Category Archives: Mathematics

Polymath Proposal: 4-folds of Mumford’s type

Posted on August 24, 2021 by Persiflage

Let \(A/K\) be an abelian variety of dimension \(g\) over a number field. If \(g \not\equiv 0 \bmod 4\) and \(\mathrm{End}(A/\mathbf{C}) = \mathbf{Z}\), then Serre proved that the Galois representations associated to \(A\) have open image in \(\mathrm{GSp}_{2g}(\mathbf{Z}_p)\). The result … Continue reading →

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59,281

Posted on August 19, 2021 by Persiflage

The target audience of this blog (especially the mathematics) is usually professional mathematicians in the Langlands program. I do sometimes, however, have posts suitable for a broader mathematical audience. Very rarely though do I have anything (possibly) interesting to say … Continue reading →

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Divisors near sqrt(n)

Posted on June 8, 2021 by Persiflage

Analytic Number Theory Alert! An even more idle question than normal (that’s because it comes from twitter). Alex Kontorovich noted with pleasure the following pictorial representation of the integers from a Veritasium youtube video, where prime numbers are represented by … 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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The Arbeitsgemeinschaft has returned!

Posted on March 26, 2021 by Persiflage

An update on this post; the Arbeitsgemeinschaft on derived Galois deformation rings and the cohomology of arithmetic groups will now be taking place the week of April 5th. Here is some practical information if you are curious. Is there somewhere … Continue reading →

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Test Your Intuition: p-adic local Langlands edition

Posted on March 9, 2021 by Persiflage

Taking a page from Gil Kalai, here is a question to test your intuition about 2-dimensional crystalline deformation rings. Fix a representation: \(\rho: G_{\mathbf{Q}_p} \rightarrow \mathrm{GL}_2(\overline{\mathbf{F}}_p)\) after twisting, let me assume that this representation has a crystalline lift of weight … Continue reading →

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Fermat Challenge

Posted on February 21, 2021 by Persiflage

A challenge inspired from a question of Doron Zeilberger. Do there exist arbitrarily large integers \(n\) with the following property: There exists an ordered field \(F\) such that \(x^n+ y^n = z^n\) has solutions in \(F\) with \(xyz \ne 0\). … Continue reading →

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Ramanujan Machine Redux

Posted on February 11, 2021 by Persiflage

I had no intention to discuss the Ramanujan Machine again, but over the past few days there has been a flurry of (attempted) trollish comments on that post, so after taking a brief look at the latest version, I thought … Continue reading →

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Hire my students!

Posted on November 18, 2020 by Persiflage

I have three students graduating this year: Shiva Chidambaram, Eric Stubley, and Noah Taylor. In light of the last post, I should give them a boost by reminding you of their (numerous) results which have been discussed on this blog. … Continue reading →

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The eigencurve is (still) proper

Posted on October 23, 2020 by Persiflage

Although I don’t think about it so much anymore, the eigencurve of Coleman-Mazur was certainly one of my first loves. I can’t quite say I learnt about \(p\)-adic modular forms at my mother’s knee, but I did spend a formative … Continue reading →

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