##### Department of Mathematics,

University of California San Diego

****************************

### Math 208 - Algebraic Geometry

## Kiran Kedlaya

#### UCSD

## The tame Belyi theorem in positive characteristic

##### Abstract:

Belyi's theorem says that on one hand, a curve over a field of characteristic 0 that admits a finite map to $\mathbf{P}^1$ ramified over at most three points must descend to a subfield algebraic over $\mathbb{Q}$, and on the other hand any curve over such a subfield does indeed admit such a morphism (without any further base extension). One might ask whether a similar statement holds over a field of characteristic $p$, replacing $\mathbb{Q}$ with $\mathbb{F}_p$. For general morphisms this is false, but it becomes true if we restrict to tamely ramified morphisms to $\mathbf{P}^1$. Such a statement was originally given by Saidi, in which the ``other hand'' assertion was made conditional on the existence of some tamely ramified morphism from the given curve to $\mathbf{P}^1$. In the pre-talk, we will discuss how to establish existence of a tamely ramified morphism in characteristic \mbox{$p\>2$}. This is ``classical'' over an infinite algebraic extension of $\mathbb{F}_p$; to do it over a fixed finite field requires a density statement in the style of Poonen's finite field Bertini theorem. In the talk proper, we will discuss work of Sugiyama-Yasuda that establishes the existence of a tamely ramified morphism when the base field is algebraically closed of characteristic 2. The case where the base field is finite of characteristic 2 requires a further geometric reinterpretation of the key construction of Sugiyama-Yasuda; this is joint work with Daniel Litt and Jakub Witaszek.

### January 31, 2020

### 3:00 PM

### AP&M 7321

****************************