Tag Archives: completed cohomology

Not quite what I meant

Posted on August 15, 2024 by Persiflage

Weibo Fu wrote an interesting paper on upper bounds for spaces of Bianchi modular forms, pushing previous results of Simon Marshall and Yongquan Hu to get more or less optimal results in the weight aspect. More generally, for any number … Continue reading →

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ArXiv x 3

Posted on March 4, 2022 by Persiflage

Three recent arXiv preprints this week caught my interest and seemed worth mentioning here. The first is a paper by Oscar Randal-Williams, which considers (among other things) the cohomology of congruence subgroups of \(\mathrm{SL}_N(\mathbf{Z})\) in the stable range. This is … Continue reading →

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The stable cohomology of SL(F_p)

Posted on June 19, 2019 by Persiflage

Back by popular demand: an actual mathematics post! Today’s problem is the following: compute the cohomology of \(\mathrm{SL}(\mathbf{F}_p)\) for a (mod-p) algebraic representation. Step 0 is to say what this problem actually is. It makes sense to talk about certain … 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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H_2(Gamma_N(p),Z)

Posted on January 30, 2015 by Persiflage

In this post (which is a follow-up to the last post), I wanted to compute the group \( H_2(\Gamma_N(p),\mathbf{Z})\), where \( \Gamma_N(p)\) is the congruence subgroup of \( \mathrm{SL}_N(\mathbf{Z})\) for large enough \( N\) and \( p\) is prime. In … Continue reading →

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Stable completed homology without Quillen-Lichtenbaum

Posted on January 21, 2015 by Persiflage

Having just made (hopefully) the final revisions on my paper on stable completed cohomology groups, I wanted to record here a few remarks which didn’t otherwise make it into the paper. The first is that, in addition to the result … 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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Virtual Congruence Betti Numbers

Posted on September 27, 2013 by Persiflage

Suppose that \(G\) is a real semisimple group and that \(X = \Gamma \backslash G/K\) is a compact arithmetic locally symmetric space. Let us call a cohomology class tautological if it is invariant under the group \(G\). For example, if … Continue reading →

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