Three modular fivefolds

Abstract: Eichler and Shimura showed that to every rational Hecke eigenform of weight 2 there is associated an isogeny class of elliptic curves with the same L-function (and Wiles, Taylor-Wiles, et al. proved a very famous converse).  Work of Elkies and Schutt gives a similar result for eigenforms of weight 3, though the result has a very different flavour since all such forms arise from imaginary quadratic fields with class group of exponent dividing 2.  There have been many attempts to associate Calabi-Yau threefolds to eigenforms of weight 4 in such a way that the interesting part of the L-function of the threefold matches that of the modular form, but in general the problem of doing so is open.  In this talk we give three examples of double covers of projective 5-space whose L-functions involve an eigenform of weight 6.  Two of the examples are proved; one is known to have a Calabi-Yau desingularization, but the other is not.  In connection with these we will describe some new results (joint with Colin Ingalls) on resolutions of singularities of double covers. The third example, still conjectural, appears to point to a previously unknown identity of hypergeometric functions.  We will also show how to use an idea of Burek to find quotients of our varieties for which the point counts can be expressed in terms of a single eigenform of weight 6.

number theory

Audience: researchers in the topic

CRM-CICMA Québec Vermont Seminar Series

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