Category Archives: Mathematics

Serre 1: Calegari 0

Posted on October 18, 2016 by Persiflage

I just spent a week or so trying to determine whether Serre’s conjecture about the congruence subgroup property was false for a very specific class of S-arithmetic groups. The punch line, perhaps not surprisingly, was that I had made an … Continue reading →

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Prime divisors of polynomials

Posted on May 29, 2016 by Persiflage

A heuristic model from the last post suggests that the “expected” order of the Galois group associated to a weight one modular form of projective type \(A_5\) is infinite. And when one tries to solve the inverse Galois problem for … Continue reading →

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Central Extensions and Weight One Forms

Posted on May 23, 2016 by Persiflage

As mentioned in the comments to the last post, Kevin Buzzard and Alan Lauder have made an extensive computation of weight one modular forms in characteristic zero (see also here). Thinking about what that data might contain, I wondered about … Continue reading →

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LMFDB!?

Posted on May 12, 2016 by Persiflage

The LMFDB has gone live! I previously expressed on this blog a somewhat muted opinion about certain aspects of the website’s functionality, and it seems that my complaints have mainly been addressed in the latest version. On the other hand, … Continue reading →

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Report From Berkeley

Posted on April 22, 2016 by Persiflage

My recent trip to Berkeley did not result in a chance to test whether the Cheeseboard pizza maintained its ranking, but did give me the opportunity to attend the latest Bay Area Number Theory and Algebraic Geometry day, on a … Continue reading →

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Tensor Products

Posted on March 28, 2016 by Persiflage

Let \(W\) be an irreducible representation of a finite group \(G.\) Say that \(W\) is tensor indecomposable if any isomorphism \(W = U \otimes V\) implies that either \(U\) or \(V\) is a character. In conversations with Matt and Toby … Continue reading →

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Hilbert Modular Forms of Partial Weight One, Part III

Posted on October 17, 2015 by Persiflage

My student Richard Moy is graduating! Richard’s work has already appeared on this blog before, where we discussed his joint work with Joel Specter showing that there existed non-CM Hilbert modular forms of partial weight one. Today I want to … Continue reading →

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Counting solutions to a_p = λ

Posted on May 15, 2015 by Persiflage

We know that the eigenvalue of \(T_2\) on \(\Delta\) is \(24.\) Are there any other level one cusp forms with the same Hecke eigenvalue? Maeda’s conjecture in its strongest form certainly implies that there does not. But what can one … Continue reading →

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144169

Posted on May 12, 2015 by Persiflage

The space of classical modular cuspforms of level one and weight 24 has dimension two — the smallest weight for which the dimension is not zero or one. What can we say about the Hecke algebra acting on this space … Continue reading →

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Chenevier on the Eigencurve

Posted on April 24, 2015 by Persiflage

Today I wanted to mention a theorem of Chenever about components of the Eigencurve. Let \(\mathcal{W}\) denote weight space (which is basically a union of discs), and let \(\pi: \mathcal{E} \rightarrow \mathcal{W}\) be the Coleman-Mazur eigencurve together with its natural … Continue reading →

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