We show that a given rational Shimura curve Y with strictly maximal Higgs field in the moduli space of g-dimensional abelian varieties does not generically intersect the Schottky locus for large g.
We achieve this by using a result of Viehweg and Zuo which says that if Y parameterizes a family of curves of genus g, then the corresponding family of Jacobians is isogenous over Y to the g-fold product of a modular family of elliptic curves. After reducing the situation from the field of complex numbers to a finite field, we will see, combining the Weil and Sato-Tate conjectures, that this is impossible for large genus g.
We prove the non-existence of Shimura subvarieties of positive dimension contained generically in the hyperelliptic Torelli locus for curves of genus at least 8, which is an analogue of Oort's conjecture in the hyperelliptic case.
For infinitely many Hurwitz spaces parametrizing cyclic covers of the projective line, we provide a method to determine the integral PEL datum of the Shimura variety that contains the image of the Hurwitz space under the Torelli morphism.
In this paper, we use a group-theoretic approach to give a concrete description of the geometric structure of the supersingular locus of unitary Shimura varieties with exotic good reduction. This approach also is a more uniform way to prove results of this form obtained previously by, for example, Vollaard-Wedhorn and Rapoport-Terstiege-Wilson.
In this paper, we study the basic locus in the fiber at $p$ of a certain unitary Shimura variety with a certain parahoric level structure. The basic locus $\widehat{\mathcal{M}^{ss}}$ is uniformized by a formal scheme $\mathcal{N}$ which is called Rapoport-Zink space. We show that the irreducible components of the induced reduced subscheme $\mathcal{N}_{red}$ of $\mathcal{N}$ are Deligne-Lusztig varieties and their intersection behavior is controlled by a certain Bruhat-Tits building. Also, we define special cycles in $\mathcal{N}$ and study their intersection multiplicities.
We study the structure of the supersingular locus of the Rapoport--Zink integral model of the Shimura variety for $\mathrm{GU}(2,2)$ over a ramified odd prime with the special maximal parahoric level. We prove that the supersingular locus equals the disjoint union of two basic loci, one of which is contained in the flat locus, and the other is not. We also describe explicitly the structures of basic loci. More precisely, the former one is purely $2$-dimensional, and each irreducible component is birational to the Fermat surface. On the other hand, the latter one is purely $1$-dimensional, and …
In this paper we study the supersingular locus of the reduction modulo p of the Shimura variety for GU(1,s) in the case of an inert prime p. Using Dieudonné theory we define a stratification of the corresponding moduli space of p-divisible groups. We describe the incidence relation of this stratification in terms of the Bruhat-Tits building of a unitary group. In the case of GU(1,2), we show that the supersingular locus is equi-dimensional of dimension 1 and is of complete intersection. We give an explicit description of the irreducible components and their intersection behaviour.
In this paper, we study the $μ$-ordinary locus of a Shimura variety with parahoric level structure. Under the axioms in \cite{HR}, we show that $μ$-ordinary locus is a union of some maximal Ekedahl-Kottwitz-Oort-Rapoport strata introduced in \cite{HR} and we give criteria on the density of the $μ$-ordinary locus.
We analyze the geometry of the supersingular locus of the reduction modulo p of a Shimura variety associated to a unitary similitude group GU(1,n-1) over Q, in the case that p is ramified. We define a stratification of this locus and show that its incidence complex is closely related to a certain Bruhat-Tits simplicial complex. Each stratum is isomorphic to a Deligne-Lusztig variety associated to some symplectic group over F_p and some Coxeter element. The closure of each stratum is a normal projective variety with at most isolated singularities. The results are analogous to those of Vollaard/…
In this paper, we give a notion of the potentially good reduction locus of a Shimura variety. It consists of the points which should be related with motives having potentially good reductions in some sense. We show the existence of such locus for a Shimura variety of preabelian type. Further, we construct a partition of the adic space associated to a Shimura variety of preabelian type, which is expected to describe degenerations of motives. Using this partition, we prove that the cohomology of the potentially good reduction locus is isomorphic to the cohomology of a Shimura variety up to non-s…
We prove that the generic point of a Hilbert modular four-fold is not a Jacobian. The proof uses degeneration techniques and is independent of properties of the mapping class group used in preceding papers on locally symmetric subvarieties of the moduli space of abelian varieties contained in the Schottky locus.
