Siegel Modular Threefolds Research Articles

A geometric Jacquet\u2013Langlands correspondence for paramodular Siegel threefolds

We study the Picard–Lefschetz formula for Siegel modular threefolds of paramodular level and prove the weight-monodromy conjecture for its middle degree inner cohomology. We give some applications to the Langlands programme: using Rapoport-Zink uniformisation of the supersingular locus of the special fiber, we construct a geometric Jacquet–Langlands correspondence between {text {GSp}}_4 and a definite inner form, proving a conjecture of Ibukiyama. We also prove an integral version of the weight-monodromy conjecture and use it to deduce a level lowering result for cohomological cuspidal automorphic representations of {text {GSp}}_4.

Antisymmetric Paramodular Forms of Weights 2 and 3

Abstract We define an algebraic set in $23$-dimensional projective space whose ${{\mathbb{Q}}}$-rational points correspond to meromorphic, antisymmetric, paramodular Borcherds products. We know two lines inside this algebraic set. Some rational points on these lines give holomorphic Borcherds products and thus construct examples of Siegel modular forms on degree 2 paramodular groups. Weight $3$ examples provide antisymmetric canonical differential forms on Siegel modular three-folds. Weight $2$ is the minimal weight and these examples, via the paramodular conjecture, give evidence for the modularity of some rank 1 abelian surfaces defined over $\mathbb{Q}$.

On isomorphisms between Siegel modular threefolds

The Satake compactification of the moduli space of principally polarized abelian surfaces with a level two structure has a degree 8 endomorphism. The aim of this paper is to show that this result can be extended to other modular threefolds. The main tools are Siegel modular forms and Satake compactifications of arithmetic quotients of the Siegel upper-half space. Indeed, the construction of the degree 8 endomorphism on suitable modular threefolds is done via an isomorphism of graded rings of modular forms. By studying the action of the Fricke involution one gets a further extension of the previous result to other modular threefolds. The possibility of a similar situation in higher dimensions is also discussed.

Some Siegel Threefolds with a Calabi-Yau Model II

In a previous paper, we described some Siegel modular threefolds which ad- mit a Calabi-Yau model. Using a different method we give in this paper an enlarged list of such varieties. Basic for this method is a paper of van Geemen and Nygaard. They study a variety X that is the complete intersection of four quadrics in P 7 (C). This is biholomorphic equivalent to the Satake compactification of H2=Γ ' for a certain subgroup Γ ' Sp(2;Z) and it will be the starting point of our investigation. It has been pointed out that a (projective) small resolution of this variety is a rigid Calabi-Yau manifold ˜ X. Then we will consider the action of quotients of modular groups on X and study possible resolutions that admits a Calabi-Yau model in the category of complex spaces.

The meromorphic continuation of the zeta function of Siegel modular threefolds over totally real fields

In this paper we prove the meromorphic continuation of the zeta function of Siegel modular threefolds over arbitrary totally real number elds.

On Siegel three-folds with a projective Calabi–Yau model

In the papers [FS1], [FS2] we described some Siegel modular threefolds which admit a weak Calabi–Yau model.*) Not all of them admit a projective model. In fact, Bert van Geemen, in a private communication, pointed out a significative example which cannot admit a projective model. His comment was a starting motivation for this paper. We mention that a weak Calabi–Yau threefold is projective if, and only if, it admits a Kaehler metric. The purpose of this paper is to exhibit criteria for the projectivity, to treat several examples, and to compute their Hodge numbers. Some of these Calabi–Yau manifolds seem to be new. For example, we obtain a rigid Calabi–Yau manifold with Euler number 4. Basic for our examples is a certain complete intersection X of four quadrics introduced the paper [GN] of van Geemen and Nygaard:

Tate conjecture for twisted Siegel modular threefolds

In this paper we prove Tate conjecture for twisted Siegel modular threefolds. 2010 Mathematics subject classification: 11F41, 11F80, 11R42, 11R80.

