MATH Seminar

Title: Non-Archimedean and Tropical Techniques in Arithmetic Geometry
Defense: Algebra
Speaker: Jackson Morrow of Emory University
Contact: David Zureick-Brown, DAVID.M.BROWN.JR@GMAIL.COM
Date: 2020-03-03 at 4:00PM
Venue: MSC W303
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Let $K$ be a number field, and let $C/K$ be a curve of genus $g \geq 2$. In 1983, Faltings famously proved that the set $C(K)$ of $K$-rational points is finite. Given this, several questions naturally arise: \begin{enumerate} \item How does this finite quantity $\#C(K)$ varies in families of curves? \item What is the analogous result for degree $d>1$ points on $C$? \item What can be said about a higher dimensional variant of Faltings result? \end{enumerate} In this thesis, we will prove several results related to the above questions. \\ In joint with with J.~Gunther, we prove, under a technical assumption, that for each positive integer $d > 1$, there exists a number $B_d$ such that for each $g > d$, a positive proportion of odd hyperelliptic curves of genus $g$ over $\mathbb{Q}$ have at most $B_d$ ``unexpected'' points of degree $d$. Furthermore, we may take $B_2 = 24$ and $B_3 = 114$. \\ Our other results concern the strong Green--Griffiths--Lang--Vojta conjecture, which is the higher dimensional version of Faltings theorem (ne\'e the Mordell conjecture). More precisely, we prove the strong non-Archimedean Green--Griffiths--Lang--Vojta conjecture for closed subvarieties of semi-abelian varieties and for projective surfaces admitting a dominant morphism to an elliptic curve. \\ Time permitting, we will introduce a new construction of the non-Archimedean Kobayashi pseudo-metric for a Berkovich analytic space $X$ and provide evidence that our definition is the ``correct'' one. In particular, if this pseudo-metric is an actual metric on $X$, then it defines the Berkovich analytic topology and $X$ does not admit a non-constant morphism from any analytic tori.

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