Jennifer Balakrishnan (Boston University): Quadratic Chabauty for modular curves (click for slides)
We describe how p-adic height pairings can be used to determine the set of rational points on curves, in the spirit of Kim’s nonabelian Chabauty program. In particular, we discuss what aspects of the quadratic Chabauty method can be made practical for certain modular curves. This is joint work with Netan Dogra, Steffen Mueller, Jan Tuitman, and Jan Vonk.
Gal Binyamini (Weizmann Institute of Science): Multiplicity estimates on general foliations
Estimating orders of zeros of polynomials on leafs of foliations is important in many transcendence proofs. I also used such estimates to prove sharper variants of the Pila Wilkie theorem. There is some overlap between these methods, but the classical zero estimates appearing e.g. in the work of Wüstholz apply specifically to translation invariant foliations on group varieties and produce sharper, essentially optimal estimates – subject only to natural obstructions related to algebraic subgroups. This leads to results such as the Wüstholz analytic subgroup theorem and the Masser-Wüstholz period theorem, which seem to lie beyond the reach of the more “general” point-counting strategy.
Toward a uniform approach to these subjects I will formulate a sharp zero estimate, generalizing the group-variety estimates to general foliations. The algebraic subgroup obstruction is replaced by general unlikely intersections. I’ll briefly discuss the proof, based on Gabrielov-Khovanskii’s calculation of intersection multiplicities in terms of Milnor fibers, and topological estimates using stratified complex Morse theory. Finally I’ll discuss some transcendence-type applications of this result in the direction of Bombieri-Schneider-Lang, the analytic subgroup theorem, etc.
Anna Cadoret (Institut de Mathématiques de Jussieu): A very small step towards the “open image problem” over higher dimensional bases (click for slides)
Let X be a smooth geometrically connected variety over a finitely generated field. Given a l-adic local system F on X, define the degeneration locus of F to be the set of closed points where the image of the attached local Galois representation is of positive codimension in the image of the generic one (e.g. if F arises from geometry, namely F=Rf*ℚl for a smooth proper morphism f:Y → X, the degeneration locus of F is closely related to the set of closed points whose corresponding fiber Yx carries additional cycles). The “open image problem” asks for “minimal assumptions” on F ensuring that the set of k-rational points in the degeneration locus is not Zariski-dense. One expects that l-adic local systems arising from geometry satisfy such “minimal assumptions”. If X is a curve, Tamagawa and I proved that the minimal assumption is that the Lie algebra of the image of the geometric fundamental group be perfect.
After recalling the general philosophy, I will discuss the case where X is the (direct!) product of two curves.
Johan Commelin (Albert–Ludwigs-Universität Freiburg): Exponential periods and o-minimality (click for slides)
Let α ∈ ℂ be an exponential period. In this talk I will present on joint work with Philipp Habegger and Annette Huber. We show that the real and imaginary part of α are up to signs volumes of sets definable in the o-minimal structure generated by ℚ, the real exponential function and
sin|[0,1]. This is a weaker analogue of the precise characterisation of ordinary periods as numbers whose real and imaginary part are up to signs volumes of ℚ-semialgebraic sets; and it points to a relation between the theory of periods and o-minimal structures.
Furthermore, we compare the definition of naive exponential periods to the existing definitions of cohomological exponential periods and periods of Nori motives and show that they all lead to the same notion. In particular, naive exponential periods are the same as periods of exponential Nori motives, which justifies that the definition of naive exponential periods singles out the correct set of complex numbers to be called exponential periods.
(Joint work with Philipp Habegger and Annette Huber.)
Laura DeMarco (Harvard University): Variation of Canonical Height in families of maps on ℙ1 (click for slides)
There are many parallels between the (arithmetic) dynamics of morphisms on algebraic varieties and the (arithmetic) geometry of abelian varieties. One example is the variation of canonical height, along a section of a family of elliptic curves or abelian varieties over a base curve, defined over a number field. Key results were proved by Tate, Silverman, Lang, Call, and Green in the 1980s. It is natural to wonder if the same properties carry over to the more general setting of families of maps, equipped with the Call-Silverman canonical height. I will talk about this problem and how it has led to interesting dynamical questions (and some answers), even in the simplest case of maps on ℙ1. This is joint work with Myrto Mavraki.
Bas Edixhoven (University of Leiden): On consequences of Pink’s unlikely intersection conjecture for mixed Shimura varieties (slides available on the speaker’s website)
This concerns conjecture 1.3 in Pink’s 2005 preprint `A common generalization of the conjectures of André-Oort, Manin-Mumford and Mordell-Lang’. In an article with Daniel Bertrand it was shown by us that the claimed consequence, conjecture 6.1, a relative Manin-Mumford conjecture for families of semiabelian varieties, is actually false. The problem lies in the proof of the implication. In this talk, I want to show that, with a little modification, Pink’s proof that conjecture 1.3 implies conjectures 5.1 and 5.2 (the unlikely intersection generalisations of Manin-Mumford and Mordell-Lang for semiabelian varieties) is correct. In order to do this, we give and use a description of the special subvarieties of certain mixed Shimura varieties.
