34 research outputs found
Rank distribution in a family of cubic twists
In 1987, Zagier and Kramarz published a paper in which they presented
evidence that a positive proportion of the even-signed cubic twists of the
elliptic curve should have positive rank. We extend their data,
showing that it is more likely that the proportion goes to zero
Regulators of rank one quadratic twists
We investigate the regulators of elliptic curves with rank 1 in some families
of quadratic twists of a fixed elliptic curve. In particular, we formulate some
conjectures on the average size of these regulators. We also describe an
efficient algorithm to compute explicitly some of the invariants of an odd
quadratic twist of an elliptic curve (regulator, order of the Tate-Shafarevich
group, etc.) and we discuss the numerical data that we obtain and compare it
with our predictions.Comment: 28 pages with 32 figure
Hall-Littlewood polynomials and Cohen-Lenstra heuristics for Jacobians of random graphs
Cohen-Lenstra heuristics for Jacobians of random graphs give rise to random
partitions. We connect these random partitions to the Hall-Littlewood
polynomials of symmetric function theory, and use this connection to give
combinatorial proofs of properties of these random partitions. In addition, we
use Markov chains to give an algorithm for generating these partitions.Comment: 10 page
-Torsion Points In Finite Abelian Groups And Combinatorial Identities
The main aim of this article is to compute all the moments of the number of
-torsion elements in some type of nite abelian groups. The averages
involved in these moments are those de ned for the Cohen-Lenstra heuristics for
class groups and their adaptation for Tate-Shafarevich groups. In particular,
we prove that the heuristic model for Tate-Shafarevich groups is compatible
with the recent conjecture of Poonen and Rains about the moments of the orders
of -Selmer groups of elliptic curves. For our purpose, we are led to de ne
certain polynomials indexed by integer partitions and to study them in a
combinatorial way. Moreover, from our probabilistic model, we derive
combinatorial identities, some of which appearing to be new, the others being
related to the theory of symmetric functions. In some sense, our method
therefore gives for these identities a somehow natural algebraic context.Comment: 24 page
Discretisation for odd quadratic twists
The discretisation problem for even quadratic twists is almost understood,
with the main question now being how the arithmetic Delaunay heuristic
interacts with the analytic random matrix theory prediction. The situation for
odd quadratic twists is much more mysterious, as the height of a point enters
the picture, which does not necessarily take integral values (as does the order
of the Shafarevich-Tate group). We discuss a couple of models and present data
on this question.Comment: To appear in the Proceedings of the INI Workshop on Random Matrix
Theory and Elliptic Curve