# Joseph Rabinoff : From Diophantine equations to *p*-adic analytic geometry

A Diophantine equation is a polynomial equation in several variables,
generally with integer coefficients, like *x ^{3} + y^{3} = z^{3}*. Provably finding
all integer solutions of a Diophantine equation is a storied mathematical
problem that is easy to state and notoriously difficult to solve. The method
of Chabauty--Coleman is one particularly successful technique for ruling out
extraneous solutions of a certain class of Diophantine equations. The method
is

*p*-adic in nature, and involves producing

*p*-adic analytic functions that vanish on all integer-valued solutions. I will discuss work with Katz and Zureick-Brown on finding uniform bounds on the number of rational points on a curve of fixed genus, defined over a number field, subject to a (conjecturally weak) restriction on its Jacobian. The same technique also makes progress on the uniform Manin-Mumford conjecture on the size of torsion packets on curves of fixed genus.

**Category**: Presentations**Duration**: 01:24:41**Date**: September 4, 2018 at 3:10 PM-
**Tags:**seminar, Department of Mathematics Seminar

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