We show that families of Pryms of abelian Galois covers of $\mathbb{P}^1$ in $A_{g-1}$ (resp. $A_g$) do not give rise to high dimensional Shimura subvareties.
In this note, we study the superspecial loci of orthogonal type Shimura varieties of signature (n-2, 2) with n>3. We prove a conjecture of Gross on the parametrizations of the superspecial locus in the special fiber of an orthogonal type Shimura variety and its lift in the integral model by certain homogeneous spaces. As applications, we provide a mass formula for the superspecial locus.
Oort has conjectured that there do not exist Shimura curves lying generically in the Torelli locus of curves of genus $g \geq 8$. We show that there do not exist one-dimensional Shimura families of semi-stable curves of genus $g\geq 5$ of Mumford type. We also show that there do not exist Shimura curves lying generically in the Torelli locus of hyperelliptic curves of genus $g\geq 8$. The first result proves a slightly weaker form of the conjecture for the case of Shimura curves of Mumford type. The second result proves the conjecture for the Torelli locus of hyperelliptic curves. We also pres…
We prove the generic exclusion of certain Shimura varieties of unitary and orthogonal types from the Torelli locus. The proof relies on a slope inequality on surface fibration due to G. Xiao, and the main result implies that certain Shimura varieties only meet the Torelli locus in dimension zero.
We determine the class of the (closure of the) Schottky locus in genus 4 in the Igusa- and the Voronoi compactification of A_4 and comment on the weight 8 modular form which vanishes on it.
We review the Newton stratification and Ekedahl-Oort stratification on the special fiber of a smooth integral model for a Shimura variety of Hodge type at a prime of good reduction. We show that the μ-ordinary locus coincides with the generic Ekedahl-Oort stratum, and that for any two geometric points in the μ-ordinary locus there is an isomorphism of the attached Dieudonne modules with additional structure. As a consequence, we proof that the μ-ordinary locus is open and dense, thus generalizing the results which were already known in the PEL-case. To prove our results we provide a method whi…
Oort has conjectured that there do not exist Shimura curves contained generically in the Torelli locus of genus-$g$ curves when $g$ is large enough. In this paper we prove the Oort conjecture for Shimura curves of Mumford type and Shimura curves parameterizing principally polarized $g$-dimensional abelian varieties isogenous to $g$-fold self-products of elliptic curves for $g>11$. We also prove that there do not exist Shimura curves contained generically in the Torelli locus of hyperelliptic curves of genus $g>7$. As a consequence, we obtain a finiteness result regarding smooth genus-$g$ curve…
We study Shimura curves of PEL type in $\mathsf{A}_g$ generically contained in the Prym locus. We study both the unramified Prym locus, obtained using étale double covers, and the ramified Prym locus, corresponding to double covers ramified at two points. In both cases we consider the family of all double covers compatible with a fixed group action on the base curve. We restrict to the case where the family is 1-dimensional and the quotient of the base curve by the group is $\mathbb{P}^1$. We give a simple criterion for the image of these families under the Prym map to be a Shimura curve. Usin…
This paper is a contribution to the general problem of giving an explicit description of the basic locus in the reduction modulo $p$ of Shimura varieties. Motivated by \cite{Vollaard-Wedhorn} and \cite{Rapoport-Terstiege-Wilson}, we classify the cases where the basic locus is (in a natural way) the union of classical Deligne-Lusztig sets associated to Coxeter elements. We show that if this is satisfied, then the Newton strata and Ekedahl-Oort strata have many nice properties.
We study Shimura (special) subvarieties in the moduli space $A_{p,D}$ of complex abelian varieties of dimension $p$ and polarization type $D$. These subvarieties arise from families of covers compatible with a fixed group action on the base curve such that the quotient of the base curve by the group is isomorphic to $\mathbb{P}^1$. We give a criterion for the image of these families under the Prym map to be a special subvariety and, using computer algebra, obtain 210 Shimura subvarieties contained in the Prym locus.
