# Mathematical Sciences Research Institute

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# Seminar

DDC - Diophantine Problems: The Existential Closedness Problem for the Modular j-function October 26, 2020 (09:00 AM PDT - 10:00 AM PDT)
Parent Program: Decidability, definability and computability in number theory: Part 1 - Virtual Semester MSRI: Online/Virtual
Speaker(s) Sebastian Eterovic (University of California, Berkeley)
Description

This seminar will focus on Diophantine problems in a broad sense, with a view towards (but not limited to) interactions between Number Theory and Logic. Particular attention will be given to topics with the potential of further developments in the context of this MSRI scientific program. This will provide an opportunity for researchers to update on new results, techniques and some of the main problems of the field.

To participate in this seminar, please register here: https://www.msri.org/seminars/25206

Video

#### The Existential Closedness Problem For The Modular J-Function

Abstract/Media

To participate in this seminar, please register here: https://www.msri.org/seminars/25206

Abstract:

The existential closedness problem for $j$ asks to find a "minimal" set of geometric conditions that an algebraic variety $V\subset\mathbb{C}^{2n}$ should satisfy in order to ensure that it has a point of the form $(z_1,\ldots,z_n,j(z_1),\ldots,j(z_n))$. Furthermore, one wants to know if for every finitely generated field $F$ there is a generic point in $V$ over $F$ of this form. In this talk I will introduce the problem, I will present some of the known results, and I will explain how it relates to some very important open conjectures such as the Zilber-Pink conjecture and the modular Schanuel conjecture.