Jordan ELLENBERG - Heights on Stacks
Channel: Institut des Hautes Études Scientifiques (IHÉS)
Date: 2019-02-01

When  I  was  Barry’s  student,  one  thing  I  learned  from  him  was  that  Diophantine  problems  were often  not  problems  about  rational  points  on varieties,  but  about  rational  points  on  stacks.  At  the  time  we were talking about generalized Fermat equations, but the point of view is much more general; for instance, a number field with Galois group G can be thought of as a rational point on the classifying stack BG.Two popular questions in number theory are:1. How many degree-d number fields are there with discriminant at most X?2. How many rational points are there on a cubic surface with height at most X?Our  expectations  about  the  first  question  are  governed  by  Malle’s  conjecture;  about  the  second,  by  the Batyrev-Manin  conjecture. The  forms  of  the  conjectures  are  very  similar,  predicting  in  both  cases  an asymptotic of the form c Xa (log  X)b,  and  this  is  no  coincidence:  in  light  of  the  above description, both are questions about rational points on a stack, and both questions should be asking us to count rational points of height at most X.A serious obstacle is that there is no definition of the height of a rational point on a stack. The main point of this talk is to propose a definition and try to convince you it’s the right one. If there’s time, I’ll also argue that when we talk about heights with respect to a line bundle we have always secretly meant “vector bundle,” or should have.(Joint work with Matt Satriano and David Zureick-Brown).

Vidéo réalisée avec le soutien du projet AAMOT de l’ERC