We construct an infinite number of Shimura curves contained in the locus of hyperelliptic Jacobians of genus 3. In the opposite direction, we show that in genus 3 the only possible non-complete (in the moduli space of abelian threefolds) Kuga curves contained in the hyperelliptic locus have the same degeneration data as that of the examples we construct. The locus of genus 3 hyperelliptic Jacobians is a divisor within the moduli space of principally polarized abelian threefolds, and our result demonstrates the techniques we develop more generally for dealing with Shimura curves contained withi…
We give an explicit weak solution to the Schottky problem, in the spirit of Riemann and Schottky. For any genus $g$, we write down a collection of polynomials in genus $g$ theta constants, such that their common zero locus contains the locus of Jacobians of genus $g$ curves as an irreducible component. These polynomials arise by applying a specific Schottky-Jung proportionality to an explicit collection of quartic identities for theta constants in genus $g-1$, which are suitable linear combinations of Riemann's quartic relations.
We study families of Galois covers of curves of positive genus. It is known that under a numerical condition these families yield Shimura subvarieties generically contained in the Jacobian locus. We prove that there are only 6 families satisfying this condition, all of them in genus 2,3 or 4. We also show that these families admit two fibrations in totally geodesic subvarieties, generalizing a result of Grushevsky and Möller. Countably many of these fibres are Shimura. Thus the Jacobian locus contains infinitely many Shimura subvarieties of positive dimension of any $g \leq 4$.
We study the locus of abelian Galois covers of $\mathbb{P}^{1}$ in $A_{g}$ and the problem of occurrence of Shimura (special) subvarieties generated by these covers in the Torelli locus $T_{g}$ inside $A_{g}$. We first investigate the existence of Shimura subvarieties in the mentioned locus by some computational methods based on Moonen-Oort works and then exclude many cases using both characteristic $p$ methods and monodromy computations.
We study the reduction of certain integral models of Shimura varieties of PEL type with Iwahori level structure. On these spaces we have the Kottwitz-Rapoport and the $p$-rank stratification. We show that the $p$-rank is constant on a KR stratum, generalizing a result of Ngô and Genestier. We prove an abstract, uniform formula for the $p$-rank on a KR stratum. In the symplectic and in the unitary case we derive explicit formulas for its value. We apply these formulas to the question of the density of the ordinary locus and to the question of the dimension of the $p$-rank 0 locus.
We show the existence of two new Shimura subvarieties of $\mathcal{A}_2, \mathcal{A}_3$ generically contained in the Torelli locus. They provide the first examples of Shimura subvarieties obtained by means of Jacobians carrying non-trivial endomorphisms not directly induced by the automorphisms of the curves. We also obtain a new example of a Shimura subvariety of $\mathcal{A}_4$ generically contained in the Prym locus.
This article has three goals. First, we generalize the result of Deuring and Serre on the characterization of supersingular locus of modular curves to all Shimura varieties given by totally indefinite quaternion algebras over totally real number fields. Second, we generalize the result of Ribet on arithmetic level raising to such Shimura varieties in the inert case. Third, as an application to number theory, we use the previous results to study the Selmer group of certain triple product motive of an elliptic curve, in the context of the Bloch--Kato conjecture.
Given a family of Galois coverings of the projective line we give a simple sufficient condition ensuring that the closure of the image of the family via the period mapping is a special (or Shimura) subvariety in A_g. By a computer program we get the list of all families in genus up to 8 satisfying our condition. There is no family in genus 8, all of them are in genus at most 7. These examples are related to a conjecture of Oort. Among them we get the cyclic examples constructed by various authors (Shimura, Mostow, De Jong-Noot, Rohde, Moonen and others) and the abelian non-cyclic examples foun…
In this paper, we present a solution to the Schottky problem in the spirit of Schottky and Jung for genus five curves. To do so, we exploit natural incidence structures on the fibers of several maps to reduce all questions to statements about the Prym map for genus six curves. This allows us to find all components of the big Schottky locus and thus, to show that the small Schottky locus introduced by Donagi is irreducible.
