Galois representations, Integral points, Unlikely intersections

This is the homepage for the workshop on Galois representations, Integral points, and Unlikely intersections held in Mainz from April 9th until April 11th.  The speakers are

  • Laura Capuano (Oxford),
  • Philipp Habegger (Basel),
  • Aaron Levin (Michigan), and
  • Daniel Litt (IAS, University of Georgia).
  • Kang Zuo (Mainz)

All talks will take place in the Hilbertraum. (Fifth floor of the mathematics building.)

The tentative schedule is as follows. Abstract and titles are below.

Tuesday April 9th

9h - 10h Philipp Habegger Unlikely Intersections, height bounds, and applications to rational points on curves I
Coffee break
10:30-11:30h Philipp Habegger Unlikely Intersections, height bounds, and applications to rational points on curves II
Lunch
1330-1430h     Daniel Litt Arithmetic and the representation theory of fundamental groups I  
Coffee break
15h-16h          Laura Capuano Unlikely intersections in families of abelian varieties I

 

 

Wednesday April 10th

9h-10h              Laura Capuano Unlikely intersections in families of abelian varieties II
Coffee break
1030h-1130h    Laura Capuano  Unlikely intersections in families of abelian varieties III
Lunch
1330h-1430h     Daniel Litt Arithmetic and the representation theory of fundamental groups II
Coffee break
15h-16h       Aaron Levin  Integral points on algebraic varieties I

 

Thursday April 11th

9h - 10h               Philipp Habegger    Unlikely Intersections, height bounds, and applications to rational points on curves III 
Coffee break
10:30h- 1130h     Aaron Levin   Integral points on algebraic varieties II 
Lunch
1330-1430h      Daniel Litt  Arithmetic and the representation theory of fundamental groups III
Coffee break
15h-16h            Kang Zuo  Motivic local systems  of rank two over the punctured projective line via an                                                 arithmetic Simpson correspondence
17h - 18h          Aaron Levin    Integral points on algebraic varieties II

Abstracts

Laura Capuano. Unlikely intersections in families of abelian varieties

Abstract. The Zilber-Pink conjectures on unlikely intersections predict the behaviour of subvarieties of families of (semi)abelian varieties and more generally of Shimura varieties when intersected with "special" subvarieties of the ambient space. These conjectures generalise many well-known conjectures/results such as Manin-Mumford, Mordell-Lang and André-Oort. In this course, I will present recent results in this framework obtained in a series of joint papers with F. Barroero for curves in families of abelian varieties defined over a number field (the case of families of abelian varieties of relative dimension 2 is nothing but the "relative version" of Manin-Mumford and has been previously proved in a series of papers by Masser and Zannier).
The proof of these results follows the now well-estabilished Pila-Zannier strategy, first introduced by the two authors in 2008 to give an alternative proof of Raynaud's Theorem (formerly the Manin-Mumford conjecture for abelian varieties), which combines theorems about counting rational points of bounded height in o-minimal structures (Pila-Wilkie and subsequent generalizations) with other diophantine ingredients. I will also present an application of these unlikely intersections results to the study of the solvability of a function field variant of the classical Pell equation.

 

Philipp Habegger. Unlikely Intersections, height bounds, and applications to rational points on curves

Abstract: This course contains an introduction to the Conjecture on Unlikely Intersections, sometimes also called the Zilber-Pink Conjecture. This conjecture generalizes many classical results in diophantine geometry such as Faltings's Theorem (the Mordell Conjecture), Raynaud's Theorem (the Manin-Mumford Conjecture), and the André-Oort Conjecture, solved now in many cases after work of Daw, Klingler, Pila, Orr, Tsimerman, Ullmo, Yafaev, and others. Important tools in this area is the point counting strategy going back to an idea of Zannier and tools from the theory of heights.  
I will present joint work with Ziyang Gao comparing the Néron-Tate and Weil heights in a one parameter family of abelian varieties. As an application we give new estimates for the number of rational points on a one-parameter family of curves of genus at least 2. The latter is more recent joint work with Vesselin Dimitrov and Ziyang Gao.

 

Aaron Levin. Integral points on algebraic varieties
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Abstract: We will give an overview of the subject of integral points on algebraic varieties, starting from introductory examples and definitions, and ending with topics drawn from current research in the area.  Topics will include points on curves, effective methods, higher-dimensional results, points of bounded degree, and related tools (primarily from Diophantine approximation).

Daniel Litt. Arithmetic and the representation theory of fundamental groups
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Abstract: Let X be an algebraic variety over the complex numbers. Which representations of the fundamental group of X come from geometry, i.e. arise inside the monodromy representation on the cohomology of a family of varieties over X? I'll discuss applications of arithmetic techniques (including p-adic Hodge theory, the geometric Langlands program, and Galois deformations) to this question, including (1) the proof of a function field analogue of Shafarevich's finiteness conjecture, (2) restrictions on the mod p structure of non-trivial monodromy representations, and (3) several open conjectures on the structure of the set of such representations.
.
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Kang Zuo: Motivic local systems  of rank two over the punctured projective line via an arithmetic Simpson correspondence
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Abstract: We propose an arithmetic Simpson correspondence for Higgs bundles over an arithmetic scheme, and speculate about relations between the arithmetic dynamical system over the projective line with four punctured points arising from the so-called rank-2 Higgs-de Rham flow, and multiplication on the associated elliptic curve as the double cover of  the projective line ramified at the four points. We predict that a fixed rank-2 graded stable Higgs bundle of degree -1 over the projective line  with logarithmic singularities at four punctured points corresponds to the Hodge structure  of an abelian scheme endowed with real multiplication over the projective line  and bad reduction precisely at the four points  if and only if the zero of the Higgs field is the image of a torsion point on the associated  elliptic curve.  
We have constructed 26  complete solutions  in the case of elliptic surfaces whose Kodaira-Spencer maps  have zeros of torsion order 1, 2, 3, 4 and 6.  Similar phenomena appear in the work by Kontsevich on rank-2  ell-adic motivic local systems on the projective line with four punctured points.
This is a joint project with J. Lu, X. Lu, R.R. Sun and J. B. Yang.

Flying to Mainz

The nearest airport is Frankfurt Airport. It's about 20 minutes from the central station of Mainz by train.

Getting to the math department

There are three possibilities.

  1. You can walk from the central station. The walk is about 35 minutes.
  2. From the central station (Mainz Hauptbahnhof) take a bus or tram to Friedrich von Pfeifferweg. From there, the mathematics building (Staudingerweg 9) is about 7 minutes walking.
  3. From the central station (Mainz Hauptbahnhof) take a bus or tram to Universität. From there, the mathematics building (Staudingerweg 9) is about 20 minutes walking.

 

 

Organizer: Ariyan Javanpeykar