unQVNTS (Vermont) Spring 2018

Sophie Gonet, Elliptic curves over the complex numbers

We introduce the projective plane, elliptic curves as abelian varieties, and the projective embedding of a complex 1-dimensional torus into projective space via the Weierstrass p-function. We then briefly discuss the endomorphism ring of a complex 1-dimensional torus. Time permitting, Christelle will discuss projective embeddings via theta functions.

Lloyd Simons, A brisk introduction to abelian varieties over the complex numbers

Taylor Dupuy, Complex tori vs abelian varieties

Taylor Dupuy, Triple product formula and universal vectorial extensions

Jonathan Sands, The Mordell-Lang conjecture on subvarieties of abelian varieties

Following Mazur's expository article from the year 2000, we present a brief outline of the proof of the Mordell-Lang conjecture based on work of Vojta, Faltings, Buium, Voloch, and others. A version of this states that under standard hypotheses, a subvariety of an abelian variety is in fact a translate of an abelian subvariety. A consequence is the classical Mordell conjecture that a smooth projective irreducible algebraic curve over a number field is the Zariski closure of its rational points over some larger number field if and only if the curve has genus 0 or 1. For the purposes of exposition, we will focus on fields of characteristic 0.

David Dummit, Complex Multiplication Abelian Varieties

This talk will review the definitions and some of the basic information for "CM abelian varieties" - those abelian varieties 'with complex multiplication(s)'. This generalizes the results in the case of elliptic curves (the abelian varieties of dimension 1) and the talk will indicate some of the results for CM elliptic curves in greater detail as motivation for the higher dimensional case, in particular attempting to address the question "why would one care about CM abelian varieties?"