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Meta
Tag Archives: Galois Representations
A public service announcement concerning Fontaine-Mazur for GL(1)
There’s a rumour going around that results from transcendence theory are required to prove the Fontaine-Mazur conjecture for \(\mathrm{GL}(1)\). This is not correct. In Serre’s book on \(\ell\)-adic representations, he defines a \(p\)-adic representation \(V\) of a global Galois group … Continue reading
Posted in Mathematics
Tagged Fontaine-Mazur Conjecture, Galois Representations, MO, Serre
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Report from Luminy
For how long has Luminy been infested with bloodthirsty mosquitoes? The combination of mosquitoes in my room with the fact that my bed was 6′ long with a completely unnecessary headboard (which meant that I had to sleep on an … Continue reading
Posted in Mathematics, Uncategorized
Tagged Ana Caraiani, Galois Representations, Jan Nekovar, Luminy, Mosquitoes, Peter Scholze, Ultrafilters, Wild Boars
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Scholze on Torsion, Part IV
This is a continuation of Part I, Part II, and Part III. I was planning to start talking about Chapter IV, instead, this will be a very soft introduction to a few lines on page 72. At this point, we … Continue reading
Scholze on Torsion, Part III
This is a continuation of Part I and Part II. Before I continue along to section V.3, I want to discuss an approach to the problem of constructing Galois representations from the pre-Scholze days. Let’s continue with the same notation … Continue reading
Posted in Mathematics
Tagged Galois Representations, Langlands, Peter Scholze, torsion, Vaporware
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Scholze on Torsion, Part II
This is a sequel to Part I. Section V.1: Today we will talk about Chapter V. We will start with Theorem V.1.4. This is basically a summary of the construction of Galois representations in the RACSDC case, which follows, for … Continue reading
Posted in Mathematics
Tagged Bianchi Groups, Determinants, Gaëtan Chenevier, Galois Representations, Peter Scholze, torsion
6 Comments
Scholze on Torsion, Part I
This is a sequel to this post, although as it turns out we still won’t actually get to anything substantial — or indeed anything beyond an introduction — in this post. Let me begin with some overview. Suppose that \(X … Continue reading
Posted in Mathematics
Tagged Galois Representations, Langlands, Peter Scholze, torsion
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Scholze on Torsion 0
This will be the first zeroth of a series of posts talking about Scholze’s recent preprint, available here. This is mathematics which will, no question, have more impact in number theory than any recent paper I can think of. The … Continue reading
Posted in Mathematics
Tagged Cohomology, David Geraghty, Galois Representations, Peter Scholze, torsion
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Finiteness of the global deformation ring over local deformation rings
(This post is the result of a conversation I had with Matt). Suppose that \(\overline{\rho}: G_{F} \rightarrow \mathrm{GL}_n(\mathbf{F})\) is a continuous mod-\(p\) absolutely irreducible Galois representation. For now, let’s assume that \(F/F^{+}\) is a CM field, and \(\overline{\rho}\) is essentially … Continue reading
Posted in Mathematics
Tagged BLGGT, David Geraghty, Galois Representations, Jack Thorne, Matthew Emerton, Vytas Paskunas
5 Comments
Galois Representations for non-self dual forms, Part III
Here are some complements to the previous remarks, considered in Part I and Part II. First, in order to deal with non-zero weights, one has to replace the Shimura varieties \(Y\), \(X\), \(W\) by Kuga-Satake varieties over these spaces. This … Continue reading
Inverse Galois Problem
My favourite group as far as the inverse Galois problem goes is \(G = \mathrm{SL}_2(\mathbf{F}_p)\). This is not known to be a Galois group over \(\mathbf{Q}\) for any \(p > 13\), the difficulty of course being that is must correspond … Continue reading
Posted in Mathematics
Tagged David Zywina, Galois Representations, Inverse Galois Problem, PSL(2), SL(2)
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