Uniform Bounds for Rational Points

Seminar: 
Colloquium
Event time: 
Wednesday, September 29, 2021 - 4:15pm
Speaker: 
Jordan Ellenberg
Speaker affiliation: 
University of Wisconsin
Event description: 

Abstract:  There is a long history in number theory of finiteness theorems for Diophantine equations, assertions that a certain equation has only finitely many solutions.  Even further, one can ask for effective finiteness theorems; a theorem is effective when it gives an upper bound for the size of any solution, reducing the problem of listing all solutions to a finite (in principle!) computation.  Effective finiteness theorems, unfortunately, are very hard to come by.  But there is an interesting interrmediate goal:  theorems which give upper bounds for the number of solutions, or for the number of solutions subject to some conditions.  I will give a general talk about the recent history of results of this kind, avoiding the technical guts of things and trying to give some general idea of strategies, finishing with a recent result of mine with Lawrence and Venkatesh, and a theorem of (2017 Yale Ph.D.) Vesselin Dimitrov with Gao and Habegger.