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On the Modularity of Three Calabi–Yau Threefolds With Bad Reduction at 11
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- Journal:
- Canadian Mathematical Bulletin / Volume 49 / Issue 2 / 01 June 2006
- Published online by Cambridge University Press:
- 20 November 2018, pp. 296-312
- Print publication:
- 01 June 2006
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- Article
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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.
