### Rhine Seminar on Transcendence Basel-Freiburg-Strasbourg

The Rhine Seminar on Transcendence is a joint initative of **Giuseppe
Ancona** and **Thomas Dreyfus** (Strasbourg),

**Annette Huber** and **Amador Martin-Pizarro** (Freiburg), **Philipp Habegger** (Basel).

We plan to meet around twice a year and discuss current research revolving around transcendence theory in a one day format.

### Second Meeting on November 4, 2022 in Basel

Speakers: **Giuseppe Ancona** (Strasbourg), **Francesco Gallinaro** (Freiburg), **Harry Schmidt** (Basel)

The venue is Kollegienhaus of the University of Basel, located at Petersplatz 1. The nearest tram stop is *Universität* (Tram 3).

Registration is closed.

**Schedule**

10.00 - 11.00 | Meet and greet (coffee and snacks) |

11.00 - 11.45 | Harry Schmidt |

12.15 - 14.00 | Lunch break |

14.00 - 14.45 | Giuseppe Ancona |

15.15 - 16.00 | Francesco Gallinaro |

All talks are 45 minutes and take place in lecture hall 120 of the Kollegienhaus located at Petersplatz 1. The coffee break at 10.00 is near the lecture hall.

**Titles and abstracts**

Speaker: **Harry Schmidt** (11.00 - 11.45)

Title: Isogeny estimates along families of abelian varieties

Abstract: 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 of \(A,B\) 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 of \(A\) 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.

Speaker: **Giuseppe Ancona** (14.00 - 14.45)

Title: *Algebraic classes in mixed characteristic and Andre's \(p\)-adic periods*

Abstract: (Joint work with D. Fratila) Motivated by the study of algebraic classes in mixed characteristic, we define a countable subalgebra of \(\mathbb{Q}_p\) which we call the algebra of “Andre's \(p\)-adic periods”. We will explain the analogy and the difference between these \(p\)-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.

Speaker: **Francesco Gallinaro** (15.15 - 16.00)

Title: *Exponential Sums Equations and Tropical Geometry*

Abstract: 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.

### Past meeting

**First meeting**on March 21, 2022