An algebraic subvariety Z of A_g is totally geodesic if it is the image via the natural projection map of some totally geodesic submanifold X of the Siegel space. We say that X is the symmetric space uniformizing Z. In this paper we determine which symmetric space uniformizes each of the low genus counterexamples to the Coleman-Oort conjecture obtained studying Galois covers of curves. It is known that the counterexamples obtained via Galois covers of elliptic curves admit two fibrations in totally geodesic subvarieties. The second result of the paper studies the relationship between these fib…
We prove a lower bound for the codimension of the Andreotti-Mayer locus N_{g,1} and show that the lower bound is reached only for the hyperelliptic locus in genus 4 and the Jacobian locus in genus 5. In relation with the boundary of the Andreotti-Mayer loci we study subvarieties of principally polarized abelian varieties (B,Theta) parametrizing points b such that Theta and the translate Theta_b are tangentially degenerate along a variety of a given dimension.
We study Shimura subvarieties of $\mathsf{A}_g$ obtained from families of Galois coverings $f: C \rightarrow C'$ where $C'$ is a smooth complex projective curve of genus $g' \geq 1$ and $g= g(C)$. We give the complete list of all such families that satisfy a simple sufficient condition that ensures that the closure of the image of the family via the Torelli map yields a Shimura subvariety of $\mathsf{A}_g$ for $g' =1,2$ and for all $g \geq 2,4$ and for $g' > 2$ and $g \leq 9$. In a previous work of the first and second author together with A. Ghigi [FGP] similar computations were done in the c…
We show that the degree of the images of the moduli space of (principally polarized) abelian varieties A_g and of the moduli space of curves M_g in the projective space under the theta constant embedding are equal to the top self-intersection numbers of one half the first Hodge class on them. This allows us to obtain an explicit formula for the degree of A_g, and an explicit upper bound for the degree of M_g. Knowing the degree of A_g allows us to effectively determine the subvariety itself, i.e. to effectively obtain all polynomial equations satisfied by theta constants. Furthermore, combinin…
In this paper we study the Oort conjecture on Shimura subvarieties contained generically in the Torelli locus in the Siegel modular variety $\mathcal{A}_g$. Using the poly-stability of Higgs bundles on curves and the slope inequality of Xiao on fibred surfaces, we show that a Shimura curve $C$ is not contained generically in the Torelli locus if its canonical Higgs bundles contains a unitary Higgs subbundle of rank at least $(4g+2)/5$. From this we prove that a Shimura subvariety of $\mathbf{SU}(n,1)$-type is not contained generically in the Torelli locus when a numerical inequality holds, whi…
Recent developments on the uniformity of the number of rational points on curves and subvarieties in a moving abelian variety rely on the geometric concept of the degeneracy locus. The first-named author investigated the degeneracy locus in certain mixed Shimura varieties. In this expository note we revisit some of these results while minimizing the use of mixed Shimura varieties while working in a family of principally polarized abelian varieties. We also explain their relevance for applications in diophantine geometry.
We prove the congruence relation for the mod-p reduction of Shimura varieties associated to a unitary similitude group GU(n-1,1), when p is inert and n odd. When n is even, this result was obtained by T. Wedhorn and O. Bültel using the density of the ordinary locus in the p-isogeny space. This condition fails in our case. A key element is the understanding of the supersingular locus, for which we refer to the two articles by I. Vollaard and T. Wedhorn. The proof makes extensive use of elementary algebraic geometry, but also some deeper results.
We complete the study of the supersingular locus in the fiber at $p$ of a Shimura variety attached to a unitary similitude group $\GU(1,n-1)$ over $\QQ$ in the case that $p$ is inert. This was started by the first author in \cite{Vo_Uni} where complete results were obtained for $n =2,3$. The supersingular locus is uniformized by a formal scheme $\Ncal$ which is a moduli space of so-called unitary $p$-divisible groups. It depends on the choice of a unitary isocrystal $\Nbf$. We define a stratification of $\Ncal$ indexed by vertices of the Bruhat-Tits building attached to the reductive group of …
Let O be a maximal order in a totally indefinite quaternion algebra over a totally real number field. In this note we study the locus Q_O of quaternionic multiplication by O in the moduli space A_g of principally polarized abelian varieties of even dimension g with particular emphasis in the two-dimensional case. We describe Q_O as a union of Atkin-Lehner quotients of Shimura varieties and we compute the number of irreducible components of Q_O in terms of class numbers of CM-fields.
