STNB2024(37th edition)
Langlands base change for GL(2) and automorphy for symmetric powers
Co-ordinators
Iván Blanco Chacón (UAH and Aalto University) and Javier Guillán Rial (CRM)
Description
The goal of this session is to explain in detail the main result in [1], namely, that given a totally real Galois number field F and a holomorphic newform f of weight at least 2 and odd level N, under mild conditions of the splitting behaviour of certain small primes at F, there is a Hilbert modular form g over F such that the restriction to $G_F$ of the $\lambda$-adic Galois representation attached to f agrees with the Galois representations attached to g. Time permitting, we will also give an overview of the proof of the automorphy of the 5-th symmetric power over GL(2,F) of certain automorphic forms as described in [2]. [1]Dieulefait, L.V.: Langlands Base Change for GL(2), Annals of Math. 176 (2012), p 1015-1038. [2]Dieulefait, L.V.: Automorphy of Symm^5(GL(2)) and base change (with Appendix A by R. Guralnick and Appendix B by L. Dieulefait and T. Gee), J. Math. Pures et Appl. 104 (2015), p 619-656.
Talks
- Preliminary facts and a (detailed) overview of the proof (Iván Blanco Chacón)
- First half of the proof of base change over Q: (micro) good dihedral primes and killing ramification (Javier Guillán Rial)
- Second half of the proof of base change over Q: weight modifications and connecting with a CM form (Javier Guillán Rial)
- Automorphy of $Sim^5(GL(2))$ and refinement of the base-change proof for GL(2) (Iván Blanco Chacón)
