Jordan S. Ellenberg
Random Dieudonne modules, random p-divisible groups, and random curves over finite fields.
Thu, 01/05/2012 - 15:01 — SomNambulAuthors: Jordan S. Ellenberg, Bryden Cais, David Zureick-Brown
Subjects: Number Theory
AbstractWe describe a probability distribution on isomorphism classes of principally
quasi-polarized p-divisible groups over a finite field k of characteristic p
which can reasonably be thought of as "uniform distribution," and we compute
the distribution of various statistics (p-corank, a-number, etc.) of
p-divisible groups drawn from this distribution. It is then natural to ask to
what extent the p-divisible groups attached to a randomly chosen hyperelliptic
curve (resp. curve, resp. abelian variety) over k are uniformly distributed in
this sense.Upper bounds for the growth of Mordell-Weil ranks in pro-p towers of Jacobians.
Tue, 01/26/2010 - 16:01 — SomNambulAuthors: Jordan S. Ellenberg
Subjects: Number Theory
AbstractWe study the variation of Mordell-Weil ranks in the Jacobians of curves in a
pro-p tower over a fixed number field. In particular, we show that under mild
conditions the Mordell-Weil rank of a Jacobian in the tower is bounded above by
a constant multiple of its dimension. In the case of the tower of Fermat
curves, we show that the constant can be taken arbitrarily close to 1. The main
result is used in the forthcoming paper of Guillermo Mantilla-Soler on the
Mordell-Weil rank of the modular Jacobian J(Np^m).Linnik's ergodic method and the distribution of integer points on spheres.
Thu, 01/07/2010 - 16:01 — SomNambulAuthors: Jordan S. Ellenberg, Akshay Venkatesh, Philippe Michel
Subjects: Number Theory
AbstractWe discuss Linnik's work on the distribution of integral solutions to
$x^2+y^2+z^2 =d$, as $d$ goes to infinity. We give an exposition of Linnik's
ergodic method; indeed, by using large-deviation results for random walks on
expander graphs, we establish a refinement of his equidistribution theorem. We
discuss the connection of these ideas with modern developments (ergodic theory
on homogeneous spaces, $L$-functions).Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields.
Thu, 12/03/2009 - 16:01 — SomNambulAuthors: Jordan S. Ellenberg, Akshay Venkatesh, Craig Westerland
Subjects: Number Theory
AbstractWe prove a homological stabilization theorem for Hurwitz spaces: moduli
spaces of branched covers of the complex projective line. This has the
following arithmetic consequence: let l>2 be prime and A a finite abelian
l-group. Then there exists Q = Q(A) so that, for q greater than Q and not
congruent to 1 modulo l, a positive fraction of quadratic extensions of F_q(t)
have the l-part of their class group isomorphic to A.Every curve is a Teichmuller curve.
Fri, 09/11/2009 - 19:12 — SomNambulAuthors: D. B. McReynolds, Jordan S. Ellenberg
Subjects: Geometric Topology
AbstractWe prove that every algebraic curve X defined over the algebraic closure of
the rationals is birational over the complex numbers to a Teichmuller curve.- 2 comments
- login or register to post comments
User login
