The isogeny estimates of Masser and Wüstholz give a bound for the minimal degree of an isogeny between two abelian varieties $A$,$B$ of dimension g. Fixing the dimension g (and assuming that $A$,$B$ are defined over a number field), their bound is polynomial in the degree of the field of definition $A$,$B$ of and the Faltings height of $A$. I am going to talk about joint work with Binyamini in which we prove an effective polynomial estimate for the minimal degree of an isogeny between an abelian variety $A$ and a member $B$ of a fixed family of abelian varieties. Our bound only depends on the Faltings height $A$ of and the family but not on the particular member of the family. This has some direct applications to problems in unlikely intersections such as an effective and uniform version of a theorem of Orr. I will go into some details of the proof that are reminiscent of those in proofs in transcendence theory.
otivated by the study of algebraic classes in mixed characteristic, we define a countable subalgebra of Qp which we call the algebra of “Andre's -$p$ adic periods”. We will explain the analogy and the difference between these -adic periods and the classical complex periods. For instance, they both contain several examples of special values of classical functions (logarithm, gamma function, ...) and they share transcendence properties. On the other hand, the classical Tannakian formalism which is used to bound the transcendence degree of complex periods has to be modified in order to be used in the p-adic setting. We will discuss concrete examples of all these instances though elliptic curves and Kummer extensions.
The Exponential-Algebraic Closedness Conjecture, due to Zilber, predicts sufficient conditions for systems of equations involving polynomials and exponentials to be solvable in the complex numbers; it is formulated geometrically, asking about the intersections between complex algebraic varieties and (Cartesian powers of) the graph of the exponential function. The conjecture originated from model theory, and it would imply a strong tameness result for subsets of the complex numbers that are definable using polynomials and exponentials. In this talk, we will briefly recall the motivation of this question and then focus on the case of varieties which split as the product of a linear space and an algebraic subvariety of the multiplicative group. These varieties correspond to systems of exponential sums equations, and the proof that the conjecture holds in this case uses tropical geometry and the theory of toric varieties.