# Masterclass: Arakelov geometry on Shimura varieties

Rational numbers share many common features with rational functions on an algebraic curve. This analogy has been a guiding principle of algebraic number theory from as early on as the 19th century. However, there is a substantial difference between them: An affine algebraic curve can always be compactified, whereas rational numbers “live” on the spectrum of integers, which has no compactification in ordinary algebraic geometry. This non-compactness presents a serious obstruction for any kind of arithmetic intersection theory. Arakelov geometry aims to overcome this obstruction by adding the archimedean places to the spectrum of integers as additional points at infinity.

Arakelov’s ingenious realization of this idea became one of the conceptual inputs in Faltings’ later proof of the Mordell conjecture. Of substantial importance has been also the arithmetic Grothendieck-Riemann-Roch theorem first proven by Faltings for arithmetic surfaces and extended to general arithmetic varieties by Gillet-Soulé. In the nineties, it became a standard tool in diophantine geometry, enabling the proof of the Mordell-Lang conjecture by Vojta and Faltings and the proof of the Bogomolov conjecture by Ullmo and Zhang.

Shimura varieties are an important class of arithmetic varieties, which are naturally endowed with a rich geometry and arithmetic, and the Arakelov geometry of automorphic line bundles on them has been of central interest in many recent investigations – for example, the proof of the averaged Colmez conjecture and in Kudla’s program. However, the natural hermitian metrics on these automorphic line bundles have logarithmic singularities at the boundary. While it is nevertheless possible to build an arithmetic intersection theory on Shimura varieties, this theory has some defects, for example, the lack of an arithmetic Grothendieck-Riemann-Roch theorem.

The central theme of this masterclass will be to introduce the relevant tools for arithmetic intersection theory on Shimura varieties and eventually aims to discuss more recent topics.

## Target group

The target audience is PhD students in arithmetic geometry in the broadest sense. While no knowledge of Arakelov geometry is presupposed, participants should ideally have a solid background in algebraic geometry, algebraic number theory, and modular curves.

## Lecturers

- Jose Burgos Gil (ICMAT - Madrid)
- Gerard Freixas i Montplet (Institut de Mathématiques de Jussieu - Paris)
- Anna von Pippich (Universität Konstanz)

The speakers are all expert researchers in arithmetic geometry and Arakelov geometry with profound experience in the Arakelov geometry of Shimura varieties.

The lecture series is planned to be supplemented with additional research talks and tutorials to provide more context and further applications as well as necessary background.

## Registration

Information about registration will follow.