Oort Conjecture Research Articles

A two‐dimensional arithmetic André–Oort problem

AbstractWe state and investigate an integral analogue of the André–Oort conjecture (in integral models of Shimura varieties). We establish an instance of this conjecture: the case of a modular curve, as a scheme over . Our approach relies on equidistribution estimates related to subconvexity in analytic number theory and our result is unconditional.

O-minimal de Rham Cohomology

AbstractO-minimal geometry generalizes both semialgebraic and subanalytic geometries, and has been very successful in solving special cases of some problems in arithmetic geometry, such as André–Oort conjecture. Among the many tools developed in an o-minimal setting are cohomology theories for abstract-definable continuous manifolds such as singular cohomology, sheaf cohomology and Čech cohomology, which have been used for instance to prove Pillay’s conjecture concerning definably compact groups. In the present thesis we elaborate an o-minimal de Rham cohomology theory for abstract-definable $C^{\infty }$ manifolds in an o-minimal expansion of the real field which admits smooth cell decomposition and defines the exponential function. We can specify the o-minimal cohomology groups and attain some properties such as the existence of Mayer–Vietoris sequence and the invariance under abstract-definable $C^{\infty }$ diffeomorphisms. However, in order to obtain the invariance of our o-minimal cohomology under abstract-definable homotopy we must work in a tame context that defines sufficiently many primitives and assume the validity of a statement related to Bröcker’s question.Abstract prepared by Rodrigo Figueiredo.E-mail: rodrigo@ime.usp.brURL: https://doi.org/10.11606/T.45.2019.tde-28042019-181150

The Coleman–Oort conjecture: Reduction to three key cases

We show that the Coleman–Oort conjecture can be reduced to three particular cases. As an application, we extend a result of Lu and Zuo, to the effect that for g ⩾ 8 $g \geqslant 8$ the Coleman–Oort conjecture is true on the hyperelliptic locus.

Lower bounds for Galois orbits of special points on Shimura varieties: a point-counting approach

Let S be a Shimura variety. We conjecture that the heights of special points in \(S(\overline{\mathbb {Q}})\) are discriminant negligible with respect to some Weil height function \(h:S(\overline{\mathbb {Q}})\rightarrow \mathbb {R}\). Assuming this conjecture to be true, we prove that the sizes of the Galois orbits of special points grow as a fixed power of their discriminant (an invariant we will define in the text). In particular, we give a new proof of a theorem of Tsimerman on lower bounds for Galois degrees of special points in Shimura varieties of abelian type. This gives a new proof of the André–Oort conjecture for such varieties that avoids the use of Masser–Wüstholz isogeny estimates, replacing them by a point-counting argument.

Point counting for foliations over number fields

AbstractLet${\mathbb M}$be an affine variety equipped with a foliation, both defined over a number field${\mathbb K}$. For an algebraic$V\subset {\mathbb M}$over${\mathbb K}$, write$\delta _{V}$for the maximum of the degree and log-height ofV. Write$\Sigma _{V}$for the points where the leaves intersectVimproperly. Fix a compact subset${\mathcal B}$of a leaf${\mathcal L}$. We prove effective bounds on the geometry of the intersection${\mathcal B}\cap V$. In particular, when$\operatorname {codim} V=\dim {\mathcal L}$we prove that$\#({\mathcal B}\cap V)$is bounded by a polynomial in$\delta _{V}$and$\log \operatorname {dist}^{-1}({\mathcal B},\Sigma _{V})$. Using these bounds we prove a result on the interpolation of algebraic points in images of${\mathcal B}\cap V$by an algebraic map$\Phi $. For instance, under suitable conditions we show that$\Phi ({\mathcal B}\cap V)$contains at most$\operatorname {poly}(g,h)$algebraic points of log-heighthand degreeg.We deduce several results in Diophantine geometry. Following Masser and Zannier, we prove that given a pair of sections$P,Q$of a nonisotrivial family of squares of elliptic curves that do not satisfy a constant relation, whenever$P,Q$are simultaneously torsion their order of torsion is bounded effectively by a polynomial in$\delta _{P},\delta _{Q}$; in particular, the set of such simultaneous torsion points is effectively computable in polynomial time. Following Pila, we prove that given$V\subset {\mathbb C}^{n}$, there is an (ineffective) upper bound, polynomial in$\delta _{V}$, for the degrees and discriminants of maximal special subvarieties; in particular, it follows that the André–Oort conjecture for powers of the modular curve is decidable in polynomial time (by an algorithm depending on a universal, ineffective Siegel constant). Following Schmidt, we show that our counting result implies a Galois-orbit lower bound for torsion points on elliptic curves of the type previously obtained using transcendence methods by David.