L-functions of [formula omitted

We prove most of B. van Geemen and D. van Straten's conjectures on the explicit description of Andrianov L-functions of Siegel cuspforms of degree 2 of weight 3 for the group Γ ( 4 , 8 ) , which are contained in [B. van Geemen, D. van Straten, The cuspform of weight 3 on Γ 2 ( 2 , 4 , 8 ) , Math. Comp. 61 (204) (1993) 849–872]. These L-functions are related to the Galois representations on the Siegel modular threefold Γ ( 4 , 8 ) ∖ H 2 as determined by B. van Geemen and N. Nygaard [B. van Geemen, N.O. Nygaard, On the geometry and arithmetic of some Siegel modular threefolds, J. Number Theory 53 (1995) 45–87].

Robert MacPherson and Arithmetic Groups

We survey contributions of Robert MacPherson to the theory of arithmetic groups. There are two main areas we discuss: (i) explicit reduction theory for Siegel modular threefolds, and (ii) constructions of compactifications of locally symmetric spaces. The former is joint work with Mark McConnell, the latter with Lizhen Ji.

Minimal Siegel modular threefolds

The starting point of this paper is the maximal extension Γ*t of Γt, the subgroup of Sp4(ℚ) which is conjugate to the paramodular group. Correspondingly we call the quotient [Ascr ]*t=Γ*t\ℍ2 the minimal Siegel modular threefold. The space [Ascr ]*t and the intermediate spaces between [Ascr ]t=Γt\ℍ2 which is the space of (1, t)-polarized abelian surfaces and [Ascr ]*t have not yet been studied in any detail. Using the Torelli theorem we first prove that [Ascr ]*t can be interpreted as the space of Kummer surfaces of (1, t)-polarized abelian surfaces and that a certain degree 2 quotient of [Ascr ]t which lies over [Ascr ]*t is a moduli space of lattice polarized K3 surfaces. Using the action of Γ*t on the space of Jacobi forms we show that many spaces between [Ascr ]t and [Ascr ]*t possess a non-trivial 3-form, i.e. the Kodaira dimension of these spaces is non-negative. It seems a difficult problem to compute the Kodaira dimension of the spaces [Ascr ]*t themselves. As a first necessary step in this direction we determine the divisorial part of the ramification locus of the finite map [Ascr ]t→[Ascr ]*t. This is a union of Humbert surfaces which can be interpreted as Hilbert modular surfaces.

Fundamental group of locally symmetric varieties

We investigate the fundamental group of a compactified locally symmetric variety after resolution of singularities. We show that it can be any finite group and give examples to show that even for Siegel modular threefolds it can be complicated. We also calculate some cases of special interest, including the fundamental group of the Barth-Nieto space.

On the Geometry and Arithmetic of Some Siegel Modular Threefolds

On the Geometry and Arithmetic of Some Siegel Modular Threefolds

Explicit reduction theory for Siegel modular threefolds

Explicit reduction theory for Siegel modular threefolds

The Picard group of Siegel modular threefolds.

The Picard group of Siegel modular threefolds.

Moduli of Abelian Varieties

On extra components in the functorial compactification of Ag.- On Mumford's uniformization and Neron models of Jacobians of semistable curves over complete rings.- Torelli theorem via Fourier-Mukai transform.- On the Andre-Oort conjecture for Hilbert modular surfaces.- Toroidal resolutions for some matrix singularities.- Formal Brauer groups and moduli of abelian surfaces.- Isogeny classes of abelian varieties with no principal polarizations.- Igusa's modular form and the classification of Siegel modular threefolds.- Mirror symmetry and quantization of abelian varieties.- Group schemes with additional structures and Weyl group cosets.- Moduli space of elliptic curves with Heisenberg level structure.- Singularities of the height strata in the moduli of K3 surfaces.- A stratification of a moduli space of abelian varieties.- Newton polygon strata in the moduli space of abelian varieties.- The dimension of Oort strata of Shimura varieties of PEL-type.- Hyperelliptic Jacobians and modular representations.- Windows for displays of p-divisible groups.