Related papers: A two dimensional arithmetic Andr\'e-Oort problem
We establish an effective version of the Andr\'e-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…
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…
We state and prove a variant of the Andr\'e-Oort conjecture for the product of 2 modular curves in positive characteristic, assuming GRH for quadratic fields.
The modular case of the Andr\'e-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)…
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…
In the proofs of most cases of the Andr\'e-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…
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…
We explore an analogue of the Andr\'e-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)…
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…
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…
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…
The object of this work is to present the status of art of an open problem: to provide an analogue for Shimura curves of the Ihara's lemma \cite{Ihara73} which holds for modular curves. We will describe our direct result towards the…
We study one-dimensional algebraic families of pairs given by a polynomial with a marked point. We prove an "unlikely intersection" statement for such pairs thereby exhibiting strong rigidity features for these pairs. We infer from this…
We present a conjecture on the geometry of the Hodge locus of a (graded polarizable, admissible) variation of mixed Hodge structure over a complex smooth quasi-projective base, generalizing to this context the Zilber-Pink Conjecture for…
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…
We give a new construction of overconvergent modular forms of arbitrary weights, defining them in terms of functions on certain affinoid subsets of Scholze's infinite-level modular curve. These affinoid subsets, and a certain canonical…
In this note, we present recent progress on rigidity problems in one-dimensional complex dynamics, including the proof of Dynamical Andr\'e-Oort conjecture for curves and generic injectivity of multiplier spectrum. The proofs combine ideas…
The purpose of this article is to explain the Pila-Zannier strategy for proving the Andr\'e-Oort conjecture. First, however, we will provide a brief introduction to the theory of Shimura varieties.
For quadratic curves over $F_p$, the number of solutions, which is governed by an analogue of the Mordell-Weil group, is expressed with the Legendre symbol of a coefficient of quadratic curves. Focusing on the number of solutions, a…
The Andr\'e-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…