Tag Archives: Jack Thorne

SL_n versus GL_n

Posted on July 18, 2024 by Persiflage

I recently wrote a paper (with Toby Gee and George Boxer, see also here) on constructing regular algebraic automorphic representations \(\pi\) of (cohomological) weight zero and level one, and therefore also cuspidal cohomology classes in the cohomology of \(\mathrm{GL}_n(\mathbf{Z})\) for … Continue reading →

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Clozel 70, Part II

Posted on September 30, 2023 by Persiflage

Many years ago, Khare asked me (as I think he asked many others at the time) whether I believed their existed an irreducible motive \(M\) over \(\mathbf{Z}\) (so good reduction everywhere) with Hodge-Tate weights \([0,1,2,\ldots,n-1]\) for any \(n > 1\). … Continue reading →

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Clozel 70, Part I

Posted on September 24, 2023 by Persiflage

I recently returned home from a trip to Paris for Clozel’s 70th birthday conference. Naturally I stayed in an airbnb downtown, and the RER B gods smiled on me with a hassle free commute for the entire week. Tekés was … Continue reading →

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30 years of modularity: number theory since the proof of Fermat

Posted on July 8, 2022 by Persiflage

It’s probably fair to say that the target audience for this blog is close to orthogonal to the target audience for my talk, but just in case anyone wants to watch it in HD (and with the audio synced to … Continue reading →

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Counting solutions to a_p = λ, Part II

Posted on March 3, 2020 by Persiflage

This is a sequel to this post where the problem of counting eigenforms with \(a_p = \lambda\) and \(\lambda \ne 0\) was considered. Here we report on recent progress in the case \(\lambda = 0\). It is a somewhat notorious … Continue reading →

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New Results in modularity, Christmas Update II

Posted on December 30, 2019 by Persiflage

Just like last year, once again saint Nick has brought us a bounty of treasures related to Galois representations and automorphic forms in the final week of the year. First there was this paper by Newton and Thorne, proving, among … Continue reading →

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Local-global compatibility for imaginary quadratic fields

Posted on January 16, 2019 by Persiflage

One of the key steps in the 10-author paper is to prove results on local-global compatibility for Galois representations associated to torsion classes. The results proved in that paper, unfortunately, fall well-short of the optimal desired local-global compatibility statement, because … Continue reading →

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New Results In Modularity, Christmas Update

Posted on December 25, 2018 by Persiflage

It’s a Christmas miracle! Keen watchers of this blog will be happy to learn that the 10 author paper discussed here and here is now available. (And just in case you also missed it, you can also find the other … 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 II

Posted on June 23, 2017 by Persiflage

This is part two of series on work in progress with Patrick Allen, Ana Caraiani, Toby Gee, David Helm, Bao Le Hung, James Newton, Peter Scholze, Richard Taylor, and Jack Thorne. Click here for Part I It has been almost … Continue reading →

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