Abstract
The divisor class group, (co)homology, and Picard group of the closed fibers of various regular proper models of the product of two elliptic curves at a place of multiplicative reduction are computed. The variation of the isomorphism class of the closed fiber with the variation of the elliptic curves is discussed. The higher direct images of the sheaf, \(\mathbb{Z}/n\), are computed when n is prime to the residue characteristic.
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Acknowledgements
The hospitality of the Isaac Newton Institute for Mathematical Sciences and of the Max-Planck-Institut für Mathematik, Bonn where some of this work was done is gratefully acknowledged as is the partial support of the NSF, DMS 99-70500, and the NSA, H98230-08-1-0027. Thanks to A. Weisse for help with the figures and to the referee for comments which led to an improvement of the exposition and to Debbie Iscoe at the Fields Institute for translating AMSTeX to LaTeX.
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Schoen, C. (2013). Invariants of Regular Models of the Product of Two Elliptic Curves at a Place of Multiplicative Reduction. In: Laza, R., Schütt, M., Yui, N. (eds) Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds. Fields Institute Communications, vol 67. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6403-7_17
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DOI: https://doi.org/10.1007/978-1-4614-6403-7_17
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