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Series: Algebra Seminar

The rational solutions to the equation describing an elliptic curve form a finitely generated abelian group, also known as the Mordell-Weil group. Detemining the rank of this group is one of the great unsolved problems in mathematics. The Shafarevich-Tate group of an elliptic curve is an important invariant whose conjectural finiteness can often be used to determine the generators of the Mordell-Weil group. In this talk, we first introduce the definition of the Shafarevich-Tate group. We then discuss the theory of visibility, initiated by Mazur, by means of which non-trivial elements of the Shafarevich-Tate group of an elliptic curve an be 'visualized' as rational points on an ambient curve. Finally, we explain how this theory can be used to give theoretical evidence for the celebrated Birch and Swinnerton-Dyer Conjecture.

Series: Algebra Seminar

Come and see!

Series: Algebra Seminar

For q a power of a prime, consider the ring \mathbb{F}_q[T].
Due to the many similarities between \mathbb{F}_q[T] and the
ring of integers \mathbb{Z}, we can define for
\mathbb{F}_q[T] objects that are analogous to elliptic curves,
modular forms, and modular curves. In particular, for
\mathfrak{p} a prime ideal in \mathbb{F}_q[T], we can define
the Drinfeld modular curve X_0(\mathfrak{p}), and study the
reduction modulo \mathfrak{p} of its Weierstrass points, as is
done in the classical case by Rohrlich, and Ahlgren and Ono. In
this talk we will present some partial results in this
direction, defining all necessary objects as we go. The first 20
minutes should be accessible to graduate students interested in
number theory.

Series: Algebra Seminar

We will discuss several instances of sequences of complex manifolds
X_n whose Betti numbers b_i(X_n) converge, when properly scaled,
to a
limiting distribution. The varieties considered have Betti
numbers
which are described in a combinatorial way making their study
possible. Interesting examples include varieties X for which
b_i(X)
is the i-th coefficient of the reliability polynomial of an
associated graph.

Series: Algebra Seminar

Which commutative groups can occur as the ideal class group (or
"Picard group") of some Dedekind domain? A number theorist naturally
thinks of the case of integer rings of number fields, in which the
class group must be finite and the question of which finite groups
occur is one of the deepest in algebraic number theory. An algebraic
geometer naturally thinks of affine algebraic curves, and in
particular, that the Picard group of the standard affine ring of an
elliptic curve E over C is isomorphic to the group of rational points
E(C), an uncountably infinite (Lie) group. An arithmetic geometer
will be more interested in Mordell-Weil groups, i.e., E(k) when k is a
number field -- again, this is one of the most notorious problems in
the field. But she will at least be open to the consideration of E(k)
as k varies over all fields.
In 1966, L.E. Claborn (a commutative algebraist) solved the "Inverse
Picard Problem": up to isomorphism, every
commutative group is the Picard group of some Dedekind domain. In the
1970's, Michael Rosen (an arithmetic geometer) used elliptic curves to
show that any countable commutative group can serve as the class group
of a Dedekind domain. In 2008 I learned about Rosen's work and showed
the following theorem: for every commutative group G there is a field
k, an elliptic curve E/k and a Dedekind domain R which is an overring
of the standard affine ring k[E] of E -- i.e., a domain in between
k[E] and its fraction field k(E) -- with ideal class group isomorphic
to G. But being an arithmetic geometer, I cannot help but ask about
what happens if one is not allowed to pass to an overring: which
commutative groups are of the form E(k) for some field k and some
elliptic curve E/k? ("Inverse Mordell-Weil Problem")
In this talk I will give my solution to the "Inverse Picard Problem"
using elliptic curves and give a conjectural answer to the "Inverse
Mordell-Weil Problem". Even more than that, I can (and will, time
permitting) sketch a proof of my conjecture, but the proof will
necessarily gloss over a plausible technicality about Mordell-Weil
groups of "arithmetically generic" elliptic curves -- i.e., I do not
in fact know how to do it. But the technicality will, I think, be of
interest to some of the audience members, and of course I am (not so)
secretly hoping that someone there will be able to help me overcome
it.

Series: Algebra Seminar

Given a nonconstant holomorphic map f: X \to Y between compact
Riemann surfaces, one of the first objects we learn to construct is its
ramification divisor R_f, which describes the locus at which f fails to be
locally injective. The divisor R_f is a finite formal linear combination of
points of X that is combinatorially constrained by the Hurwitz formula.
Now let k be an algebraically closed field that is complete with respect to
a nontrivial non-Archimedean absolute value. For example, k = C_p. Here the
role of a Riemann surface is played by a projective Berkovich analytic
curve. As these curves have many points that are not algebraic over k, some
new (non-algebraic) ramification behavior appears for maps between them. For
example, the ramification locus is no longer a divisor, but rather a closed
analytic subspace. The goal of this talk is to introduce the Berkovich
projective line and describe some of the topology and geometry of the
ramification locus for self-maps f: P^1 \to P^1.

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.