Category Archives: Mathematics

Galois Representations for non-self dual forms, Part III

Posted on April 27, 2013 by Persiflage

Here are some complements to the previous remarks, considered in Part I and Part II. First, in order to deal with non-zero weights, one has to replace the Shimura varieties \(Y\), \(X\), \(W\) by Kuga-Satake varieties over these spaces. This … Continue reading →

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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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Torsion in the cohomology of co-compact arithmetic lattices

Posted on February 6, 2013 by Persiflage

Various authors (including Bergeron and Venkatesh) have shown that the cohomology of certain arithmetic groups have a lot of torsion. For example, if \(\Gamma\) is a co-compact arithmetic lattice in \(\mathrm{SL}_2(\mathbf{C})\), and \(\mathcal{L}\) is an acyclic local system, then \(\log … Continue reading →

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A scandal in Romania

Posted on January 28, 2013 by Persiflage

I was invited to review some research proposals for the CNCS. They offered a modest remuneration for my time (something like €168, I believe). For privacy reasons I won’t comment on the proposals I read, suffice to say that they … Continue reading →

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Small Cyclotomic Integers

Posted on January 26, 2013 by Persiflage

Julia Robinson is a famous mathematician responsible for fundamental work in logic and in particular on Hilbert’s Tenth problem. Less well known nowadays is that her husband, Raphael Robinson, was a number theorist at Berkeley. One question R.Robinson asked concerned … Continue reading →

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Random p-adic Matrices

Posted on January 23, 2013 by Persiflage

Does anyone know if the problem of random matrices over (say) \(\mathbf{Z}_p\) have been studied? Here I mean something quite specific. One could do the following, namely, since \(\mathbf{Z}_p\) is compact with a natural measure, look at random elements in … Continue reading →

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NT Seminar: A haruspicy

Posted on January 18, 2013 by Persiflage

Following JSE’s advice, I will blog on something that I know absolutely nothing about. Apologies in advance for mathematical errors! SLM gave a number theory seminar this week about the first Betti number of \(\Gamma(n)\) — as \(n\) varies — … Continue reading →

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Number theory and 3-manifolds

Posted on January 13, 2013 by Persiflage

It used to be the case that the Langlands programme could be used to say something interesting about arithmetic 3-manifolds qua hyperbolic manifolds. Now, after the work of Agol, Wise, and others has blown the subject to smithereens, this gravy … Continue reading →

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