- You are here:
- GT Home
- Home
- News & Events

Series: Algebra Seminar

An elliptic curve over the integer ring of a p-adic field whose
special fiber is ordinary has a canonical line contained in its
p-torsion. This fact has many arithmetic applications: for instance,
it shows that there is a canonical partially-defined section of the
natural map of modular curves X_0(Np) -> X_0(N). Lubin was the first
to notice that elliptic curves with "not too supersingular" reduction
also contain a canonical order-p subgroup. I'll begin the talk by
giving an overview of Lubin and Katz's theory of the canonical
subgroup of an elliptic curve. I'll then explain one approach to
defining the canonical subgroup of any abelian variety (even any
p-divisible group), and state a very general existence result. If
there is time I'll indicate the role tropical geometry plays in its
proof.

Series: Algebra Seminar

Grothendieck's anabelian conjectures say that hyperbolic curves over
certain fields should be K(pi,1)'s in algebraic geometry. It follows
that points on such a curve are conjecturally the sections of etale pi_1
of the structure map. These conjectures are analogous to equivalences
between fixed points and homotopy fixed points of Galois actions on
related topological spaces. This talk will start with an introduction to
Grothendieck's anabelian conjectures, and then present a 2-nilpotent
real section conjecture: for a smooth curve X over R with negative Euler
characteristic, pi_0(X(R)) is determined by the maximal 2-nilpotent
quotient of the fundamental group with its Galois action, as the kernel
of an obstruction of Jordan Ellenberg. This implies that the set of real
points equipped with a real tangent direction of the smooth
compactification of X is determined by the maximal 2-nilpotent quotient
of Gal(C(X)) with its Gal(R) action, showing a 2-nilpotent birational
real section conjecture.

Series: Algebra Seminar

In classical holomorphic dynamics, rational self-maps of the
Riemann sphere whose critical points all have finite forward orbit under iteration are known as post-critically finite (PCF) maps. A deep result of Thurston shows that if one excludes examples arising from endomorphisms of elliptic curves, then PCF maps are in some sense sparse, living in a countable union of zero-dimensional subvarieties of the appropriate moduli space (a result offering dubious comfort to number theorists, who tend to work over countable fields). We show that if one restricts attention to polynomials, then the set of PCF points in moduli space is actually a set of algebraic points of bounded height. This allows us to give an elementary proof of the appropriate part of Thurston's result, but it also provides an effective means of listing all PCF polynomials of a given degree, with coefficients of bounded algebraic degree (up to the appropriate sense of equivalence).

Series: Algebra Seminar

Let k be a field (not of characteristic 2) and let t be an
indeterminate. Legendre's elliptic curve is the elliptic curve over k(t)
defined by y^2=x(x-1)(x-t). I will discuss the arithmetic of this curve
(group of solutions, heights, Tate-Shafarevich group) over the
extension fields k(t^{1/d}). I will also mention several variants and
open problems which would make good thesis topics.

Series: Algebra Seminar

We begin this talk by discussing four different problems that arenumber theoretic or combinatorial in nature. Two of these problems remainopen and the other two have known solutions. We then explain how these seeminglyunrelated problems are connected to each other. To disclose a little more information,one of the problems with a known solution is the following: Is it possible to find anirrational number $q$ such that the infinite geometric sequence $1, q, q^{2}, \dots$has 4 terms in arithmetic progression?

Series: Algebra Seminar

I will discuss some recent results, obtained jointly with Sam Payne and Joe Rabinoff, on tropicalizations of elliptic curves.

Series: Algebra Seminar

I will discuss D-module type invariants on hyperplane arrangements and
their relation to the intersection lattice (when known).

Series: Algebra Seminar

A smooth quartic curve in the projective plane has 36 representations as a symmetric determinant of
linear forms and 63 representations as a sum of three squares. We report on joint work with Daniel Plaumann and Cynthia Vinzant regarding the explicit computation of these objects. This lecture offers a gentle introduction to the 19th century theory of plane quartics from the current perspective of convex algebraic geometry.

Series: Algebra Seminar

In this talk we recall some modular techniques (chinese remaindering,rational reconstruction, etc.) that play a crucial role in manycomputer algebra applications, e.g., for solving linear systems over arational function field, for evaluating determinants symbolically,or for obtaining results by ansatz ("guessing"). We then discuss howmuch our recent achievements in the areas of symbolic summation andintegration and combinatorics benefited from these techniques.

Series: Algebra Seminar

The polynomials mentioned in the title were introduced
by Cauchy and Liouville in 1839 in connection with early attempts
at a proof of Fermat's Last Theorem. They were subsequently studied
by Mirimanoff who in 1903 conjectured their irreducibility over the
rationals. During the past fifteen years it has become clear that
Mirimanoff's conjecture is closely related to properties of certain
special functions and to some deep results in diophantine approximation.
Apparently, there is also a striking connection to hierarchies of certain
evolution equations (which this speaker is not qualified to address). We
will present and discuss a number of recent results on this problem.