p-adic equidistribution of CM points and applications

[Math. Dept.]

August 09, 2018  10:00-11:00

E302(2)  School of Mathematics

[seminar]20180809Daniel Disegni-01.png

 SPEAKER

Daniel Disegni (Université Paris-Sud)

 ABSTRCT

Let $X$ be a modular curve. It is a curve over the integers, whose complex points are a quotient of the (compactified) Poincaré upper-half plane $H$ by a subgroup of $SL(2,Z)$. The curve $X$ has a natural supply of (almost) rational points called CM points. Following Heegner, these are useful to construct rational solutions to cubic equations in two variables (elliptic curves). Let $z_n$ be a sequence of CM points of increasing “p-adic complexity” for some prime $p$. A theorem of Duke says that the images of the $z_n$ in $X(C)$ equidistribute to the quotient of the Haar measure on $H$. A theorem of Cornut—Vatsal describes the equidistribution properties of the $z_n$ modulo primes different from $p$. I will talk about a p-adic equidistribution result for the $z_n$, and sketch its application to the study of Heegner’s solutions.

 SUPPORTED BY

School of Mathematics, Sichuan University