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 $F_q[T]$-modules.
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 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 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 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 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 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 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…
Let X be a product of Drinfeld modular curves over a general base ring A of odd characteristic. We classify those subvarieties of X which contain a Zariski-dense set of CM points. This is an analogue of the André-Oort conjecture. As an application, we construct non-trivial families of higher Heegner points on modular elliptic curves over global function fields.
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 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.
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…
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 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.
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 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…
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-Oo…
The André-Pink-Zannier conjecture concerns the intersection of subvarieties and the generalized Hecke orbit of a given point in mixed Shimura varieties. 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 conjecture in this case and prove several different cases for this conjecture: its overlap with the André-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.
The goal of this paper is to study a $p$-adic analog of the joint of the conjectures of André--Oort and André--Pink. More precisely, on a product of ordinary Siegel formal moduli schemes, we study the distribution of points whose components are either CM points or points in Hecke orbits. We use linearity of formal subschemes of the product as the $p$-adic analog of geodesicness over complex numbers. Moreover, we relax the usual incidence relations by using $p$-adic distance. We also study a $p$-adic formal scheme theoretic analog of the Ax--Lindemann theorem.
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.
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 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.
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.
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…
Consider a fibered power of an elliptic surface. We characterize its subvarieties that contain a Zariski dense set of points that are torsion points in fibers with complex multiplication. This result can be viewed as a mix of the Manin-Mumford and André-Oort Conjecture and is related to a conjecture of Pink. The main technical tool is a new height inequality. We also use it to give another proof of a case of Gubler's result on the Bogomolov Conjecture over function fields.
Let s be a special point on a Shimura variety, and x a pre-image of s in a fixed fundamental set of the associated Hermitian symmetric domain. We prove that the height of x is polynomially bounded with respect to the discriminant of the centre of the endomorphism ring of the corresponding Z-Hodge structure. Our bound is the final step needed to complete a proof of the Andre-Oort conjecture under the conjectural lower bounds for the sizes of Galois orbits of special points, using a strategy of Pila and Zannier.
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 …
Let $S$ be a Shimura variety and let $h$ be a Weil height function on $S$. We conjecture that the heights of special points in $S$ are discriminant negligible. 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 the case of Shimura varieties of abelian type, the height bound holds by the recently proved averaged Colmez formula, and our theorem gives a new proof of Tsimerman's Galois lower bound in this case. The main novelty is that our approach avoids the u…
Let $F/\mathbb{Q}$ be a totally real field and $A$ a modular $\GL_2$-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 $L'(1, A, χ) \neq 0$ 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 $\GL_{2}(\BA_{F})$. Let $χ$ be a …
