# Model theory of absolute Galois groups with a view towards arithmetic geometry

Lead Research Organisation:
University of Oxford

Department Name: Mathematical Institute

### Abstract

In his famous letter to Faltings, Grothendieck sketched the programme which came to be known as anabelian geometry. It is in the spirit of this programme that we endeavour to analyse the model theory of absolute Galois groups, in particular of absolute Galois groups of number fields.

The absolute Galois group GK of a field K is the group of automorphisms of the algebraic closure of K that fix K elementwise. GK is an important invariant of K ; one may think this group as being the analogue of a fundamental group in the context of fields. This is actually more than a mere analogy once we introduce the so called étale topology. Typically GK encodes arithmetic information about K in a computationally accessible manner, due to the machinery of Galois cohomology. This gives rise to the hope for effective versions of Faltings' celebrated Theorem about the finiteness of the number of rational points on algebraic curves of genus >1 over number fields.

More concretely, we will work on two conjectures about the absolute Galois group of the field Q of rational numbers: the Absolute Galois Conjecture (AGC) that any field K whose absolute Galois group is isomorphic to that of Q carries a henselian valuation with divisible value group and with residue field isomorphic to Q; and the Elementary Galois Conjecture (EGC) that any field K whose absolute Galois group is elementarily equivalent to that of Q carries a henselian valuation with divisible value group and with residue field elementarily equivalent to Q. It is not hard to see that EGC implies AGC and it is a nontrivial theorem of Koenigsmann that AGC is equivalent to the so-called birational section conjecture in Grothendieck's anabelian geometry.

The techniques for approaching these conjectures include general valuation theory, a study of the structure of absolute Galois groups with a focus on how they "see" valuations (this is already well developed by the earlier work of Koenigsmann), recent progress on the model theory of absolute Galois groups by Philip Dittmann (2018), refined techniques from algebraic number theory including Galois cohomology and possibly some inputs from stability theory.

The project combines the three EPSRC research areas: Algebra, Logic/Combinatorics and Number Theory.

The absolute Galois group GK of a field K is the group of automorphisms of the algebraic closure of K that fix K elementwise. GK is an important invariant of K ; one may think this group as being the analogue of a fundamental group in the context of fields. This is actually more than a mere analogy once we introduce the so called étale topology. Typically GK encodes arithmetic information about K in a computationally accessible manner, due to the machinery of Galois cohomology. This gives rise to the hope for effective versions of Faltings' celebrated Theorem about the finiteness of the number of rational points on algebraic curves of genus >1 over number fields.

More concretely, we will work on two conjectures about the absolute Galois group of the field Q of rational numbers: the Absolute Galois Conjecture (AGC) that any field K whose absolute Galois group is isomorphic to that of Q carries a henselian valuation with divisible value group and with residue field isomorphic to Q; and the Elementary Galois Conjecture (EGC) that any field K whose absolute Galois group is elementarily equivalent to that of Q carries a henselian valuation with divisible value group and with residue field elementarily equivalent to Q. It is not hard to see that EGC implies AGC and it is a nontrivial theorem of Koenigsmann that AGC is equivalent to the so-called birational section conjecture in Grothendieck's anabelian geometry.

The techniques for approaching these conjectures include general valuation theory, a study of the structure of absolute Galois groups with a focus on how they "see" valuations (this is already well developed by the earlier work of Koenigsmann), recent progress on the model theory of absolute Galois groups by Philip Dittmann (2018), refined techniques from algebraic number theory including Galois cohomology and possibly some inputs from stability theory.

The project combines the three EPSRC research areas: Algebra, Logic/Combinatorics and Number Theory.

## People |
## ORCID iD |

Jochen Koenigsmann (Primary Supervisor) | |

Konstantinos Kartas (Student) |

### Studentship Projects

Project Reference | Relationship | Related To | Start | End | Student Name |
---|---|---|---|---|---|

EP/R513295/1 | 01/10/2018 | 30/09/2023 | |||

2099876 | Studentship | EP/R513295/1 | 01/10/2018 | 30/09/2021 | Konstantinos Kartas |