Tag Archives: Akshay Venkatesh

The class number 100 problem

Posted on January 19, 2017 by Persiflage

Some time ago, Mark Watkins busted open the “class number n” problem for smallish n, finding all imaginary quadratic fields of class number at most 100 (the original paper is here) Although the paper describes the method in detail, it … Continue reading →

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Artin No-Go Lemma

Posted on December 13, 2016 by Persiflage

The problem of constructing Galois representations associated to Maass forms with eigenvalue 1/4 is, by now, a fairly notorious problem. The only known strategy, first explained by Carayol, is to first transfer the representation to a unitary group over an … Continue reading →

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Ventotene, Part II

Posted on March 17, 2016 by Persiflage

I promised to return to a more mathematical summary of the conference in Ventotene, and indeed I shall do so in the next two posts. One of the themes of the conference was bounding the order of the torsion subgroup … Continue reading →

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Derived Langlands

Posted on December 22, 2014 by Persiflage

Although it has been in the air for some time, it seems as though ideas from derived algebraic geometry have begun to inform developments in the Langlands program. (A necessary qualifier: I am talking about reciprocity in the classical arithmetic … Continue reading →

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Harris 60

Posted on December 15, 2014 by Persiflage

I’ve just returned from the excellent MSRI workshop which honored Michael Harris’ 60th birthday, and here is a brief summary of some of the gossip and mathematics I picked up when I was there. First, let me take note of … 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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Gross Fugue

Posted on January 11, 2014 by Persiflage

Here are some variations on the theme of the last post, which is also related to a problem of Dick Gross. In this post, I want to discuss weight one modular forms where the level varies in the “vertical” aspect … Continue reading →

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En Passant III

Posted on August 31, 2013 by Persiflage

Question: When you are sick in bed, can you do any mathematics? I just spent the past few weeks with a sinus infection and was completely unable to do anything productive, that is, apart from writing an NSF grant (which … 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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