From 56137c3fe571cf5fcae6d2a6f1e8e676b3030850 Mon Sep 17 00:00:00 2001 From: Kevin Buzzard Date: Thu, 4 Apr 2024 23:45:55 +0100 Subject: [PATCH] (tex) more bestiary ideas for ch5 --- blueprint/src/chapter/ch04overview.tex | 3 ++- blueprint/src/chapter/ch05bestiary.tex | 13 ++++++++++++- 2 files changed, 14 insertions(+), 2 deletions(-) diff --git a/blueprint/src/chapter/ch04overview.tex b/blueprint/src/chapter/ch04overview.tex index 46f367f8..35c09cf6 100644 --- a/blueprint/src/chapter/ch04overview.tex +++ b/blueprint/src/chapter/ch04overview.tex @@ -50,7 +50,8 @@ \section{Compatible families, and reduction at 3} We now use Khare--Wintenberger to lift $\rho$ to a potentially modular $\ell$-adic Galois representation of conductor 2, and put it into an $\ell$-adic famiily using the Brauer's theorem trick in \cite{blggt}. Finally we look at the 3-adic specialisation -of this family. Reducing mod 3 we get a representation which must be reducible because +of this family. Reducing mod 3 we get a representation which is flat at 3 and tame at 2, +so must be reducible because of the techniques introduced in Fontaine's paper on abelian varieties over $\Z$ (an irreducible representation would cut out a number field whose discriminant violates the Odlyzko bounds). One can now go on to deduce that the 3-adic representation must be reducible, which diff --git a/blueprint/src/chapter/ch05bestiary.tex b/blueprint/src/chapter/ch05bestiary.tex index 0a0302c9..42c8e19f 100644 --- a/blueprint/src/chapter/ch05bestiary.tex +++ b/blueprint/src/chapter/ch05bestiary.tex @@ -10,6 +10,10 @@ \chapter{A collection of results which are needed in the proof.} mult 1, +cyclic base change + +automorphic induction from GL_1(quad extn) to GL_2 + Galois rep associated to an auto rep, Definition of an automorphic representation for the units of a quaternion algebra over a totally real field (including situations where the algebra is split at one or two infinite places). @@ -20,4 +24,11 @@ \chapter{A collection of results which are needed in the proof.} Moret-Bailly -mor to come +Artin symbol of local class field theory + +Existence of solvable extension avoiding a global extension and with prescribed local behaviour + +Poitou-Tate + +local Tate duality +