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On Hilbert modular threefolds of discriminant 49

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Abstract

Let \(K\) be the totally real cubic field of discriminant \(49\), let \({\fancyscript{O}}\) be its ring of integers, and let \(p\subset {\fancyscript{O}}\) be the prime over \(7\). Let \(\Gamma (p)\subset \Gamma = SL_{2} ({\fancyscript{O}})\) be the principal congruence subgroup of level \(p\). This paper investigates the geometry of the Hilbert modular threefold attached to \(\Gamma (p) \) and some related varieties. In particular, we discover an octic in \(\mathbb{P }^3\) with \(84\) isolated singular points of type \(A_2\).

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Acknowledgments

We thank Igor Dolgachev for referring us to Edge’s paper [5] (Remark 9.2), and we thank Don Zagier for his encouragement of this work. We thank an anonymous referee for a very careful reading and for many helpful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd, Piscataway, NJ, 08854, USA

    Lev A. Borisov

  2. Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA, 01003, USA

    Paul E. Gunnells

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  1. Lev A. Borisov
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  2. Paul E. Gunnells
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Correspondence to Paul E. Gunnells.

Additional information

The authors were partially supported by the NSF, through grants DMS–1003445 (LB) and DMS–0801214 (PG)

Dedicated to Don Zagier, on the occasion of his 60th birthday.

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Borisov, L.A., Gunnells, P.E. On Hilbert modular threefolds of discriminant 49. Sel. Math. New Ser. 19, 923–947 (2013). https://doi.org/10.1007/s00029-012-0108-0

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