Algebraic Geometry Seminar

Here are two popular questions in number theory:

 

1.  How many degree-d number fields are there with discriminant at most X?

2.  How many solutions are there to x^3 + y^3 + z^3 + w^3 = 0 with x,y,z,w coprime integers of absolute value 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, and this is no coincidence: I will explain how to think of both questions in a common framework, that of counting points of bounded height on an algebraic stack.  A serious obstacle is that there is no definition of the height of a rational point on a stack.  I will propose a definition and try to convince you it’s the right one.

 

(joint work with Matt Satriano and David Zureick-Brown)