Georgia Institute of Technology College of Sciences

In the last decades, path following methods have become a very popular strategy to solve systems of polynomial equations. Many of the advances are due to the correct understanding of the geometrical properties of an algebraic object, the so-called solution variety for polynomial system solving. I summarize here some of the most recent advances in the understanding of this object, focusing also on the certifcation and complexity of the numerical procedures involved in path following methods.

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.

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.

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.