Rational and Integral Points via Analytic and Geometric Methods (18w5012)
The Casa Matemática Oaxaca (CMO) will host the "Rational and Integral Points via Analytic and Geometric Methods" workshop from May 27th to June 1st, 2018. The study of rational or integral solutions to systems of polynomial equations is a topic that is almost as old as mathematics itself. Such systems define algebraic varieties and a driving force, historically, has been the decidability question for the existence (or non-existence) of rational or integral points on varieties. In the event that such points exist, furthermore, it is natural to try and understand their density. The aim of this meeting is to bring together researchers in analytic number theory and arithmetic geometry to push the boundaries of these fundamental questions. The Casa Matemática Oaxaca (CMO) in Mexico, and the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Banff, are collaborative Canada-US-Mexico ventures that provide an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station in Banff is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT). The research station in Oaxaca is funded by CONACYT.
23
2018
42
13 Stunden 19 Minuten
23 Ergebnisse
32:30
1Top, JaapAlready 190 years ago Jacobi in a paper in Crelle's journal described the celebrated closure theorem of Poncelet as a ``bekanntes Problem der Elementargeometrie''. Some natural number theoretical questions arising in this context, will be discussed during this talk.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
25:26
Morgan, AdamFor an abelian variety A over a number field K, a consequence of the Birch and Swinnerton-Dyer conjecture is the 2-parity conjecture: the global root number agrees with the parity of the 2-infinity Selmer rank. It is a standard result that the root number may be expressed as a product of local terms and we show that, over any quadratic extension of K, the same holds true for the parity of the 2-infinity Selmer rank. Using this we prove several new instances of the 2-parity conjecture for general principally polarised abelian varieties by comparing the local contributions arising. Somewhat surprisingly, the local comparison relies heavily on results from the theory of quadratic forms in characteristic 2.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
28:47
Newton, RachelYongqi Liang has shown that for rationally connected varieties over a number field K, sufficiency of the Brauer-Manin obstruction to the existence of rational points over all finite extensions of K implies sufficiency of the Brauer-Manin obstruction to the existence of zero-cycles of degree 1 over K. I will discuss joint work with Francesca Balestrieri where we extend Liang's result to Kummer varieties.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
30:38
4Vishe, PankajWe will prove that smooth Quartic hypersurfaces satisfy the Hasse Principle as long as they are defined over at least 30 variables. The key tool here is employing Kloosterman's version of circle method. This is a joint work with Oscar Marmon (U Lund).
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
59:26
4Zureick-Brown, DavidThis is joint with Jordan Ellenberg and Matt Satriano.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
28:51
1Bright, MartinThere has been much interest recently in bounding the Brauer groups of K3 surfaces over number fields. On the other hand, the arithmetic of integral points on log K3 surfaces appears to share some features with that of rational points on K3 surfaces. Some of the simplest examples of log K3 surfaces are the open surfaces obtained by starting with a projective del Pezzo surface and removing a smooth anticanonical divisor. We use Merel's boundedness of torsion on elliptic curves to prove boundedness of the Brauer groups of such log K3 surfaces over a number field. This is joint work with Julian Lyczak.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
32:35
1Schindler, DamarisWe discuss small solutions to ternary diagonal inequalities of any degree where all of the variables are assumed to be of size P. We study this problem on average over a one-parameter family of forms and discuss a generalization of work of Bourgain on generic ternary diagonal quadratic forms to higher degree. In particular we discuss how these Diophantine inequalities are related to counting rational points close to varieties.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
59:20
2Heath-Brown, Roger2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
31:06
2Voloch, FelipeIt is widely expected that, if a curve over a global field has no rational points, that there is an obstruction to existence of rational points coming from the Brauer group. One piece of evidence for this is an heuristic due to Poonen. We show that Poonen's argument also applies to p-primary subgroups of the Brauer group (for any prime p) but that there are examples of curves with no rational point but not having an obstruction coming from the p-primary subgroups of the Brauer group. Joint work with B. Creutz and B. Viray.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
30:37
Berg, JenniferAfter fixing numerical invariants such as dimension, it is natural to ask which birational classes of varieties fail the Hasse principle, and moreover whether the Brauer group (or certain distinguished subsets) explains this failure. In this talk, we will focus on K3 surfaces (e.g. a double cover of the plane branched along a smooth sextic curve), which have been a testing ground for many conjectures on rational points. In 2014, Ieronymou and Skorobogatov asked whether any odd torsion in the Brauer group of a K3 surface could obstruct the Hasse principle. We answer this question in the affirmative for transcendental classes; via a purely geometric approach, we construct a 3-torsion transcendental Brauer class on a degree 2 K3 surface over the rationals with geometric Picard rank 1 (hence with trivial algebraic Brauer group) which obstructs the Hasse principle. Moreover, we do this without needing a central simple algebra representative. This is joint work with Tony Varilly-Alvarado.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
27:04
Gunther, JosephMasser and Vaaler gave an asymptotic formula for the number of algebraic numbers of given degree and increasing height. This problem was solved by counting lattice points in an expanding star body. We'll explain how to estimate the volumes of slices of star bodies, which allows one to count algebraic integers, algebraic integers of given norm and/or trace, and more. There will be pictures.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
29:41
1Frei, ChristopherLet l be a positive integer. We discuss improved average bounds for the l-torsion of the class groups for some families of number fields, including degree-d-fields for d between 2 and 5. The improvements are based on refinements of a technique due to Ellenberg, Pierce and Wood.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
