Workshop

I will survey on some issues concerning effectivity in diophantine problems. After a preamble of general nature, and a (very) brief summary of some of the known results, focusing mainly on integral points on curves,  I shall present some new examples, especially concerning integral points on curves of genus 2. (Part of this concerns work in progress with P. Corvaja.)

We consider the set $\mathcal{M}_n(\mathbb{Z}; H))$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain upper and lower bounds on the number of $s$-tuples of matrices from $\mathcal{M}_n(\mathbb{Z}; H)$, satisfying various multiplicative relations, including multiplicative dependence and bounded generation of a subgroup of $\mathrm{GL}_n(\mathbb{Q})$. These problems generalise those studied in the scalar case $n=1$ by F. Pappalardi, M. Sha, I. E. Shparlinski and C. L. Stewart (2018) with an obvious distinction due to the non-commutativity of matrices. As a part of our method, we obtain a new upper bound on the number of matrices from $\mathcal{M}_n(\mathbb{Z}; H)$ with a given characteristic polynomial $f \in\mathbb{Z}[X]$, which is uniform with respect to $f$. This complements the asymptotic formula of A. Eskin, S. Mozes and N. Shah (1996) in which $f$ has to be fixed and irreducible.

Joint work with Igor Shparlinski.

In this talk, I will discuss the applications of counting techniques to unlikely intersection problems  related to families of abelian varieties. After a very brief historical introduction, I will focus on questions related to effectivity and uniformity and, time permitting, extensions to fields of positive characteristics.  In the course of my talk, I will cover joint work with Binyamini, Jones, and Thomas as well as, time permitting, Griffon. 

I will give a finite-time algorithm that, on input (g,K,S) with g > 0, K a totally real number field of odd degree, and S a finite set of places of K, outputs the finitely many g-dimensional abelian varieties A/K which are of GL_2-type over K and have good reduction outside S.

The point of this is to effectively compute the S-integral K-points on a Hilbert modular variety, and the point of that is to be able to compute all K-rational points on complete curves inside such varieties.

This gives effective height bounds for rational points on infinitely many curves and (for each curve) over infinitely many number fields. For example one gets effective height bounds for odd-degree totally real points on x^6 + 4y^3 = 1, by using the hypergeometric family associated to the arithmetic triangle group \Delta(3,6,6).

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 to 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. In particular, I will discuss finiteness in the case of a sequence of isogenies of pairwise coprime order.

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.

 While this talk will have some overlap with the one given at the introductory workshop it will *not* be the same.

We prove in all generality that on a smooth complex quasi-projective variety $X$,   rigid connections  yield $F$-isocrystals on almost all good reductions $X_{\mathbb F_q}$ and that rigid local systems yield crystalline local systems on  $X_K$ for $K$ the field of fractions of the Witt vectors of a finite field $\mathbb F_q$, for almost all $X_{\mathbb F_q}$. This improves our earlier work where, if $X$ was not projective, we assumed a strong cohomological condition (which is fulfilled for Shimura varieties of real rank $\geq 2$), and we obtained only infinitely many $\mathbb F_q$ of growing characteristic. While the earlier proof was via characteristic $p$, the new one is purely $p$-adic and uses $p$-adic topology. 

Let X be a curve of genus at least 2 over a number field K. By Mordell's Conjecture, X(K) is known to be finite. I will discuss a (very slow) algorithmic approach to determining the set X(K), conditional on the Hodge, Tate, and Fontaine--Mazur conjectures. Joint work in progress with Levent Alpöge.

Motivated by the “Mordell–Lang + Bogomolov” (proved by Poonen and independently by S. Zhang), I hope to relax the usual incidence relations in unlikely intersections by using certain distance relations. The focus of this talk is a joint of the conjectures of Andr ́e–Oort and Andr ́e–Pink on a product of ordinary Siegel formal moduli schemes. I will make a conjecture and present partial progress obtained via a perfectoid approach. A closely related analog of the Ax–Lindemann principle will also be discussed. 

Recently, P. Dolce and F. Zucconi proved that Roth's theorem holds over adelic curves, under certain hypotheses.

Here, an adelic curve is a field $K$, together with a collection of absolute values parameterized by a measure space, such that an analogue of the product formula holds for all elements of $K^{*}$.  (This definition was formulated by H. Chen and A. Moriwaki, and encompasses number fields, function fields, and Moriwaki's concept of arithmetic function fields.)

We will briefly describe adelic curves and some examples, and then summarize the proof of the theorem of Dolce and Zucconi.

In this talk, I will explain the impacts of some work of Szpiro, Edixhoven, and others who have recently left us.