Francesco Amoroso: "On fields with the Bogomolov property"
Abstract:
In 2001 Bombieri and Zannier introduced the notion of fields having the so-called Bogomolov
property (where the height is bounded below outside torsion points), denoted Property (B). In
this talk we discuss some recent results on this subject.
Victor Beresnevich: "Badly approximable points on manifolds and Schmidt's conjecture in higher dimensions"
Abstract:
In 1980 Schmidt conjectured the existence of points in the Euclidean plane which are badly
approximable for two different collections of `weights of approximation'. The conjecture
was proved by Badziahin, Pollington and Velani in 2011 by fibering the set of interest
with a family of lines in the plane. In this talk I will give an overview of recent developments
that involve badly approximable points on non-degenerate manifolds and in higher dimensions.
Yuri Bilu: "Effective proof of the theorem of André on the complex multiplication points on curves"
Abstract:
A complex multiplication point (hereinafter CM-point) on the complex
affine plane C^2 is a point of the form (j(a), j(b)), where a and b are
imaginary quadratic irrationalities and j denotes the modular invariant.
In 1998, Yves André proved that the irreducible plane curve f(x,y)=0 can
contain only finitely many CM-points, except when the curve is a
horizontal or vertical line, or a modular curve. It was the first proven
case of the famous André-Oort hypothesis about special points on Shimura
varieties.
Later several other proofs of the the Theorem of André were discovered;
mention especially a remarkable proof by Pila, which readily extends to
the multidimensional case. But, until recently, all known proof of the
Theorem of André were ineffective; that is, they did not allow, in
principle, to determine all CM-points on the curve. This was due to the
use of the Siegel-Brauer inequality on the class number of an imaginary
quadratic field, which is known to be ineffective.
Recently Lars Kühne and others suggested two new approaches to the
Theorem of André, which are both effective. One approach uses the method
of Baker and completely avoids the inequality of Siegel-Brauer. In the
other approach, the Siegel-Brauer inequality is replaced by the
"semi-effective" theorem of Siegel-Tatuzawa.
In my talk I will discuss these new approaches to the Theorem of André.
Yann Bugeaud: "Root separation of integer polynomials"
Abstract:
We survey recent results on root separation of integer polynomials of degree
at least four. In a joint work with A. Dujella, we have constructed parametric
families of reducible polynomials of degree d having two roots very close to
each other, but many questions remain largely open.
Emmanuel Breuillard: "Uniform height lowers bounds on character varieties of reductive groups, and the uniform Tits alternative"
Lecture series (3 talks)
Abstract:
The main focus of these lectures will be to describe a somewhat unexpected connection between
certain uniformity problems in geometric group theory and height lower bounds in diophantine geometry.
We introduce a notion of normalized height on the set of k tuples of invertible matrices taken up to
conjugation, and prove a Bogomolov type lower bound for k tuples generating a non virtually solvable
subgroup. This is then applied to prove a uniform version of the Tits alternative regarding the existence
of free subgroups in linear groups. If time permits I will discuss further applications such as expander
graphs and diophantine approximation in Lie groups.
Sara Checcoli: "On the Northcott and Bogomolov properties for fields and groups"
Abstract:
We will discuss some problems concerning the Northcott and Bogomolov properties, introduced by Bombieri
and Zannier: a set of algebraic numbers A has the Northcott property (N) if it contains finitely many elements
of bounded logarithmic Weil height; it has the Bogomolov property (B) if there exists a constant c>0 such that
every element in A has height either 0 or at least c.
Following Amoroso, David and Zannier, we say that a profinite group G has property (B) (resp. (N)) if for
any number field K and for any Galois extension L/K of Galois group G, the field L satisfies (B) (resp. (N)).
It is a natural and quite studied problem to investigate which subfields of $\bar\Q$ and which groups have
property (N) or (B). After a brief survey, I will discuss some results obtained in collaboration with M. Widmer
and some open questions.
