Tag Archives: Serre

Schur-Siegel-Smyth-Serre-Smith

Posted on November 25, 2021 by Persiflage

If \(\alpha\) is an algebraic number, the normlized trace of \(\alpha\) is defined to be \( \displaystyle{T(\alpha):=\frac{\mathrm{Tr}(\alpha)}{[\mathbf{Q}(\alpha):\mathbf{Q}].}}\) If \(\alpha\) is an algebraic integer that is totally positive, then the normalized trace is at least one. This follows from the AM-GM … Continue reading →

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Potential Automorphy for GL(n)

Posted on April 29, 2021 by Persiflage

Fresh on the arXiv, a nice new paper by Lie Qian proving potential automorphy results for ordinary Galois representations \(\rho: G_F \rightarrow \mathrm{GL}_n(\mathbf{Q}_p)\) of regular weight \([0,1,\ldots,n-1]\) for arbitrary CM fields \(F\). The key step in light of the 10-author … Continue reading →

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Update on Sato-Tate for abelian surfaces

Posted on July 19, 2018 by Persiflage

Various people have asked me for an update on the status of the Sato-Tate conjecture for abelian surfaces in light of recent advances in modularity lifting theorems. My student Noah Taylor has exactly been undertaking this task, and this post … Continue reading →

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New Results In Modularity, Part I

Posted on June 23, 2017 by Persiflage

I usually refrain from talking directly about my papers, and this reticence stems from wishing to avoid any appearance of tooting my own horn. On the other hand, nobody else seems to be talking about them either. Moreover, I have … Continue reading →

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Correspondance Serre-Tate, Part I

Posted on December 24, 2016 by Persiflage

Reading the correspondence between Serre and Tate has been as delightful as one could expect. What is very nice to see — although perhaps not so surprising — is the utter delight that both Serre and Tate find in discussing … Continue reading →

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Indulgences

Posted on December 15, 2016 by Persiflage

Today is Persiflage’s birthday, but let us not also forget to wish many happy returns to friends of the blog Ana and Vytas, who share the same birthday! Here is how I plan to celebrate: I will spare you with the … Continue reading →

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Serre 1: Calegari 0

Posted on October 18, 2016 by Persiflage

I just spent a week or so trying to determine whether Serre’s conjecture about the congruence subgroup property was false for a very specific class of S-arithmetic groups. The punch line, perhaps not surprisingly, was that I had made an … 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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Review of Buzzard-Gee

Posted on March 3, 2015 by Persiflage

This is a review of the paper “Slopes of Modular Forms” submitted for publication in a Simons symposium proceedings volume. tl;dr: This paper is a nice survey article on questions concerning the slopes of modular forms. Buzzard has given a … Continue reading →

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Derived Langlands

Posted on December 22, 2014 by Persiflage

Although it has been in the air for some time, it seems as though ideas from derived algebraic geometry have begun to inform developments in the Langlands program. (A necessary qualifier: I am talking about reciprocity in the classical arithmetic … Continue reading →

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