Author Archives: Persiflage

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 →

Posted in Mathematics, Students | Tagged , | 2 Comments

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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Why Blog?

Posted on October 6, 2012 by Persiflage

The reasons for running a blog are various; they run the gamut from narcissism to simple self-aggrandizement. Time will tell which type of blog this will be. I’m not really looking to find an audience beyond those people I already know … Continue reading →

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