Workshop

The space Mor_d^N of degree d morphisms f from P^N to itself is parameterized by the space of coefficients of the polynomials that define f. This space has a natural structure as an affine variety. Elements R of Aut(P^N), or equivalently of PGL_{N+1}, act on Mor_d^N via conjugation, f --> R^{-1} o f o R. Conjugation commutes with iteration, so it is the natural action to consider when studying the dynamics of f, since conjugate maps determine the same dynamical system up to simultaneous change of variables of the domain and range. In this talk I will discuss how geometric invariant theory yields a nice quotient Mor_d^N//Aut(P^N), how this quotient can be embedded in various larger spaces with even nicer properties, what happens when one adds dynamical level structure, some structure theorems and open problems, and, as time permits, various related topics such as field of moduli versus field of definition problems for dynamical systems.

Let f be a rational function of degree greater than 1 defined over a number field k and let x in k.  Iterating f, we obtain
a sequence of field extensions K_n generated by the inverse images of x under f^n. Passing to the inverse limit one obtains an iterated Galois group G_x.  There is a natural "generic" Galois group G attached to f in which G_x sits. Conjecturally, G_x will have finite index in G unless x is "special" in some fashion.  While in its earliest form, this conjecture (originally made in a special case by Boston and Jones) was made in analogy with the Serre finite index theorem, it seems to have a very different flavor.  We will discuss various results towards this conjecture, the diophantine techniques used in these results, a more general higher-dimensional conjecture, and a very simple conjecture on the irreducibility of iterates of polynomials that is completely unresolved.

I'll report on a project with Matt Satriano and David Zureick-Brown (who will be a Research Member later this semester) which in some sense started out in the MSRI "Rational and Integral Points on Higher-Dimensional Varieties" semester in 2006. Some natural arithmetic objects (like numbers) are parametrized by an algebraic variety, but many others (like elliptic curves, or number fields) are parametrized by more general algebraic objects called "stacks." Rational points on stacks are just as interesting as rational points on varieties, but a very useful tool for the study of rational points, a theory of *heights*, was missing in this case. I'll talk about the definition of heights we developed for this situation, make the case that it's a good definition, and discuss some of the refinements that have come in the time since we proposed it.

Given a l-adic local system on a variety over a number field, one can define its toric locus, whic is the analogue of the CM locus of a variation of Hodge structure. I will discuss a work in progress on the sparcity of the points of bounded degree in the toric locus of local systems arising from geometry.

Let A be an abelian variety over an algebraic closure of Q. A conjecture of Mocz asserts that there are only finitely many isomorphism classes of abelian varieties isogenous A, and of height less than some fixed constant c.

In this talk, I will sketch a proof of the conjecture when the Mumford-Tate conjecture - which is known in many cases - holds for A. This result should be compared with Faltings' famous theorem, which is about finiteness for abelian varieties defined over a fixed number field.

This is joint work with Lucia Mocz.

Tate proved (amongst other things) that the action of Frobenius on the Tate-module of an abelian variety over a finite field is semisimple. I will discuss the analogue of this question in the context of exceptional Shimura varieties. This is joint work with Ben Bakker and Jacob Tsimerman.

In this talk, I will present various results, established in collaboration with François Charles, concerning formal-analytic arithmetic surfaces. These play the role, in Arakelov geometry, of germs of analytic or formal surfaces in complex algebraic geometry.

Firstly I will discuss various motivations for the study of formal-analytic arithmetic surfaces, notably algebraization results in Diophantine geometry and their recent developments by Calegari-Dimitrov-Tang. I will also recall classical theorems in complex geometry, due to Andreotti and Nori, of which our results are arithmetic analogues. I will present in some detail the arithmetic counterpart, in Arakelov geometry, of Nori’s degree bound. This arithmetic bound notably involves a new archimedean invariant, the overflow, attached to an analytic map from a pointed compact Riemann surface to another Riemann surface. Explicit expressions for this invariant play a key role in various applications of our results in Diophantine geometry.

Further applications of our arithmetic Nori’s bound will be presented in the talk of François Charles

In this talk, I will discuss several results obtained in collaboration with Jean-Benoît Bost on the geometry of formal-analytic surfaces, as introduced in Bost's talk, of which this one is a follow-up.

Using numerical invariants of those surfaces, I will discuss finiteness results under suitable positivity conditions, with applications to qualitative finiteness results for certain algebras of power series satisfying "twisted" integrality conditions, motivated by concrete situations in the arithmetic geometry of modular curves.

I will also explain how to use formal-analytic arithmetic surfaces to obtain finiteness results for fundamental groups of certain arithmetic surfaces, including some integral models of modular curves.

The Zilber-Pink conjectures on unlikely intersections deal with intersections of subvarieties of a (semi)abelian variety or, more in general, of a Shimura variety, with “special” subvarieties of the ambient space. These conjectures generalize many classical results such as Faltings’ Theorem (Mordell Conjecture), Raynaud’s Theorem (Manin-Mumford Conjecture) and André-Oort Conjecture and have been studied by several authors in the last two decades.

Most proofs of results in this area follow the well-established Pila-Zannier strategy, first introduced by the two authors in 2008 to give an alternative proof of Raynaud’s theorem as a combination of results coming from o-minimality (Pila-Wilkie’s theorem) with other diophantine ingredients. The talk will focus on a general introduction to these problems, on some results for semi-abelian varieties and families of abelian varieties, and on applications to other problems of diophantine nature.

A classical question in analytic number theory is: given a polynomial with integer coefficients, how often does it take squarefree values? In arithmetic statistics, we are particularly interested in the case of discriminant polynomials. In this talk, I will present several different cases of this question. First, we will consider a classical result of Davenport--Heilbronn which considers the case of discriminants of binary cubic forms. Motivated by this case, I will discuss two works, joint with Manjul Bhargava and Xiaoheng Wang, in which we consider the spaces of degree-n polynomials and degree-n binary forms. We will prove that a positive proportion of these polynomials and forms have squarefree discriminant.