Category Archives: Mathematics

Classic Papers in Number Theory

Posted on December 27, 2012 by Persiflage

One of my students came to me with the idea of having a reading course on “classic papers in number theory”. The idea is for everyone to spend the week reading a particular paper, and then have one student lead … Continue reading →

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The Two Cultures of Mathematics: A Rebuttal

Posted on December 12, 2012 by Persiflage

Gowers writes thoughtfully about combinatorics here, in an essay which references Snow’s famous lectures (or famous amongst mathematicians – I’ve never met anyone else who has ever heard of them). The trouble, however, starts (as it often does) with the … Continue reading →

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Hilbert Modular Forms of Partial Weight One, Part II

Posted on December 8, 2012 by Persiflage

Anyone who spends any time thinking about Hilbert modular forms of partial weight one — see part I — should, at some point, wonder whether there actually exist any examples, besides the “trivial” examples arising as inductions of Grossencharacters. Fred … Continue reading →

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There are no unramified abelian extensions of Q (almost)

Posted on November 27, 2012 by Persiflage

In my class on modularity, I decided to explain what Wiles’ argument (in the minimal case) would look like for \(\mathrm{GL}(1)/F\). There are two ways one can go with this. On the one hand, one can try to prove (say) … Continue reading →

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Jacobi by pure thought

Posted on October 26, 2012 by Persiflage

JB asks whether there is a conceptual proof of Jacobi’s formula: \(\Delta = q \prod_{n=1}^{\infty}(1 – q^n)^{24}\) Here (to me) the best proof is one that requires the least calculation, not necessarily the “easiest.” Here is my attempt. We use … Continue reading →

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En Passant

Posted on October 24, 2012 by Persiflage

Several times in NYC, I’ve had the chance to visit Eataly, an italian food court/upscale delicatessen run by Mario Batali. You can either sit down at one of the various restaurants for an antipasto plate with a glass of wine, … Continue reading →

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Remarks on Buzzard-Taylor

Posted on October 18, 2012 by Persiflage

Let \(\rho: G_{\mathbf{Q},S} \rightarrow \mathrm{GL}_2(\overline{\mathbf{Q}}_p)\) be continuous and unramified at \(p\). The Fontaine-Mazur conjecture predicts that \(\rho\) has finite image and is automorphic. Buzzard and Taylor proved this result under the assumption the natural assumption that \(\rho\) is odd, that … Continue reading →

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Why it is good to be Pure

Posted on October 15, 2012 by Persiflage

There do not exist any regular pure motives \(M\) over \(\mathbf{Q}\) which are not essentially self dual. Here is why. \(M\) gives rise to a compatible family of Galois representations for each rational prime \(v\) such that the characteristic polynomial … Continue reading →

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Hilbert Modular Forms of Partial Weight One, Part I

Posted on October 13, 2012 by Persiflage

Let \(\pi\) be an algebraic Hilbert modular cuspform for some totally real field \(F^{+}\). Then, associated to \(\pi\), one has a compatible family of Galois representations: \(r_{\lambda}(\pi): G_{F^{+}} \rightarrow \mathrm{GL}_2(\mathcal{O}_{\lambda})\) which are unramified outside finitely many primes (this is the … Continue reading →

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Even Galois Representations mod p

Posted on October 7, 2012 by Persiflage

Suppose that \(\overline{\rho}: G_{\mathbf{Q}}: \rightarrow \mathrm{GL}_2(\overline{\mathbf{F}}_p)\) is a continuous  irreducible Galois representation. What does the Langlands program say about such \(\overline{\rho}\)? When \(\overline{\rho}\) is odd, the situation is quite satisfactory, the answer being given by Serre’s conjecture. For example, having fixed a … Continue reading →

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