Some evidence for the Coleman–Oort conjecture

The Coleman–Oort conjecture says that for large g there are no positive-dimensional Shimura subvarieties of $${\mathsf {A}}_g$$ generically contained in the Jacobian locus. Counterexamples are known for $$g\le 7$$ . They can all be constructed using families of Galois coverings of curves satisfying a numerical condition. These families are already classified in cases where: (a) the Galois group is cyclic, (b) it is abelian and the family is 1-dimensional, or c) $$g\le 9$$ . By means of carefully designed computations and theoretical arguments excluding a large number of cases we are able to prove that for $$g\le 100$$ there are no other families than those already known.

Mass formula and Oort's conjecture for supersingular abelian threefolds

Katsura and Oort obtained an explicit description of the supersingular locus S3,1 of the Siegel modular variety of degree 3 in terms of class numbers. In this paper we study an alternative stratification of S3,1, the so-called mass stratification. We show that when p≠2, there are eleven strata (one of a-number 3, two of a-number 2 and eight of a-number 1). We give an explicit mass formula for each stratum and classify possible automorphism groups on each stratum of a-number one. On the largest open stratum we show that every automorphism group is {±1} if and only if p≠2; that is, we prove that Oort's conjecture on the automorphism groups of generic supersingular abelian threefolds holds precisely when p>2.

Effective André–Oort for non-compact curves in Hilbert modular varieties

In the proofs of most cases of the André–Oort conjecture, there are two different steps whose effectivity is unclear: the use of generalizations of Brauer–Siegel and the use of Pila–Wilkie. Only the case of curves in ℂ 2 is currently known effectively (by other methods).

On the geometric André–Oort conjecture for variations of Hodge structures

Abstract Let 𝕍 {{\mathbb{V}}} be a polarized variation of integral Hodge structure on a smooth complex quasi-projective variety S. In this paper, we show that the union of the non-factor special subvarieties for ( S , 𝕍 ) {(S,{\mathbb{V}})} , which are of Shimura type with dominant period maps, is a finite union of special subvarieties of S. This generalizes previous results of Clozel and Ullmo (2005) and Ullmo (2007) on the distribution of the non-factor (in particular, strongly) special subvarieties in a Shimura variety to the non-classical setting and also answers positively the geometric part of a conjecture of Klingler on the André–Oort conjecture for variations of Hodge structures.

Degree gaps for multipliers and the dynamical André–Oort conjecture

AbstractWe demonstrate how recent work of Favre and Gauthier, together with a modification of a result of the author, shows that a family of polynomials with infinitely many post-critically finite specializations cannot have any periodic cycles with multiplier of very low degree, except those that vanish, generalizing results of Baker and DeMarco, and Favre and Gauthier.

Triples of singular moduli with rational product

We show that all triples [Formula: see text] of singular moduli satisfying [Formula: see text] are “trivial”. That is, either [Formula: see text]; some [Formula: see text] and the remaining [Formula: see text] are distinct, of degree [Formula: see text], and conjugate over [Formula: see text]; or [Formula: see text] are pairwise distinct, of degree [Formula: see text], and conjugate over [Formula: see text]. This theorem is best possible and is the natural three-dimensional analogue of a result of Bilu, Luca and Pizarro-Madariaga in two dimensions. It establishes an explicit version of the André–Oort conjecture for the family of subvarieties [Formula: see text] defined by an equation [Formula: see text].

