Talks
Unlikely Intersections in certain families of abelian varieties and the polynomial Pell equation
Abstract: Given n independent points on the Legendre family of elliptic curves of equation Y^2=X(X-1)(X-c) with coordinates algebraic over Q(c), we will see that there are at most finitely many specializations of c such that two independent relations hold between the n points on the specialized curve. This fits in the framework of the so-called Unlikely Intersections. We will then see an higher-dimensional analogue of this result and explain how it applies to the problem of studying the solvability of the (almost-)Pell equation in polynomials.
This is joint work with Laura Capuano.
Special Points on Straight Lines and Hyperbolas
Abstract: I will speak on the recent joint work with Bill Allombert, Florian Luca, David Masser and Amalia Pizarro-Madariaga about the special points on the simplest algebraic curves. Call a point "special" if its coordinates are j-invariants of lattices with complex multiplication. I will state the general theorem of Yves André about finiteness of the number of such point of a "general" algebraic curve. Next, I will show that one can say much more in the case of the "simple" curves, like straight lines and hyperbolas.
Height of rational points on quadratic twists of a given elliptic curve
Abstract: Given a family of quadratic twists Ed of a fixed elliptic curve defined over Q, one expects in general that the non-torsion rational points on the curves Ed with positive rank are "high" in terms of d. We will discuss several topics related to this problem.
Strong approximation and a conjecture of Harpaz and Wittenberg
Abstract: In joint work with Damaris Schindler, strong approximation off a finite set of places is studied for some algebraic varieties which are defined using norm forms over the rational numbers. This permits the verification of (a special case of) a recent conjecture due to Harpaz and Wittenberg. The key input comes from analytic number theory and, in particular, an earlier counting argument of Browning and Heath-Brown.
On Singular Moduli that are Algebraic Units
Abstract: A singular moduli is the j-invariant of an elliptic curve with complex multiplication, e.g. 0 or 1728 or -3375. A classical result states that all singular moduli are algebraic integers. It seems to be unknown whether there are singular moduli that are algebraic units. But there can be at most finitely many, as we will see in this talk. Moreover, I will present a related finiteness statement reminiscent of Siegel's Theorem on integral points on curves of positive genus.
Rational points on intersections of quadrics
Abstract: This talk will survey what is known about the Hasse principle and weak approximation for intersections of quadrics in higher dimension. This will lead in to a progress report about ongoing work on intersections of three quadrics.
mini-course: On the principle of Batyrev and Manin
Abstract: These lectures are devoted to the study of the distribution of rational points of bounded height on algebraic varieties. One of the basic fact in the program of Batyrev and Manin is that rational points of bounded height may accumulate on subvarieties.
I shall start by describing various particular cases of accumulation, then introduce an adelic measure which seems to describe the distribution of points once the accumulating subsets are removed. Then I shall explain the geometric analog and the relationship to the notion of free rational curves. The notion of slopes as defined by Bost enables us to define a similar notion of freeness in the arithmetic setting.
Applying these slopes to the examples of the beginning reveals a strong link between slopes and accumulation.
On a modular Fermat equation
Abstract: I will talk about some diophantine problems suggested by the analogy between multiplicative groups and powers of the modular curve in problems of "unlikely intersections''. I will describe how the same circle of ideas enables a proof of a special case of the Zilber-Pink conjecture for curves.
Isogenies and torsion
Abstract: I give a sufficient condition for a cyclic isogeny between two abelian varieties to be of minimal degree. This allows to produce an upper bound for the cardinality of the torsion group of an abelian variety over a number field when an isogeny theorem is known.
Counting rational points with the determinant method
Abstract: Heath-Brown developed his p-adic determinant method in order to count rational points of bounded height on algebraic varieties. We then discovered a slightly sharper version of his p-adic method by making use of the theory of volumes of line bundles and we also found a global version of his method based on congruences modulo all primes. But it was only recently that we have been able to combine these two improvements to a new even stronger version of the determinant method. We shall in our talk present this method and some applications to counting functions for surfaces.
Higher order expansions via the circle method
Abstract: Given a homogeneous polynomial F in s variables, it is a natural question to study the number N(P) of representations of an integer n by this form, where all the variables are restricted to a box of side length P, for P tending to infinity. In the case where F is not too singular and the number of variables large enough compared to the degree, Birch has successfully applied the circle method to give an asymptotic formula for N(P). In this talk we discuss higher order expansions for N(P) as a refinement of Birch's result, which only gives the main term in the asymptotic for N(P). Previously very little was known in this direction, with the exception of a result of Vaughan and Wooley in the special case of Waring's problem. In this talk we present a much more general approach and derive a multi-term asymptotic for general forms F in sufficiently many variables, together with some interpretation for the occurring lower order terms.
Distribution of Points on Varieties over Finite Fields
Abstract: We discuss some recent results on the distribution of rational points on algebraic varieties over finite fields that belong to a given box. Generally, these results fall into two categories:
-large boxes, where asymptotic results are possible;
-small boxes, where only upper bounds are possible.
We discuss results of both types and also indicate the underlying methods. Some of these work for very general varieties, some apply only to very specific varieties such as hypersurfaces x1-1+...+xn-1=a that appear in the Erdös-Graham problem or "hyperbolas'' x1...xn=a.
Besides intrinsic interest, the aforementioned results have a wide scope of applications in number theory and beyond. These include: bounds of Kloosterman sums, shifted power identity testing, distribution of elliptic curves in isogeny and isomorphism classes, polynomial dynamical systems in finite fields and several others.
mini-course: Weakly admissible lattices, Diophantine approximation and counting
Abstract: Admissible lattices are those elements of SLn(R)/SLn(Z) whose orbit under the diagonal flow is bounded. Extending results of Hardy and Littlewood Skriganov showed around 1994 that for admissible lattices the points inside aligned boxes are extremely uniformly distributed, more precisely, the error term can be bounded by a power of the logarithm of the volume of the box (independently of its shape!). Examples of admissible lattices include ideal lattices from totally real number fields. Unfortunately, almost no lattice is admissible but slightly weakening the notion of admissibility one can show that almost every lattice is``weakly admissible". This raises the question to what extent Skriganov's result can be extended to weakly admissible lattices. This is one of the problems we shall consider in this mini-course. As shown by Dani admissible and weakly admissible lattices are tightly connected to problems in Diophantine approximation. As an application we will deduce some counting results motivated by the Littlewood conjecture.
The McKay correspondence over local fields and number fields
Abstract: I will talk about certain generalizations of the McKay correspondence to arithmetic situations. Given a linear representation of a finite group, the McKay correspondence relates a geometric invariant of the quotient variety with an algebraic invariant of the representation. For instance, one version says that the stringy Euler number of the quotient variety is equal to the number of conjugacy classes in the group. In arithmetic situations, we formulate the McKay correspondence as a comparison of two counting problems; counting rational points, and counting extensions of a local or number field. Over a local field, this leads to a geometric interpretation of mass formulas by Serre and Bhargava. Over a number field, this relates Manin’s conjecture on rational points of Fano variety with Malle’s conjecture on extensions of a number field.