Tag Archives: Galois Representations

Inverse Galois Problem

Posted on April 24, 2013 by Persiflage

My favourite group as far as the inverse Galois problem goes is \(G = \mathrm{SL}_2(\mathbf{F}_p)\). This is not known to be a Galois group over \(\mathbf{Q}\) for any \(p > 13\), the difficulty of course being that is must correspond … Continue reading →

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Galois Representations for non self-dual forms, Part II

Posted on April 21, 2013 by Persiflage

(Now with updates!) Let’s recap from part I. We have a Shimura variety \(Y\), a minimal projective compactification \(X\), and a (family of) smooth toroidal compactifications \(W\). We also have Galois representations of the correct shape associated to eigenclasses in … Continue reading →

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Galois Representations for non self-dual forms, Part I

Posted on March 26, 2013 by Persiflage

This is the first of a series of posts discussing the recent work of Harris, Lan, Taylor, and Thorne on constructing Galois representations associated to regular algebraic automorphic forms for \(\mathrm{GL}(n)\) over a CM field \(F/F^{+}\). I will dispense with … 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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