Horizontal non-vanishing of Heegner points and toric periods

Let F be a totally real number field and A a modular GL2-type abelian variety over F. Let K/F be a CM quadratic extension. Let χ be a class group character over K such that the Rankin-Selberg convolution L(s,A,χ) is self-dual with root number −1. We show that the number of class group characters χ with bounded ramification such that L′(1,A,χ)≠0 increases with the absolute value of the discriminant of K.We also consider a rather general rank zero situation. Let π be a cuspidal cohomological automorphic representation over GL2(AF). Let χ be a Hecke character over K such that the Rankin–Selberg convolution L(s,π,χ) is self-dual with root number 1. We show that the number of Hecke characters χ with fixed ∞-type and bounded ramification such that L(1/2,π,χ)≠0 increases with the absolute value of the discriminant of K.The Gross–Zagier formula and the Waldspurger formula relate the question to horizontal non-vanishing of Heegner points and toric periods, respectively. For both situations, the strategy is geometric relying on the Zariski density of CM points on self-products of a quaternionic Shimura variety. The recent result [26,31,1] on the André–Oort conjecture is accordingly fundamental to the approach.

The Oort conjecture on Shimura curves in the Torelli locus of curves

Oort has conjectured that there do not exist Shimura varieties of dimension >0 contained generically in the Torelli locus of genus-g curves when g is sufficiently large. 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. As a consequence, we obtain a finiteness result regarding smooth genus-g curves with completely decomposable Jacobians, which is related to a question of Ekedahl and Serre.

Linear Equations in Singular Moduli

Abstract We establish an effective version of the André–Oort conjecture for linear subspaces of $Y(1)^n_{\mathbb{C}} \approx \mathbb{A}_{\mathbb{C}}^n$. This gives the first effective nontrivial results of André–Oort type for higher-dimensional varieties in products of modular curves.

The Oort Conjecture for Shimura Curves of Small Unitary Rank

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 with partial corestriction upon a suitable choice of parameters, which generalizes a construction due to Mumford.

A generalization of the Oort conjecture

The Oort conjecture (now a theorem of Obus–Wewers and Pop) states that if k is an algebraically closed field of characteristic p , then any cyclic branched cover of smooth projective k -curves lifts to characteristic zero. This is equivalent to the local Oort conjecture, which states that all cyclic extensions of k[[t]] lift to characteristic zero. We generalize the local Oort conjecture to the case of Galois extensions with cyclic p -Sylow subgroups, reduce the conjecture to a pure characteristic p statement, and prove it in several cases. In particular, we show that D_9 is a so-called local Oort group .

The Oort conjecture on Shimura curves in the Torelli locus of hyperelliptic curves

Oort has conjectured that there do not exist Shimura varieties of dimension >0 contained generically in the Torelli locus of genus-g curves when g is sufficiently large. In this paper we prove the analogue of this conjecture for Shimura curves with respect to the hyperelliptic Torelli locus of genus g>7.

A special point problem of André-Pink-Zannier in the universal family of abelian varieties

The Andre-Pink-Zannier conjecture predicts that a subvariety of a mixed Shimura variety is weakly special if its intersection with the generalized Hecke orbit of a given point is Zariski dense. It is part of the Zilber-Pink conjecture. In this paper we focus on the universal family of principally polarized Abelian varieties. We explain the moduli interpretation of the Andr´e-Pink- Zannier conjecture in this case and prove several different cases for this conjecture: its overlap with the Andr´e-Oort conjecture; when the subvariety is contained in an Abelian scheme over a curve and the point is a torsion point on its fiber; when the subvariety is a curve.

About the mixed André–Oort conjecture: Reduction to a lower bound for the pure case

We prove that the mixed André–Oort conjecture holds for any mixed Shimura variety if a lower bound for the size of Galois orbits of special points in pure Shimura varieties exists. This generalizes the current results for mixed Shimura varieties of Abelian type.

On the Oort conjecture for Shimura varieties of unitary and orthogonal types

In this paper we study the Oort conjecture concerning the non-existence of 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 fibered surfaces, we show that a Shimura curve$C$is not contained generically in the Torelli locus if its canonical Higgs bundle 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, which involves the genus$g$, the dimension$n+1$, the degree$2d$of CM field of the Hermitian space, and the type of the symplectic representation defining the Shimura subdatum. A similar result holds for Shimura subvarieties of$\mathbf{SO}(n,2)$type, defined by spin groups associated to quadratic spaces over a totally real number field of degree at least$6$subject to some natural constraints of signatures.