Tag Archives: Galois Representations

Is Serre’s conjecture still open?

Posted on August 10, 2014 by Persiflage

The conjecture in this paper has indeed been proven. But that isn’t the entire story. Serre was fully aware of Katz modular forms of weight one. However, Serre was too timid was prudently conservative and made his conjecture only for … Continue reading →

Posted in Mathematics | Tagged , , , , , , , , , , , , , | Leave a comment

A public service announcement concerning Fontaine-Mazur for GL(1)

Posted on July 12, 2014 by Persiflage

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 , , , | Leave a comment

Report from Luminy

Posted on July 8, 2014 by Persiflage

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 , , , , , , , | Leave a comment

Scholze on Torsion, Part IV

Posted on June 29, 2013 by Persiflage

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 →

Posted in Mathematics | Tagged , , , , , , , , , , | Leave a comment

Scholze on Torsion, Part III

Posted on June 22, 2013 by Persiflage

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 , , , , | Leave a comment

Scholze on Torsion, Part II

Posted on June 19, 2013 by Persiflage

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 , , , , , | 6 Comments

Scholze on Torsion, Part I

Posted on June 16, 2013 by Persiflage

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 , , , | Leave a comment

Scholze on Torsion 0

Posted on June 13, 2013 by Persiflage

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 , , , , | Leave a comment

Finiteness of the global deformation ring over local deformation rings

Posted on May 18, 2013 by Persiflage

(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 , , , , , | 5 Comments

Galois Representations for non-self dual forms, Part III

Posted on April 27, 2013 by Persiflage

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 →

Posted in Mathematics | Tagged | Leave a comment