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On the Modularity of Three Calabi–Yau Threefolds With Bad Reduction at 11

Published online by Cambridge University Press:  20 November 2018

Matthias Schütt
Matthias Schütt*
Affiliation:
Institut für Mathematik (C), Universität Hannover, Welfengarten 1, 30060 Hannover, Germany e-mail: schuett@math.uni-hannover.de
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Abstract

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This paper investigates the modularity of three non-rigid Calabi–Yau threefolds with bad reduction at 11. They are constructed as fibre products of rational elliptic surfaces, involving the modular elliptic surface of level 5. Their middle $\ell$-adic cohomology groups are shown to split into two-dimensional pieces, all but one of which can be interpreted in terms of elliptic curves. The remaining pieces are associated to newforms of weight 4 and level 22 or 55, respectively. For this purpose, we develop a method by Serre to compare the corresponding two-dimensional 2-adic Galois representations with uneven trace. Eventually this method is also applied to a self fibre product of the Hesse-pencil, relating it to a newform of weight 4 and level 27.

Keywords

14J3211F1111F2320C12
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 49 , Issue 2 , 01 June 2006 , pp. 296 - 312
Copyright
Copyright © Canadian Mathematical Society 2006

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