# Joint IAS/PU Number Theory

## Arithmetic and geometry of Picard modular surfaces

Dinakar Ramakrishnan
California Institute of Technology; Visitor, School of Mathematics
December 8, 2016
Of interest are (i) the conjecture of Bombieri (and Lang) that for any smooth projective surface $X$ of general type over a number field $k$, the set $X(k)$, of $k$-rational points is not Zariski dense, and (ii) the conjecture of Lang that $X(k)$, is even finite if in addition $X$ is hyperbolic, i.e., there is no non-constant holomorphic map from the complex line $C$ into $X(C)$. We can verify them for the Picard modular surfaces $X$ which are smooth toroidal compactifications of congruence quotients $Y$ of the unit ball in $\mathbb C^2$.

## Nonabelian Cohen-Lenstra heuristics and function field theorems

Melanie Wood
November 17, 2016
The Cohen-Lenstra Heuristics conjecturally give the distribution of class groups of imaginary quadratic fields. Since, by class field theory, the class group is the Galois group of the maximal unramified abelian extension, we can consider the Galois group of the maximal unramified extension as a non-abelian generalization of the class group. We will explain non-abelian analogs of the Cohen-Lenstra heuristics due to Boston, Bush, and Hajir and work, some joint with Boston, proving cases of the non-abelian conjectures in the function field analog.

## Rational curves on elliptic surfaces

Douglas Ulmer
Georgia Institute of Technology
May 5, 2016
Given a non-isotrivial elliptic curve $E$ over $K = \mathbb F_q(t)$, there is always a finite extension $L$ of $K$ which is itself a rational function field such that $E(L)$ has large rank. The situation is completely different over complex function fields: For "most" $E$ over $K = \mathbb C(t)$, the rank $E(L)$ is zero for any rational function field $L = \mathbb C(u)$. The yoga that suggests this theorem leads to other remarkable statements about rational curves on surfaces generalizing a conjecture of Lang.

## Iwasawa theory for the symmetric square of a modular form

David Loeffler
University of Warwick
March 10, 2016
Iwasawa theory is a powerful technique for understanding the link between the special values of L-functions and arithmetic objects (such as class groups of number fields, or Mordell-Weil groups of elliptic curves). In this talk I'll discuss what Iwasawa theory predicts for the symmetric square L-function attached to a modular form; and I'll describe some recent results (joint with Sarah Zerbes) confirming some of these conjectures, using the method of Euler systems.