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Jordan S. Ellenberg

  1. Random Dieudonne modules, random p-divisible groups, and random curves over finite fields.

    Чт, 01/05/2012 - 15:01 — SomNambul
    Authors: Jordan S. Ellenberg, Bryden Cais, David Zureick-Brown
    Subjects: Number Theory
    Abstract

    We 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.

  2. Upper bounds for the growth of Mordell-Weil ranks in pro-p towers of Jacobians.

    Втр, 01/26/2010 - 16:01 — SomNambul
    Authors: Jordan S. Ellenberg
    Subjects: Number Theory
    Abstract

    We 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).

  3. Linnik's ergodic method and the distribution of integer points on spheres.

    Чт, 01/07/2010 - 16:01 — SomNambul
    Authors: Jordan S. Ellenberg, Akshay Venkatesh, Philippe Michel
    Subjects: Number Theory
    Abstract

    We 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).

  4. Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields.

    Чт, 12/03/2009 - 16:01 — SomNambul
    Authors: Jordan S. Ellenberg, Akshay Venkatesh, Craig Westerland
    Subjects: Number Theory
    Abstract

    We 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.

  5. Every curve is a Teichmuller curve.

    Пт, 09/11/2009 - 19:12 — SomNambul
    Authors: D. B. McReynolds, Jordan S. Ellenberg
    Subjects: Geometric Topology
    Abstract

    We prove that every algebraic curve X defined over the algebraic closure of
    the rationals is birational over the complex numbers to a Teichmuller curve.

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