On Hilbert modular threefolds of discriminant 49
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On Hilbert modular threefolds of discriminant 49 Lev A. Borisov · Paul E. Gunnells
Published online: 9 November 2012 © Springer Basel 2012
Abstract Let K be the totally real cubic field of discriminant 49, let O be its ring of integers, and let p ⊂ O be the prime over 7. Let ( p) ⊂ = S L 2 (O) be the principal congruence subgroup of level p. This paper investigates the geometry of the Hilbert modular threefold attached to ( p) and some related varieties. In particular, we discover an octic in P3 with 84 isolated singular points of type A2 . Mathematics Subject Classification (1991)
Primary 11F41; Secondary 14G35
1 Introduction Let K be the totally real cubic field of discriminant 49, let O be its ring of integers, and let p ⊂ O be the prime over 7. Let ( p) ⊂ = S L 2 (O) be the principal congruence subgroup of level p. This paper investigates the geometry of the Hilbert modular threefold X ◦ = ( p)\H3 and some related varieties: (1) Let X be the minimal compactification of X ◦ , and let X ch be the singular toroidal compactification built using the fans determined by taking the cones on the faces
Dedicated to Don Zagier, on the occasion of his 60th birthday. The authors were partially supported by the NSF, through grants DMS–1003445 (LB) and DMS–0801214 (PG). L. A. Borisov Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd, Piscataway, NJ 08854, USA e-mail: [email protected] P. E. Gunnells (B) Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA 01003, USA e-mail: [email protected]
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L. A. Borisov, P. E. Gunnells
of the convex hulls of the totally positive lattice points in the cusp data. Then X ch is the canonical model of X (Theorem 8.4). (2) We construct parallel weight 1 Eisenstein series F0 , F1 , F2 , F4 and a parallel weight 2 Eisenstein series E 2 that generate the ring of symmetric Hilbert modular forms of level p and parallel weight (i.e., the subring of the parallel weight Hilbert modular forms invariant under the action of the Galois group Gal(K /Q) Z/3Z) (Theorem 5.3). (3) There is a weighted homogeneous polynomial P of degree 8 with 42 terms such that P(F0 , F1 , F2 , F4 , E 2 ) = 0 (Proposition 6.7). This polynomial generates the ideal of relations on the Fi and E 2 , and the symmetric Hilbert modular threefold X Gal = X/Gal(K /Q) is the hypersurface cut out by P = 0 in the weighted projective space P(1, 1, 1, 1, 2) (Theorem 7.6). (4) Let Q be the polynomial obtained from P by setting the weight 2 variable to zero. Then Q has 24 terms and defines a degree 8 hypersurface in P3 with singular locus being 84 quotient singularities of type A2 (Proposition 9.1). These results can be considered part of the venerable tradition of writing explicit equations for modular varieties, a tradition including (i) the Klein quartic, which presents the modular curve X (7) as an explicit quartic in P2 , (ii) the Igusa quartic, which is the minimal compactification of the Siegel modular threefold of leve
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