In 2014, Pila and Tsimerman gave a proof of the Ax–Schanuel conjecture for the-function
and, with Mok, have recently announced a proof of its generalization to any (pure) Shimura …

Point-counting results for sets in real Euclidean space have found remarkable applications
to diophantine geometry, enabling significant progress on the André–Oort and Zilber–Pink …

This short survey is part of a minicourse I gave during the CMI-HIMR Summer School
“Unlikely Intersections in Diophantine Geometry” on the Zilber–Pink conjecture, formulated …

Let S be a smooth irreducible curve defined over a number field k and consider an abelian
scheme A over S and a curve C inside A, both defined over k. In previous works, we proved …

Motivated by recent work of Masser and Zannier on simultaneous torsion on the Legendre
elliptic curve E λ of equation Y 2= X (X− 1)(X− λ), we prove that, given n linearly independent …

For positive integers and, we introduce and study the notion of-multiplicative dependence
over the algebraic closure of a finite prime field, as well as-linear dependence of points on …

A divisibility sequence is a sequence of integers {dn} such that dm divides dn if m divides n.
Results of Bugeaud, Corvaja, Zannier, among others, have shown that the gcd of two …

In this paper, we study the finiteness problem of torsion points on an elliptic curve whose
coordinates satisfy some multiplicative dependence relations. In particular, we prove that on …

Let E 1 and E 2 be elliptic curves in Legendre form with integer parameters. We show there
exists a constant C such that for almost all primes, for all but at most C pairs of points on the …

We prove a global local rigidity result for character varieties of 3-manifolds into SL 2 (C).
Given a 3-manifold with toric boundary M satisfying some technical hypotheses, we prove …