## STNB2024(37th edition)

### First half of the proof of base change over Q: (micro) good dihedral primes and killing ramification

#### Presenters

Javier Guillán Rial

#### Abstract

In this session we will give the first details of the proof of Langlands base change over Q given by Dieulefait in his work from 2015 (with some minor changes). In this first half of the proof we will explain in a detailed way the philosophy of the method of "safe chains" in a setting of level refinement: we will find suitable congruences between a base modular form of odd level and another one with suitable level. For this purpose, the notion of good dihedral primes and micro good dihedral primes is introduced: these are primes that are added to the level in order to assure big image of residual representations along all the congruences.

#### Files

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