Category Archives: Mathematics

What the slopes are

Posted on February 25, 2023 by Persiflage

Let \(f\) be a classical modular eigenform of weight \(k\), for example, \(f = \Delta\). The Ramanujan conjecture states that the Hecke eigenvalues \(a_p\) satisfy the bound \(|a_p| \le 2 p^{(k-1)/2}.\) A slightly fancier but cleaner way of saying this … Continue reading →

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Check the arXiV regularly!

Posted on February 18, 2023 by Persiflage

In a previous post, I discussed a new result of Smith which addressed the following question: given a measure \(\mu\) on \(\mathbf{R}\) supported on some finite union of intervals \(\Sigma\), under what conditions do there exist polynomials of arbitrarily large … Continue reading →

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Potential Modularity of K3 surfaces

Posted on November 15, 2022 by Persiflage

This post is to report on results of my student Chao Gu who is graduating this (academic) year. If \(A/F\) is an abelian surface, then one can associate to \(A\) a K3 surface \(X\) (the Kummer surface) by blowing up … Continue reading →

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The future is now; recap from Cetraro

Posted on July 31, 2022 by Persiflage

I’ve just returned from the second Journal of Number Theory biennial conference in Italy. It’s always nice to get a chance to see slices of number theory one wouldn’t otherwise see at the conferences I usually go to (although this … 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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Locally induced representations

Posted on June 16, 2022 by Persiflage

Today is a post about work of my student Chengyang Bao. Recall that Lehmer’s conjecture asks whether \(\tau(p) \ne 0\) for all primes \(p\), where \(\Delta = q \prod_{n=1}^{\infty} (1 – q^n)^{24} = \sum \tau(n) q^n\) is Ramanujan’s modular form. … Continue reading →

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Joël Bellaïche

Posted on June 14, 2022 by Persiflage

Very sad to hear that Joël Bellaïche has just died. He got his PhD at the same time as me, and I first got to know him during the Durham conference in 2004 and later at the eigenvarieties semester at … Continue reading →

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Murphy’s Law for Galois Deformation Rings

Posted on April 23, 2022 by Persiflage

Today’s post is about work of my student Andreea Iorga! A theorem of Ozaki from 2011, perhaps not as widely known as expected, says the following: Theorem: Let \(p\) be prime, and let \(G\) be a finite \(p\)-group. Then there … Continue reading →

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What would Deuring do?

Posted on April 13, 2022 by Persiflage

This is an incredibly lazy post, but why not! Matt is running a seminar this quarter on the Weil conjectures. It came up that one possible way to prove the Weil conjectures for elliptic curves over finite fields is to … Continue reading →

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A random curve over Q

Posted on April 2, 2022 by Persiflage

Let \(X/\mathbf{Q}\) be a smooth projective curve. I would like to be able to say that the motive \(M\) associated to \(X\) “generally” determines \(X\). That is, I would like to say it in a talk without feeling like I’m … Continue reading →

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