Version 3 fixes a few typos and adds a discussion of uniform versions of the main results. Version 2 implements significant changes. An error was found in version 1 which undermined the result. To amend the error required the introduction of a Schanuel-type conjecture. Version 2 also adds, under the Schanuel-type conjecture, a new and stronger version of the main theorem
发表信息：
Proc. Edinb. Math. Soc., Volume 62, Issue 2 May 2019 , pp. 323-365
In unpublished notes, Pila discussed some theory surrounding the modular function $j$ and its derivatives. A focal point of these notes was the statement of two conjectures regarding $j$, $j'$ and $j"$: a Zilber-Pink type statement incorporating $j$, $j'$ and $j"$, which was an extension of an apparently weaker conjecture of Andre-Oort type. In this paper, I first cover some background regarding $j$, $j'$ and $j"$, mostly covering the work already done by Pila. Then I use a seemingly novel adaptation of the o-minimal Pila-Zannier strategy to prove a weakened version of Pila's "Modular Andre-Oort with Derivatives" conjecture. Under the assumption of a Schanuel-type conjecture, the central theorem of the paper implies Pila's conjecture in full generality, as well as a more precise statement on the same lines.
The modular case of the André-Oort Conjecture is a theorem of Andre and Pila, having at its heart the well-known modular function j. I give an overview of two other `nonclassical' classes of modular function, namely the quasimodular (QM) and almost holomorphic modular (AHM) functions. These are perhaps less well-known than j, but have been studied by various authors including for example Masser, Shimura and Zagier. It turns out to be sufficient to focus on a particular QM function $χ$ and its dual AHM function $χ^*$, since these generate the relevant fields. After discussing some of the proper…
We state and prove a variant of the André-Oort conjecture for the product of 2 modular curves in positive characteristic, assuming GRH for quadratic fields.
We provide an unconditional proof of the André-Oort conjecture for the coarse moduli space $\mathcal{A}_{2,1}$ of principally polarized Abelian surfaces, following the strategy outlined by Pila-Zannier.
The purpose of this article is to explain the Pila-Zannier strategy for proving the André-Oort conjecture. First, however, we will provide a brief introduction to the theory of Shimura varieties.
In this paper we prove, assuming the Generalized Riemann Hypothesis, the Andr?e-Oort conjecture on the Zariski closure of sets of special points in a Shimura variety. In the case of sets of special points satisfying an additional assumption, we prove the conjecture without assuming the GRH.
We consider the analogue of the André-Oort conjecture for Drinfeld modular varieties which was formulated by Breuer. We prove this analogue for special points with separable reflex field over the base field by adapting methods which were used by Klingler and Yafaev to prove the André-Oort conjecture under the generalized Riemann hypothesis in the classical case. Our result extends results of Breuer showing the correctness of the analogue for special points lying in a curve and for special points having a certain behaviour at a fixed set of primes.
We 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 Z. It is a scheme of dimension two and, already in this case, our conjecture is highly non-trivial. Our approach relies on equidistribution estimates related to subconvexity in analytic number theory and our result is unconditional.
In this paper we develop a strategy and some technical tools for proving the Andre-Oort conjecture. We give lower bounds for the degrees of Galois orbits of geometric components of special subvarieties of Shimura varieties, assuming the Generalised Riemann Hypothesis. We proceed to show that sequences of special subvarieties whose Galois orbits have bounded degrees are equidistributed in a suitable sense.
Let $X\subset Y(1)^n$ be a subvariety defined over a number field $\mathbb F$ and let $(P_1,\ldots,P_n)\in X$ be a special point not contained in a positive-dimensional special subvariety of $X$. We show that the if a coordinate $P_i$ corresponds to an order not contained in a single exceptional Siegel-Tatuzawa imaginary quadratic field $K_*$ then the associated discriminant $|Δ(P_i)|$ is bounded by an effective constant depending only on $\operatorname{deg} X$ and $[{\mathbb F}:{\mathbb Q}]$. We derive analogous effective results for the positive-dimensional maximal special subvarieties. From…
We give a proof of the André-Oort conjecture for $\mathcal{A}_g$ - the moduli space of principally polarized abelian varieties. In particular, we show that a recently proven `averaged' version of the Colmez conjecture yields lower bounds for Galois orbits of CM points. The André-Oort conjecture then follows from previous work of Pila and the author.