32:51
Luijk, Ronald van2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
32:39
2Mitankin, VladimirA classical result of Colliot-Thélène and Sansuc states that the only obstruction to the Hasse principle and weak approximation for generalised Châtelet surfaces is the Brauer-Manin one, conditionally on Schinzel's hypothesis. Inspired by their work, we study the analogous questions concerning the existence and the density of integral points on the corresponding affine surfaces, again under Schinzel's hypothesis. To be precise, we show that the Brauer-Manin obstruction is the only obstruction to the integral Hasse principle for an infinite family of generalised affine Châtelet surfaces. Moreover, we show that the set of integral points on any surface in this family satisfies a strong approximation property off infinity with Brauer-Manin obstruction.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
32:24
2Huang, ZhizhongWe propose a 2-cover method to study rational points on elliptic surfaces. We apply it to several isotrivial Kummer-type families whose generic Mordell-Weil ranks are 0 so that geometric argument may fail and we show that for these families rational points are Zariski dense and even dense in real topology, which is thereby in favor of a conjecture of Mazur.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
30:43
13Etropolski, AnastassiaGiven a curve of genus at least 2, it was proven in 1983 by Faltings that it has only finitely many rational points. Unfortunately, this result is ineffective, in that it gives no bound on the number of rational points. 40 years earlier, Chabauty proved the same result under the condition that the rank of the Jacobian of the curve is strictly smaller than the genus. While this is obviously a weaker result, the methods behind that proof could be made effective, and this was done by Coleman in 1985. Coleman's work led to a procedure known as the Chabauty-Coleman method, which has shown to be extremely effective at determining the set of rational points exactly, particularly in the case of hyperelliptic curves. In this talk I will discuss how we implement this method using Magma and Sage to provably determine the set of rational points on a large set of genus 3, rank 1 hyperelliptic curves, and how these calculations fit into the context of the state of the art conjectures in the field.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
25:58
1Loughran, DanielA famous theorem due to Erdős and Kac states that the number of prime divisors of an integer N behaves like a normal distribution. In this talk we consider analogues of this result in the setting of arithmetic geometry, and obtain probability distributions for questions related to local solubility of algebraic varieties. This is joint work with Efthymios Sofos.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
26:50
3Pieropan, MartaIn the 1950s Lang studied the properties of C1 fields, that is, fields over which every hypersurface of degree at most n in a projective space of dimension n has a rational point. Later he conjectured that every smooth proper rationally connected variety over a C1 field has a rational point. I will explain how to find rational points on rationally connected threefolds over C1 fields of characteristic 0.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
51:42
3Sofos, EfthymiosA topic of current interest regards 'how often' a variety has a rational point. This topic was initially studied by Serre who gave upper bounds in the case of families of conics parametrised by a projective space. In the last few years this topic has been significantly enriched by Loughran and others. I will begin by giving an overview of the latest developments and finish by discussing joint work with Erik Visse, where asymptotics are given for a family of conics parametrised by arbitrary smooth hypersurfaces of low degree.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
23:01
1Morrow, Jackson S.Let d and g be positive integers with 10 such that a positive proportion of odd genus g hyper elliptic curves over Q have at most B(d) points of degree d. If d is even, we similarly bound the degree d points not pulled back from degree d/2 points of the projective line. Our proof proceeds by refining Park’s recent application of tropical geometry to symmetric power Chabauty, and then applying results of Bhargava and Gross on average ranks of Jacobians of hyperelliptic curves.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
36:06
1Desjardins, JulieWhat can we say about the variation of the rank in a family of elliptic curves ? We know in particular that if infinitely many curves in the family have non-zero rank, then the set of rational points is Zariski dense in the associated elliptic surface. We use a “conjectural substitute” for the geometric rank (or rather for its parity) : the root number. For a non-isotrivial family, under two analytic number theory conjectures I show that the root number is -1 (resp. +1) for infinitely many curves in the family. On isotrivial families however, the root number may be constant : I describe its behaviour in this case.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
1:00:34
Hindry, MarcWe consider a family of abelian varieties over a number field K , i.e. a variety X with a map to a curve B whose fibres are abelian varieties (the interesting cases are when B is the projective line or an elliptic curve with positive rank). The generic fibre is an abelian variety over the function field K(B) and the group of K(B)-rational points has a rank r. For almost all points t in B(K) the fibre is an abelian variety Xt over K and the group of K-rational point has rank r(t). A specialisation theorem of Silverman says that for or almost all points t in B(K) the rank r(t) is greater or equal to r. We want to understand the distribution of r(t), in particular we ask wether there are infinitely many t's 1) with r(t)=r, 2) with r(t)>r. The problem looks very hard in general, but, under specific geometric conditions, we will settle the second question, and provide interesting example where much more can be proven. This is a joint work with Cecília Salgado.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
30:04
Skorobogatov, AlexeiWe show that the uniform boundedness of the transcendental Brauer group of K3 surfaces and abelian varieties of bounded dimension defined over number fields of bounded degree is a consequence of a conjecture of Coleman about rings of endomorphisms of abelian varieties. We also show that this conjecture of Coleman implies the conjecture of Shafarevich about the N\'eron-Severi lattices of K3 surfaces. This is a joint work with Martin Orr and Yuri Zarhin.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery