Author Archives: Persiflage

The Abelian House is not closed

Posted on August 3, 2014 by Persiflage

Today I will talk about \(\displaystyle{\frac{97 + 26 \sqrt{13}}{27} = 7.064604\ldots}\) For an algebraic integer \(\alpha\), the house \(\overline{|\alpha|}\) is the absolute value of the largest conjugate of \(\alpha.\) Kronecker proved the following: If \(\overline{|\alpha|} \le 1\), then either \(\alpha … Continue reading →

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The distribution of Hecke eigenvalues, part II

Posted on August 1, 2014 by Persiflage

Here are some numbers from KB promised in my last post. “For the first 61595 newforms of squarefree level coprime to 15 here’s the field extension of Z/3Z generated by the \(a_5\) field extensions:” \([\mathbf{F}_3(a_5):\mathbf{F}_3]\) Total Number Number of Galois … Continue reading →

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The distribution of Hecke eigenvalues, Part I

Posted on July 29, 2014 by Persiflage

Here is a question I raised at the Puerto-Rico conference during one of the “problem sessions.” Toby Gee seems to remember that I had some half-baked heuristics that predicted both A and B below, but perhaps one of my readers … Continue reading →

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En Passant IV

Posted on July 19, 2014 by Persiflage

My students Richard Moy and Joel Specter have uploaded their paper on partial weight one Hilbert modular forms, previously discussed here, to the ArXiv. Germany now leads the world in both soccer and perfectoid spaces. This is a recipe for … Continue reading →

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A public service announcement concerning Fontaine-Mazur for GL(1)

Posted on July 12, 2014 by Persiflage

There’s a rumour going around that results from transcendence theory are required to prove the Fontaine-Mazur conjecture for \(\mathrm{GL}(1)\). This is not correct. In Serre’s book on \(\ell\)-adic representations, he defines a \(p\)-adic representation \(V\) of a global Galois group … Continue reading →

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Report from Luminy

Posted on July 8, 2014 by Persiflage

For how long has Luminy been infested with bloodthirsty mosquitoes? The combination of mosquitoes in my room with the fact that my bed was 6′ long with a completely unnecessary headboard (which meant that I had to sleep on an … Continue reading →

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An Obvious Claim

Posted on July 6, 2014 by Persiflage

It’s been a while since I saw Serre’s “how to write mathematics badly” lecture, but I’m pretty sure there would have been something about the dangers of using the word “obvious.” After all, if something really is obvious, then it … Continue reading →

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Huuuuuge piles of cash

Posted on June 23, 2014 by Persiflage

As widely reported today, the first of the “Breakthrough” prizes in mathematics have been announced. Leaving aside the question as to whether such awards are sensible (Persiflage is more sympathetic to capitalist principles than your average pinko marxist mathematician), I … Continue reading →

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There are non-liftable weight one forms modulo p for any p

Posted on June 10, 2014 by Persiflage

Let \(p\) be any prime. In this post, we show that there is an integer \(N\) prime to \(p\) such that \(H^1(X_1(N),\omega_{\mathbf{Z}})\) has a torsion class of order \(p\). Almost equivalently, there exists a Katz modular form of level \(N\) … Continue reading →

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Math and Genius

Posted on June 6, 2014 by Persiflage

Jordan Ellenberg makes a compelling case, as usual, on the pernicious cultural notion of “genius.” Jordan’s article also brought to mind a thought provoking piece on genius by Moon Duchin here (full disclosure: the link on Duchin’s website has the … Continue reading →

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