Tag Archives: Class Field Theory

Quadratic Reciprocity

Posted on May 2, 2023 by Persiflage

I accidentally proved quadratic reciprocity in class today, or at least three quarters of a proof. Can you finish it off? Here’s the proof: start with a real quadratic field \(K\), and the sequence \(1 \rightarrow \mathcal{O}^{\times}_K \rightarrow K^{\times} \rightarrow … 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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En Passant V

Posted on July 7, 2019 by Persiflage

(warning: today’s persiflage comes with possible extra snark due to sleep deprivation) The Ramanujan Machine: I learnt from John Baez on twitter about The Ramanujan Machine, a project designed to “help reveal [the] underlying structure” of the “fundamental constants” of … Continue reading →

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Schaefer and Stubley on Class Groups

Posted on October 28, 2017 by Persiflage

I talked previously about work of Wake and Wang-Erickson on deformations of Eisenstein residual representations. In that post, I also mentioned a paper of Emmanuel Lecouturier who has also proved some very interesting theorems. Today, I wanted to talk about … Continue reading →

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Who proved it first?

Posted on April 25, 2017 by Persiflage

During Joel Specter’s thesis defense, he started out by remarking that the \(q\)-expansion: \(\displaystyle{f = q \prod_{n=1}^{\infty} (1 – q^n)(1 – q^{23 n}) = \sum a_n q^n}\) is a weight one modular forms of level \(\Gamma_1(23),\) and moreover, for \(p\) … Continue reading →

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