379 research outputs found

### Preperiodic points and unlikely intersections

In this article, we combine complex-analytic and arithmetic tools to study
the preperiodic points of one-dimensional complex dynamical systems. We show
that for any fixed complex numbers a and b, and any integer d at least 2, the
set of complex numbers c for which both a and b are preperiodic for z^d+c is
infinite if and only if a^d = b^d. This provides an affirmative answer to a
question of Zannier, which itself arose from questions of Masser concerning
simultaneous torsion sections on families of elliptic curves. Using similar
techniques, we prove that if two complex rational functions f and g have
infinitely many preperiodic points in common, then they must have the same
Julia set. This generalizes a theorem of Mimar, who established the same result
assuming that f and g are defined over an algebraic extension of the rationals.
The main arithmetic ingredient in the proofs is an adelic equidistribution
theorem for preperiodic points over number fields and function fields, with
non-archimedean Berkovich spaces playing an essential role.Comment: 26 pages. v3: Final version to appear in Duke Math.

### Torsion of elliptic curves and unlikely intersections

We study effective versions of unlikely intersections of images of torsion
points of elliptic curves on the projective line.Comment: 19 page

### Rational points on Grassmannians and unlikely intersections in tori

In this paper, we present an alternative proof of a finiteness theorem due to
Bombieri, Masser and Zannier concerning intersections of a curve in the
multiplicative group of dimension n with algebraic subgroups of dimension n-2.
The proof uses a method introduced for the first time by Pila and Zannier to
give an alternative proof of Manin-Mumford conjecture and a theorem to count
points that satisfy a certain number of linear conditions with rational
coefficients. This method has been largely used in many different problems in
the context of "unlikely intersections".Comment: 16 page

### Unlikely intersections with isogeny orbits

This thesis consists of six chapters and two appendices. The first two chapters contain the introduction and some preliminaries.
Chapter 3 contains a characterization of curves in abelian schemes, defined over $\bar{\mathbb{Q}}$, that intersect certain (enlarged) isogeny orbits infinitely often. An isogeny orbit is the set of all images of a fixed finite-rank subgroup of a fixed abelian variety, both of which we assume to be defined over $\bar{\mathbb{Q}}$, under all isogenies between the fixed abelian variety and some fiber of the abelian scheme. It is enlarged (depending on a parameter $k$) if the finite-rank subgroup is replaced by the union of its translates by abelian subvarieties of codimension at least $k$. The obtained characterization yields a stronger version of the so-called André-Pink-Zannier conjecture for curves in the case where everything is defined over $\bar{\mathbb{Q}}$.
Chapter 4 contains a characterization of subvarieties of arbitrary dimension of abelian schemes, defined over $\bar{\mathbb{Q}}$, that intersect a (non-enlarged) isogeny orbit in a Zariski dense set, under technical restrictions on the abelian scheme and the fixed abelian variety. The restrictions are satisfied for example if the abelian scheme is a fibered power of a non-isotrivial elliptic scheme and the fixed abelian variety is a power of an elliptic curve without CM that is defined over $\bar{\mathbb{Q}}$. This again proves a stronger version of the André-Pink-Zannier conjecture in certain cases. The proof combines the Pila-Zannier method with a generalized Vojta-Rémond inequality.
Chapter 5 contains (among other results) a characterization of semiabelian schemes over a curve, defined over $\bar{\mathbb{Q}}$, with infinitely many pairwise isogenous fibers. It also contains an extension of the approach to the Manin-Mumford conjecture through use of the Galois action (developed and applied by Serre, Tate, Lang, and Hindry) to the problem of studying torsion points on pairwise isogenous fibers in abelian schemes.
Chapter 6 consists of joint work with Fabrizio Barroero, where we show that the Zilber-Pink conjecture for complex abelian varieties follows from the same conjecture for abelian varieties defined over $\bar{\mathbb{Q}}$. Moreover, the conjecture holds for a curve in a complex abelian variety and it holds in any complex abelian variety that contains no abelian subvariety of dimension larger than $4$ that can be defined over $\bar{\mathbb{Q}}$.
Appendix A contains the proof of the generalized Vojta-Rémond inequality (which draws on unpublished work by Ange) mentioned above. Appendix B contains asymptotic results on counting algebraic numbers of fixed degree and fixed height

### Unlikely Intersections with Bruhat Strata

Let $\mathcal{A}_{g}$ be the moduli space of $g$-dimensional principally
polarized abelian varieties over $\mathbb{Z}$, and let $\mathcal{T} \subset
\mathcal{A}_{g}$ be a closed locus, also defined over $\mathbb{Z}$. Motivated
by unlikely intersection conjectures, we study the intersection of
$\mathcal{T}_{\mathbb{F}_{p}}$ with the Bruhat strata in
$\mathcal{A}_{g,\mathbb{F}_{p}}$ as $p$-varies; these are strata characterized
by the existence of certain subgroup schemes inside the $p$-torsion of the
fibres. We find that, away from a finite set of primes, positive-dimensional
``unlikely'' intersections of $\mathcal{T}_{\mathbb{F}_{p}}$ with such strata
are all accounted for by intersections of $\mathcal{T}$ with special loci
inside $\mathcal{A}_{g}$. This result generalizes to all abelian-type Shimura
varieties, and variations of Hodge structures equipped with certain motivic
data. It moreover gives another example of how functional transcendence
principles in characteristic zero can be used to study unlikely intersections
in positive characteristic, building on recent work by the author

### Unlikely intersections in semi-abelian surfaces

We consider a family, depending on a parameter, of multiplicative extensions
of an elliptic curve with complex multiplications. They form a 3-dimensional
variety $G$ which admits a dense set of special curves, known as Ribet curves,
which strictly contains the torsion curves. We show that an irreducible curve
$W$ in $G$ meets this set Zariski-densely only if $W$ lies in a fiber of the
family or is a translate of a Ribet curve by a multiplicative section. We
further deduce from this result a proof of the Zilber-Pink conjecture (over
number fields) for the mixed Shimura variety attached to the threefold $G$,
when the parameter space is the universal one.Comment: 20 pages. Appendix added, with a proof of the Zilber-Pink for the
Poincar\'e-biextension over a CM elliptic curv

### Some problems of unlikely intersections

Fixing X in the multiplicative algebraic group G^n_m(\Q), we are interested in the intersection of X with the union of Y where Y runs through the algebraic subgroups of G^n_m restricted only by dimension. We shall prove that, if X is an irreducible subvariety in G^n_m and the considered algebraic subgroups have codimension at least dim X, then this intersection has a bounded height by removing the anomalous sub varieties of X. Furthermore, we recover the finiteness by considering the algebraic subgroups of codimension at least 1 + dim X

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