Pietro Corvaja: "Perfect powers with few non-zero digits"
Abstract:
I will explain the relations between Vojta's conjecture on integral points on
algebraic varieties and certain conjectural finiteness statements about
integers which are perfect powers and have few non-zero digits. I will also
present old and recent unconditional results obtained in joint works with
Zannier and in a recent paper of Bennett-Bugeaud-Mignotte.
Sinnou David: "Points of small height on abelian varieties over function fields"
Abstract:
An old conjecture of Lang (for elliptic curves) generalized by Silverman, asserts that the Néron-Tate height
of a rational point of an abelian variety defined over a number field can be bounded below linearly in
terms of the Faltings height of the underlying abelian variety. We shall explore the function field analogue
of this problem.
Rainer Dietmann: "Quantitative versions of Hilbert's Irreducibility Theorem, and Probabilistic Galois Theory"
Abstract:
In this talk we want to show how recent advances on bounding the number of integral points on curves or
surfaces can be used to obtain new quantitative forms of Hilbert's irreducibility Theorem. As a special case,
we get new results on `Probabilistic Galois Theory': As van der Waerden has shown, `almost' all monic integer
polynomials of degree n have the full symmetric group S_n as Galois group. The strongest quantitative form of
this statement known so far is due to Gallagher, who made use of the Large Sieve and obtained a saving of 1/2
over the trivial bound in the exponent, which we can improve to a saving of -\sqrt{2}+2.
Christopher Frei: "Manin's conjecture for a singular del Pezzo surface over number fields"
Abstract:
Manin's conjecture predicts an asymptotic formula for the
number of rational points of bounded height on Fano varieties over
number fields. In particular, the case of del Pezzo surfaces (Fano
varieties of dimension two) has raised much interest. In recent years,
Manin's conjecture was proved, using methods from analytic number
theory, for several specific singular del Pezzo surfaces over the field
of rational numbers. We discuss a case where a stronger focus on the
geometry of numbers leads to a proof of Manin's conjecture over
arbitrary number fields.
Bobby Grizzard: "Relative Bogomolov extensions"
Abstract:
A field K of algebraic numbers has the Bogomolov property if it
has "no small points" (except torsion points). We introduce a
generalization of this to relative extensions: we say an extension L/K is
Bogomolov if all small points of L are already contained in K. We'll
discuss some basic properties of this definition and give some explicit
examples. Our main result implies that if K is a Galois extension of the
rationals with finite ramification index at some (finite) rational prime,
then there exist Bogomolov extensions L/K. This is reminiscent of the
theorem of Bombieri and Zannier that the Bogomolov property is enjoyed by
any Galois extension with a finite local degree.
Walter Gubler: "Chambert-Loir’s measures"
Abstract:
Chambert-Loir introduced measures on Berkovich spaces which occur as equidistribution measures of arithmetic
dynamical systems at non-archimedean places. In the recent work of Chambert-Loir and Ducros on forms and
currents on Berkovich spaces, these measures are described as Monge-Ampère measures using Bedford-Taylor
theory. In work in progress with Klaus Künnemann, we consider a more general notion of delta-forms which
allows us to write Chambert-Loir’s measures directly as a wedge product of delta-forms. This approach is useful
for the star-product of Green currents for divisors at non-archimedean places.
Kálmán Győry: "Effective bounds for the solutions of Diophantine equations over finitely generated domains"
Abstract:
I will present effective finiteness results in quantitative
form among others for unit equations, Thue equations, superelliptic
equations and discriminant equations over arbitrary finitely generated
domains over Z .This is a joint work with J.H.Evertse and partly
with A. Berczes.
Benjamin Klopsch: "Zeta functions of groups - representations of arithmetic groups"
Abstract:
In my talk I will give a brief introduction to zeta functions of groups, with an emphasis on representation
zeta functions of arithmetic groups. In particular, I will discuss joint results with Avni, Onn and Voll that shed
light on a conjecture of Larsen on Lubotzky in the context of the Congruence Subgroup Problem. My aim is to
survey key results and problems in the subject that are of general interest.