We describe the structure of the supersingular locus of a Shimura variety for a quaternionic unitary similitude group of degree $2$ over a ramified odd prime $p$ if the level at $p$ is given by a special maximal compact open subgroup. More precisely, we show that such a locus has purely $2$-dimensional, and every irreducible component is birational to the Fermat surface. Furthermore, we have an estimation of the numbers of connected and irreducible components. To prove these assertions, we completely determine the structure of the underlying reduced scheme of the Rapoport--Zink space for the q…
We prove that a Shimura curve in the Siegel modular variety is not generically contained in the open Torelli locus as long as the rank of unitary part in its canonical Higgs bundle satisfies a numerical upper bound. As an application we show that the Coleman-Oort conjecture holds for Shimura curves associated to partial corestriction upon a suitable choice of parameters, which generalizes a construction due to Mumford.
In this paper we construct infinitely many Shimura curves contained in the locus of Jacobians of genus four curves. All Jacobians in these families are ${\mathbb Z}/3$ covers of varying elliptic curves that appear in a geometric construction of Pirola, and include an example of a Shimura-Teichmuller curve that parameterizes Jacobians that are suitable ${\mathbb Z}/6$ covers of ${\mathbb P}^1$. We compute explicitly the period matrices of the Shimura curves we construct using the original construction of Shimura for moduli spaces of abelian varieties with automorphisms.
It is well known that, fixed an even, unimodular, positive definite quadratic form, one can construct a modular form in each genus; this form is called the theta series associated to the quadratic form. Varying the quadratic form, one obtains the ring of stable modular forms. We show that the differences of theta series associated to specific pairs of quadratic forms vanish on the locus of hyperelliptic Jacobians in each genus. In our examples, the quadratic forms have rank 24, 32 and 48. The proof relies on a geometric result about the boundary of the Satake compactification of the hyperellip…
We show that a smooth projective curve of genus $g$ can be reconstructed from its polarized Jacobian $(X, Θ)$ as a certain locus in the Hilbert scheme $\mathrm{Hilb}^d(X)$, for $d=3$ and for $d=g+2$, defined by geometric conditions in terms of the polarization $Θ$. The result is an application of the Gunning--Welters trisecant criterion and the Castelnuovo--Schottky theorem by Pareschi--Popa and Grushevsky, and its scheme theoretic extension by the authors.
We study Shimura curves of PEL type in the space of polarised abelian varieties $A^δ_p$ generically contained in the ramified Prym locus. We generalise to ramified double covers, the construction done in [10] in the unramified case and in the case of two ramification points. Namely, we construct families of double covers which are compatible with a fixed group action on the base curve. We only consider the case of one-dimensional families and where the quotient of the base curve by the group is ${\mathbb P}^1$. Using computer algebra we obtain 184 Shimura curves contained in the (ramified) Pry…
The supersingular locus in the fiber at p of a Shimura variety attached to a unitary similitude group GU(1,n-1) over Q is uniformized by a formal scheme \Cal N. In the case when p is inert, we define special cycles Z(x) in \Cal N, associated to a collection x of m `special homomorphisms' with fundamental matrix T in Herm_m(OK). When m=n and T is nonsingular, we show that the cycle Z(x) is a union of components of the Ekedahl-Oort stratification, and we give a necessary and sufficient conditions, in terms of T, for Z(x) to be irreducible. When Z(x) is zero dimensional -- in which case it reduce…
We give a solution to the weak Schottky problem for genus five Jacobians with a vanishing theta null, answering a question of Grushevsky and Salvati Manni. More precisely, we show that if a principally polarized abelian variety of dimension five has a vanishing theta null with a quadric tangent cone of rank at most three, then it is in the Jacobian locus, up to extra irreducible components. We employ a degeneration argument, together with a study of the ramification loci for the Gauss map of a theta divisor.
This paper is devoted to the study of the Higgs bundle associated with the universal abelian variety over the good reduction of a Shimura curve of PEL type. Due to the endomorphism structure, the Higgs bundle decomposes into the direct sum of Higgs subbundles of rank two. They are basically divided into two type: uniformizing type and unitary type. As the first application we obtain the mass formula counting the number of geometric points of the degeneracy locus in the Newton polygon stratification. We show that each Higgs subbundle is Higgs semistable. Furthermore, for each Higgs subbundle of…
