Georgia Institute of Technology College of Sciences

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

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.

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.

We will explicitly construct twists of elliptic curves with an arbitrarily large set of integral points over $\mathbb{F}_q(t)$. As a motivation to our main result, we will discuss a conjecture of Vojta-Lang concerning the behavior of integral points on varieties of log-general type over number fields and present a natural translation to the function field setting. We will use our construction to provide an isotrivial counter-example to this conjecture. We will also show that our main result also provides examples of elliptic curves with arbitrarily large set of independent points and of function fields with large $m$-class rank.