Tag Archives: torsion

Jacquet-Langlands and a new R=T conjecture

Posted on January 8, 2019 by Persiflage

It is somewhat mysterious how one should formulate the Jacquet-Langlands correspondence integrally, particularly in the presence of torsion classes. Even the classical case has many subtleties including for example some results in this paper of Ribet. In the case of … Continue reading →

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Ventotene, Part II

Posted on March 17, 2016 by Persiflage

I promised to return to a more mathematical summary of the conference in Ventotene, and indeed I shall do so in the next two posts. One of the themes of the conference was bounding the order of the torsion subgroup … Continue reading →

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Horizontal Vanishing Conjectures.

Posted on August 8, 2014 by Persiflage

Let \(F\) be a number field, and let \(\mathbf{G}\) be a reductive group over \(F\), and let \(\Gamma\) be a congruence subgroup of \(\mathbf{G}(\mathcal{O}_F)\). I can hear BC objecting that this doesn’t make sense without extra choices; if you have … Continue reading →

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I don’t know how to prove Serre’s conjecture.

Posted on May 30, 2014 by Persiflage

I find it slightly annoying that I don’t know how to prove Serre’s conjecture for imaginary quadratic fields. In particular, I don’t even see any particularly good strategy for showing that a surjective Galois representation — say finite flat with … Continue reading →

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Local representations occurring in cohomology

Posted on November 5, 2013 by Persiflage

Michael Harris was in town for a few days, and we chatted about the relationship between my conjectures on completed cohomology groups with Emerton and the recent work of Scholze. The brief summary is that Scholze’s results are not naively … Continue reading →

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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 →

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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 →

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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 →

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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 →

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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 →

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