We prove in this paper the Ax-Lindemann-Weierstrass theorem for all mixed Shimura varieties and discuss the lower bounds for Galois orbits of special points of mixed Shimura varieties. In particular we reprove a result of Silverberg in a different approach. Then combining these results we prove the André-Oort conjecture for any mixed Shimura variety whose pure part is a subvariety of A_6^n.
In this short paper we discuss a number of effective and/or explicit results of Andre-Oort type for the nonholomorphic function "$χ^*$", which I have discussed in a number of other papers. After working in a rather ad-hoc manner to get some good estimates on the tails of the $q$-expansions involved, we prove weak effective André-Oort results for $χ^*$, which mimic but are not full analogues of effective André-Oort results known due to Kühne/Bilu-Masser-Zannier for the classical modular function $j$. Then we go on to discuss what we call an "explicit" result; that certain triples of special poi…
We prove, assuming the generalized Riemann hypothesis, the Andre-Oort conjecture for Hilbert modular surfaces. More precisely, let K be a real quadratic field and let S be the coarse moduli space of complex abelian surfaces with multiplications by the ring of integers of K. Let C be an irreducible closed curve in S, and suppose that C contains infinitely many complex multiplication points. Then we prove, assuming GRH, that C is of Hodge type, meaning, in this case, that it parametrizes abelian varieties with more endomorphisms. Also, if we assume that C has infinitely many CM points that corre…
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 ${\bf C}^2$ is currently known effectively (by other methods). We give an effective proof of André-Oort for non-compact curves in every Hilbert modular surface and every Hilbert modular variety of odd genus (under a minor generic simplicity condition). In particular we show that in these cases the first step may be replaced by the endomorphism estimates of Wüstholz and the second …
We prove a $p$-adic analogue of the André-Oort conjecture for subvarieties of the universal abelian varieties containing a dense set of special points. Let $g$ and $n$ be integers with $n \geq 3$ and $p$ a prime number not dividing $n$. Let $R$ be a finite extension of $W[{\mathbb F}_p^{\mathrm alg}]$, the ring of Witt vectors of the algebraic closure of the field of $p$ elements. The moduli space $\cA = \cA_{g,1,n}$ of $g$-dimensional principally polarized abelian varieties with full level $n$-structure as well as the universal abelian variety $π:\cX \to \cA$ over $\cA$ may be defined over $R…
We 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 which vanish, generalizing results of Baker and DeMarco, and Favre and Gauthier.
Let $Z=X_1\times...\times X_n$ be a product of Drinfeld modular curves. We characterize those algebraic subvarieties $X \subset Z$ containing a Zariski-dense set of CM points, i.e. points corresponding to $n$-tuples of Drinfeld modules with complex multiplication (and suitable level structure). This is a characteristic $p$ analogue of a special case of the André-Oort conjecture. We follow closely the approach used by Bas Edixhoven in characteristic zero, see math.NT/0302138. Note that in this paper we assume that the characteristic $p$ is odd, and we only treat the case of Drinfeld $F_q[T]$-mo…
Using linear forms in logarithms we prove an explicit result of André-Oort type for $\mathbb{P}^1(\mathbb{C}) \times \mathbb{G}_m(\mathbb{C})$. In this variation the special points of $\mathbb{P}^1(\mathbb{C}) \times \mathbb{G}_m(\mathbb{C})$ are of the form $(α, λ)$, with $α$ a singular modulus and $λ$ a root of unity. The qualitative version of our result states that if $\mathcal{C}$ is a closed algebraic curve in $\mathbb{P}^1(\mathbb{C}) \times \mathbb{G}_m(\mathbb{C})$, defined over a number field, not containing a horizontal or vertical line, then $\mathcal{C}$ contains only finitely man…
We prove a special case of the Dynamical Andre-Oort Conjecture formulated by Baker and DeMarco. For any integer d>1, we show that for a rational plane curve C parametrized by (t, h(t)) for some non-constant polynomial h with complex coefficients, if there exist infinitely many points (a,b) on the curve C such that both z^d+a and z^d+b are postcritically finite maps, then h(z)=uz for a (d-1)-st root of unity u. As a by-product of our proof, we show that the Mandelbrot set is not the filled Julia set of any polynomial with complex coefficients.
