches quite well with our conjectures, though there appears to be a small “bias” for class number divisible by powers of 3.

Elliptic aliquot cycles of fixed length

Silverman and Stange define the notion of an aliquot cycle of length L for a fixed elliptic curve E defined over the rational numbers, and conjecture an order of magnitude for the function which counts such aliquot cycles. In the present note, we combine heuristics of Lang-Trotter with those of Koblitz to refine their conjecture to a precise asymptotic formula by specifying the appropriate constant. We give a criterion for positivity of the conjectural constant, as well as some numerical evidence for our conjecture.