Category Archives: Mathematics

The nearly ordinary deformation ring is (usually) torsion over weight space

Posted on September 19, 2014 by Persiflage

Let \(F/{\mathbf{Q}}\) be an arbitrary number field. Let \(p\) be a prime which splits completely in \(F\), and consider an absolutely irreducible representation: \(\rho: G_{F} \rightarrow {\mathrm{GL}}_2({\overline{\mathbf{Q}}}_p)\) which is unramified outside finitely many primes. If one assumes that \(\rho\) is … Continue reading →

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The Artin conjecture is rubbish

Posted on September 10, 2014 by Persiflage

Let \(\rho: G_{\mathbf{Q}} \rightarrow \mathrm{GL}_N(\mathbf{C})\) be a continuous irreducible representation. Artin conjectured that the L-function \(L(\rho,s)\) is analytically continues to an entire function on \(\mathbf{C}\) (except for the trivial representation where the is a simple pole at one) and satisfies … Continue reading →

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K_2(O_F) for number fields F

Posted on September 4, 2014 by Persiflage

Belabas and Gangl have a nice paper ( Generators and relations for \(K_2({\mathcal{O}}_F)\), which can be found here) where they compute \(K_2({\mathcal{O}}_E)\) for a large number of quadratic fields \(E\). There main result is a method for proving upper bounds … Continue reading →

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The decline and fall of Publications Mathématiques de l’lHÉS

Posted on August 15, 2014 by Persiflage

I want to discuss the decline of a once great journal. How did IHES go from this: and this: to this: It is a sad state of affairs. To be clear, I am talking about the typesetting here. The old … Continue reading →

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Is Serre’s conjecture still open?

Posted on August 10, 2014 by Persiflage

The conjecture in this paper has indeed been proven. But that isn’t the entire story. Serre was fully aware of Katz modular forms of weight one. However, Serre was too timid was prudently conservative and made his conjecture only for … Continue reading →

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Horizontal Vanishing Conjectures.

Posted on August 8, 2014 by Persiflage

Let \(F\) be a number field, and let \(\mathbf{G}\) be a reductive group over \(F\), and let \(\Gamma\) be a congruence subgroup of \(\mathbf{G}(\mathcal{O}_F)\). I can hear BC objecting that this doesn’t make sense without extra choices; if you have … Continue reading →

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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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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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