We establish the equidistribution with respect to the bifurcation measure of post-critically finite maps in any one-dimensional algebraic family of unicritical polynomials. Using this equidistribution result, together with a combinatorial analysis of certain algebraic correspondences on the complement of the Mandelbrot set $M_2$ (or generalized Mandelbrot set $M_d$ for degree $d>2$), we classify all complex plane curves $C$ with Zariski-dense subsets of points $(a,b)\in C$, such that both $z^d+a$ and $z^d+b$ are simultaneously post-critically finite for a fixed degree $d\geq 2$. Our result is …
In the moduli space of degree d polynomials, the special subvarieties are those cut out by critical orbit relations, and then the special points are the post-critically finite polynomials. It was conjectured that in the moduli space of degree d polynomials, the subvarieties containing a Zariski-dense set of special points are exactly these special subvarieties. In this article, we prove the first non-trivial case for this conjecture: the case of cubic polynomials.
A characterization of subvarieties of Shimura varieties which contain a Zariski dense subset of weakly special subvarieties has been proved by the second author, by combining o-minimality results and functional transcendence results. In this paper, we obtain a new proof of this statement by dynamics techniques on homogeneous spaces in the spirit of the earlier work of Clozel and the second author. The proof combines ergodic theory à la Ratner, with a statement on the dimension of a Hausdorff limit of a sequence of definable subsets (in an o-minimal theory) extracted from a definable family. On…
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 Oor…
In the paper four stratifications in the reduction modulo $p$ of a general Shimura variety are studied: the Newton stratification, the Kottwitz-Rapoport stratification, the Ekedahl-Oort stratification and the Ekedahl-Kottwitz-Oort-Rapoport stratification. We formulate a system of axioms and show that these imply non-emptiness statements and closure relation statements concerning these various stratifications. These axioms are satisfied in the Siegel case.
We prove an effective version of the Pila-Wilkie Theorem for sets definable using Pfaffian functions, providing effective estimates for the number of algebraic points of bounded height and degree lying on such sets. We also prove effective versions of extensions of this result due to Pila and Habegger-Pila . In order to prove these counting results, we obtain an effective version of Yomdin-Gromov parameterization for sets defined using restricted Pfaffian functions. Furthermore, for sets defined in the restricted setting, as well as for unrestricted sub-Pfaffian sets, our effective estimates d…
We consider mod p Hilbert modular forms associated to a totally real field of degree d in which p is unramified. We prove that every such form arises by multiplication by partial Hasse invariants from one whose weight (a d-tuple of integers) lies in a certain cone contained in the set of non-negative weights, answering a question of Andreatta and Goren. The proof is based on properties of the Goren-Oort stratification on mod p Hilbert modular varieties established by Goren and Oort, and Tian and Xiao.
We establish an effective version of the André-Oort conjecture for linear subspaces of $Y(1)^n_{\mathbb{C}} \approx \mathbb{A}_{\mathbb{C}}^n$. Apart from the trivial examples provided by weakly special subvarieties, this yields the first algebraic subvarieties in a Shimura variety of dimension $> 1$ whose CM-points can be (theoretically) determined.