Philipp Habegger (Universität Basel): Uniformity for the Number of Rational Points on a Curve (click for slides)
By Faltings’s Theorem, formerly known as the Mordell Conjecture, a smooth projective curve of genus at least 2 that is defined over a number field K has at most finitely many K-rational points. Votja later gave a new proof. Several authors, including Bombieri, David Philippon, de Diego, Parshin, Rémond, and Vojta, obtained upper bounds for the number of K-rational points. I will discuss joint work with Vesselin Dimitrov and Ziyang Gao where we show that the number of K-rational points is bounded from above as a function of K, the genus, and the rank of the Mordell-Weil group of the curve’s Jacobian. Our work is based on Vojta’s approach. I will describe the new technical tool, an inequality for the Néron-Tate height in a family of abelian varieties.
Daniel Hast (Boston University): Functional transcendence for the unipotent Albanese map (click for slides)
The Chabauty-Kim method is a p-adic analytic method for studying integral (or, in the projective case, rational) points on varieties over the field of rational numbers. The method works by constructing p-adic Coleman functions, built from iterated p-adic integrals, whose zero locus contains the set of integral points of the variety. In the case of a curve, a single such Coleman function suffices to deduce Diophantine finiteness. In higher dimensions, to reach conclusions such as finiteness or non-Zariski-density of integral points, we need a p-adic functional transcendence statement to show that there are no unexpected algebraic relations between the Coleman functions. We will discuss how to prove such a statement via comparison with complex Hodge theory, and how to use this to extend results in the Chabauty-Kim method to higher number fields.
Gareth Jones (University of Manchester): An effective Pila-Wilkie Theorem for pfaffian functions and some diophantine applications
I’ll discuss some joint work, still in progress, with Gal Binyamini, Harry Schmidt, and Margaret Thomas, in which we prove effective forms of the Pila-Wilkie Theorem for various structures involving pfaffian functions. I’ll also discuss applications to effective results on some unlikely intersection problems.
Bruno Klingler (Humboldt Universität, Berlin): Typical and atypical intersections in Hodge theory (click for slides)
After some reminders on variational Hodge theory, I will discuss typical and atypical intersections in this framework. Based on joint work with A.Otwinowska, and with G.Baldi and E.Ullmo (in progress).
Ronnie Nagloo (City University of New York): A differential approach to functional transcendence
In this talk I will discuss recent series of work, joint with D. Blázquez-Sanz, G. Casale, and J. Freitag around proving several Ax-Lindemann-Weierstrass and/or Ax-Schanuel type results for uniformizers of geometric structures. One of the goal will be to highlight the role played, in our work/proof, by the model theory of differentially closed fields. For the talk, I will mainly focus on the case of curves – including Shimura curves and other non arithmetic hyperbolic curves – where the geometric structures can be taken to be the Schwarzian equations satisfied by the uniformizers.
Martin Orr (University of Manchester): PEL type unlikely intersections and quantitative reduction theory (click for slides)
The strategy of Pila and Zannier has been very successful in tackling questions of unlikely intersections. One of the ingredients in this strategy is bounded parametrisation of special subvarieties. In this talk, I will discuss how we can obtain bounded parametrisation for special subvarieties of PEL type in the moduli space of abelian varieties (that is, special subvarieties corresponding to abelian varieties with extra endomorphisms). This leads to a proof of the Zilber-Pink conjecture for intersections with such special subvarieties, conditional on a large Galois orbits hypothesis. The method is based on quantitative results on the reduction theory of arithmetic groups. This is joint work with Christopher Daw.
Tom Scanlon (University of California, Berkeley): What does Zilber-Pink have to do with the (un)decidability of the theory of the rational functions? (click for slides)
Jacob Tsimerman (University of Toronto): Abelian Varieties not Isogenous to Jacobians – in arbitrary characteristic! (click for slides)
Emmanuel Ullmo (Institut des Hautes Études Scientifiques): Special subvarieties of non-arithmetic ball quotients and Hodge Theory (click for slides)
The lecture will report on a joint work with Gregorio Baldi.
Let ℾ ⊂ PU(n,1) be a lattice. Then ℾ acts on the complex unit ball of dimension n and the quotient Sℾ = ℾ \ Bn is a quasi projective variety by a result of Baily-Borel if ℾ is arithmetic and by a result of Mok if ℾ is not arithmetic. We prove that, if Sℾ contains infinitely many maximal complex totally geodesic subvarieties, then ℾ is arithmetic. We first show that Sℾ can be embedded in a period domain for polarised integral variations of Hodge structures and interpret totally geodesic subvarieties as unlikely intersections, then we use some Ax-Schanuel type result in this context due to Bakker and Tsimermann to conclude the proof. A similar result is also obtained by Bader, Fisher, Miller and Stover using super-rigidity techniques. We also prove a version of Ax-Schanuel Conjecture for Sℾ viewed as hermitian locally symmetric space by itself.