Michel Laurent: "Diophantine approximation by primitive points"
Abstract:
We are interested in solving systems of linear, or affine, inequalities seeking for integer solutions satisfying certain
constraints of coprimality. We shall review the few results presently available on the topic, both for a given
system and for a `generic' one, in other words from a metric point of view. With that regard, we shall
present a refined version of the classical Khintchine-Groshev Theorem in metric diophantine approximation.
Thái Hoàng Lê: "Intersective polynomials and Diophantine approximation"
Abstract: pdf
David Masser: "Unlikely intersections for algebraic curves in positive characteristic"
Abstract: pdf
Patrice Philippon: "Essential minima of toric varieties"
Abstract:
I will present a work in progress with J.Burgos Gil and
M.Sombra, giving combinatorial descriptions of arithmetic quantities
such as metrics, heights, algebraic successive minima, ... associated
to toric divisors on toric varieties. I will also show how the
obtained dictionary enable one to compute effectively these quantities
on specific examples.
Lukas Pottmeyer: "Heights and Free Abelian Groups"
Abstract:
It is known that the multiplicative group of K, an algebraic
extension of the rationals, modulo its roots of unity is free abelian if K
satisfies the Bogomolov property; i.e. the absolute logarithmic
Weil-height on elements of K is either zero or bounded from below by a
positive constant. In this talk we will give an example to show that the
converse is not true. This is a joint work with Grizzard and Habegger.
Gael Rémond: "Generalized Lehmer problems"
Abstract:
We propose a new conjecture extending Lehmer's problem and involving a
finite rank subgroup of nonzero algebraic numbers. The rank zero case
corresponds to the already studied so-called "relative" Lehmer's problem.
Here a generalization of Dobrowolski's result has been established by
Amoroso and Zannier. No such thing is known for higher rank subgroups. We
will also describe higher dimensional problems and the case of abelian
varieties.
Igor Shparlinski: "Effective Hilbert's Nullstellensatz and Finite Fields"
Abstract:This talk is based on a series of recent joint works with
J. Bourgain & M. Z. Garaev & S. V. Konyagin and
M.-C. Chan & B. Kerr & U. Zannier.
We give an overview of recent applications of effective
versions of Hilbert's Nullstellensatz to various problems
in the theory of finite fields.
In particular we show that almost all points on algebraic
varieties over finite fields avoid Cartesian products of
small order groups. This result is a step towards Vojta's
conjecture.
We also present some results about the size of the set
generated by s-fold products of some rational fractions in
a finite field. This result has some algorithmic applications.
We will also outline some open problems.
Cam Stewart: "A refinement of the abc conjecture"
Abstract:
We shall discuss joint work with Robert and Tenenbaum on a proposed refinement
of the well known abc conjecture.
Jeffrey Thunder: "On sums of quotients of certain L-series over function fields"
Abstract:
In his paper Northcott's Theorem on Heights II. The Quadratic Case,
W.M. Schmidt gave asymptotic estimates for the number of points in
projective space that have height no greater than a given bound and that
generate a quadratic extension of the rational number field. For the special case
where the points lie in projective 2-space, he needed to prove a result
on sums of quotients of certain L-series (coming from quadratic extensions
of the rational number field). These sums have independent interest.
One can prove asymptotic estimates when the rational number field is replaced
by a function field that are analogous to the results Schmidt obtained.
We will discuss the corresponding analog of Schmidt's sums of L-series,
which are used in the same manner as Schmidt
to prove estimates on the number of points of given height in
projective 2-space that generate a quadratic extension.
Jeffrey Vaaler: "Schauder bases and heights of algebraic numbers"
Abstract: pdf
Shou-Wu Zhang: "Congruent number problem and L-functions"
Abstract:
A thousand years old problem is to determine which positive integers are congruent numbers,
i,e, those numbers which could be the areas of right angled triangle with sides of rational lengths.
This problem has some beautiful connections with elliptic curves and L-functions.
In fact by the Birch and Swinnerton-Dyer conjecture, all n= 5, 6, 7 mod 8 should congruent numbers,
and most of n=1, 2, 3 mod 8 should not congruent numbers.
In this lecture, I will explain these connections and then some recent joint work with Ye Tian and Xinyi Yuan
based on the Waldspurger formula and the Gross--Zagier formula.