We show that the strategy of point counting in o-minimal structures can be applied to various problems on unlikely intersections that go beyond the conjectures of Manin-Mumford and André-Oort. We verify the so-called Zilber-Pink Conjecture in a product of modular curves on assuming a lower bound for Galois orbits and a sufficiently strong modular Ax-Schanuel Conjecture. In the context of abelian varieties we obtain the Zilber-Pink Conjecture for curves unconditionally when everything is defined over a number field. For higher dimensional subvarieties of abelian varieties we obtain some weaker …
These are expanded notes of the mini-courses on Pila's work that Yuri Bilu gave in Basel in April 2011, Yaroslavl in August 2011 and Chennai in February 2012. The topics covered include the Bombieri-Pila theorem, its extensions and applications to some prominent conjectures (Manin-Mumford, André-Oort).
In this paper, we prove the bounded case of the Andre-Oort conjecture for special subvarieties in a mixed Shimura variety. This generalizes previous results of L. Clozel, E. Ullmo, and A. Yafaev. The proof is reduced to a special case of the equidistribution conjecture for mixed Shimura varieties. The main tool is a theorem of S. Mozes and N. Shah for certain measures on lattice spaces.
In this paper we prove a special case of the André-Oort conjecture for Kuga varieties. If $M$ is a Kuga variety fibred over a pure Shimura variety $S$ as an abelian scheme, and $(M_n)$ is a sequence of special subvarieties in $M$ which are faithfully flat over $S$, then the Zariski closure of the union of the $M_n$'s is a finite union of special subvarieties faithfully flat over $S$. The proof is reduced to a variant of the Manin-Mumford conjecture for abelian schemes.
In unpublished notes Pila proposed a Modular Zilber-Pink with Derivatives (MZPD) conjecture, which is a Zilber-Pink type statement for the modular $j$-function and its derivatives. In this article we define D-special varieties, then state and prove two functional (differential) analogues of the MZPD conjecture for those varieties. In particular, we prove a weak version of MZPD. As a special case of our results, we obtain a functional Modular André-Oort with Derivatives statement. The main tools used in the paper come from (model theoretic) differential algebra and complex analytic geometry, an…
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…
A conjecture by Yves Andre and Frans Oort says that closed subvarieties of Shimura varieties that contain a Zariski dense subset of special points are subvarieties of Hodge type. We prove this in the case where the subvariety is a curve that contains infinitely many special points that lie in one Hecke orbit. Via work of Juergen Wolfart, Paula Cohen and Gisbert Wuestholz, this result has an application to algebraicity of values of hypergeometric functions.
We explore an analogue of the André-Oort conjecture for subvarieties of Drinfeld modular varieties. The conjecture states that a subvariety $X$ of a Drinfeld modular variety contains a Zariski-dense set of complex multiplication (CM) points if and only if $X$ is a "special" subvariety (i.e. $X$ is defined by requiring additional endomorphisms). We prove this conjecture in two cases. Firstly when $X$ contains a Zariski-dense set of CM points with a certain behaviour above a fixed prime (which is the case if these CM points lie in one Hecke orbit), and secondly when $X$ is a curve containing inf…
We prove the Andre-Oort conjecture on special points of Shimura varieties for arbitrary products of modular curves, assuming the Generalized Riemann Hypothesis. More explicitly, this means the following. Let n be a positive integer, and let S be a subset of C^n (with C the complex numbers) consisting of points all of whose coordinates are j-invariants of elliptic curves with complex multiplications. Then we prove (under GRH) that the irreducible components of the Zariski closure of S are ``special subvarieties'', i.e., determined by isogeny conditions on coordinates and pairs of coordinates. A…
We develop a theory of enlarged mixed Shimura varieties, putting the universal vectorial bi-extension defined by Coleman into this framework to study some functional transcendental results of Ax type. We study their bi-algebraic systems, formulate the Ax-Schanuel conjecture and explain its relation with the Ax logarithmique theorem and the Ax-Lindemann theorem which we shall prove. We also prove the whole Ax-Schanuel conjecture for the unipotent part. All these bi-algebraic and transcendental results generalize their counterparts for mixed Shimura varieties. In the end we briefly discuss about…
