DIOPHANTINE GEOMETRY

CIRM, LUMINY, 21 - 25 MAY, 2018

Abstracts
Francesco AmorosoThéorèmes de spécialisations et équations aux normes
Abstract : [travail en cours avec D. Masser et U. Zannier] We develop a new approach to treat
families of norm form equations. We do not use Baker’s method, applying instead the recent specialisation theorem of [1]. This allows us to prove that, under suitable assumptions, all solutionsof a norm form diophantine equation over an algebraic function field come from specialisationof functional equations. For instance for Thomas cubic equation we get :All diophantine solutions (t, x, y) ∈ Z3 of Thomas cubic equation
X(X − A1 (T )Y)(X − A2 (T )Y) + Y 3 = 1(with A1 , A2 ∈ Z[T ], 0 < deg(A1 ) < deg(A2 )) for t ∈ N (effectively) large enough, are specialisationsof a functional solution (T , X, Y).
Références[1] F. Amoroso, D. Masser and U. Zannier, “Bounded Height Problems and Silverman Specialization Theorem” , ”Bounded Height Problems and Silverman Specialization Theorem”. Duke Math J., 166, no. 13(2017), 2599–2642. Preprint. https ://hal.archives-ouvertes.fr/hal-01200626.

Fabrizio Barroero
Unlikely Intersections in families of abelian varieties and applications
Abstract : The Zilber-Pink Conjecture for families of abelian varieties describes the intersectionsof a subvariety of an abelian scheme with subgroup schemes of appropriate codimension. We
will consider the case of a curve in an abelian scheme (everything defined over the algebraic
numbers). In joint work with Laura Capuano we proved that the intersection of the curve withthe union of all flat subgroup schemes of codimension at least 2 is finite, unless the curve iscontained in a proper subgroup scheme. To have the full conjecture one would have to consider
subgroup schemes that are not contained in flat ones and these arise in fibers with endomorphismring larger than the generic one.
It turns out that one can deal with this problem when the abelian scheme is a fibered powerof an elliptic scheme.
Finally, we will see applications of this kind of results to Diophantine equations in polynomials.

1Daniel Bertrand
Unlikely intersections in a family of semi-abelian surfaces
Abstract : Let E0 be an elliptic curve with CM, and let G be a one-dimensional family of extensionsof E0 by the multiplicative group. Then, G contains a dense set C of special curves, known asRibet curves. In this joint work with H. Schmidt (see ArXiV :1803.04835v1), we consider a nonfibral irreducible curve W in G, and show that W meets C Zariski densely if and only if it is atranslate of a Ribet curve by a multiplicative section. The proof involves the Pila-Masser-Zannier
strategy, a new type of bounded height, and the Mordell-Lang theorem.

Nicolas Billerey
On the modularity of reducible Galois representations
Abstract : In this talk we extend the usual definition of modularity to reducible mod ` Galois
representations. By analogy with the irreducible case, we address questions such as Serre’s modularity conjecture (in various forms) and the level-raising problem. We present partial resultson these questions that illustrate the differences with the classical situation. This is joint workwith Ricardo Menares.

Laura Capuano
Linear relations in families of elliptic curves
Abstract : Let Eλ denote the Legendre elliptic curve of equation Y 2 = X(X − 1)(X − λ). Motivatedby the work of Masser and Zannier on simultaneous torsion, we proved in a joint work withF. Barroero that given n linearly independent points P1 (λ), . . . , Pn (λ) on Eλ with coordinates inQ(λ), there exist at most finitely many λ0 ∈ C such that the specialized points P1 (λ0 ), . . . , Pn (λ0 )satisfy two independent relations with integer coefficients on Eλ0 . This is a special case of conjectures about unlikely intersections in families of abelian varieties stated by different authors. In
my talk, I will give an overview on these problems and present an outline of the proof, whichuses the Pila-Zannier strategy combining o-minimality with other Diophantine ingredients.

Antoine Chambert-LoirGéométrie diophantienne et espaces de Berkovich
Abstract : Les espaces analytiques au sens de Berkovich sont une des variantes des diverses
théories géométriques d’espaces non archimédiens. Leur utilisation est assez pertinente en géométrie
diophantienne, en ce qu’ils permettent d’étudier divers aspects non archimédiens, en particulier
les théorèmes d’équidistribution de points de petite hauteur en géométrie d’Arakelov et leurapplication à la conjecture de Bogomolov sur les corps de fonctions (travaux de Yuan, Gubler
et Yamaki). J’évoquerai aussi une notion de formes différentielles réelles et courants sur ces espaces, développée avec Antoine Ducros, qui permet d’envisager une théorie d’Arakelov nonarchimédienne proche de la situation complexe.

Christopher Daw
Unlikely intersections in Shimura varieties
Abstract : In this talk, we will give our perspective on the Zilber-Pink conjecture for Shimura varieties, following our recent attempts to obtain conditional results in that direction. In particular,
we will discuss the recent result of Mok, Pila, and Tsimerman, which we refer to as the hyperbolic
Ax-Schanuel conjecture, as well as analogues of the structure theorems of Bombieri, Masser, andZannier, and arithmetic problems arising from attempts to adapt the usual Pila-Zannier method.
This is based on joint work with Jinbo Ren.
2Gabriel Dill
Unlikely intersections between isogeny orbits and curves
Abstract : In the spirit of the Mordell-Lang conjecture, we consider the intersection of a curve ina family of abelian varieties with the images of a finite-rank subgroup of a fixed abelian variety
A0 under all isogenies between A0 and some member of the family. After excluding certain
degenerate cases, we can prove that this intersection is finite if everything is defined over thefield of algebraic numbers. This proves a slightly modified version of the so-called André-PinkZannier conjecture over the algebraic numbers in the case of curves. We can even allow translatesof the finite-rank subgroup by abelian subvarieties of controlled dimension if we strengthen thedegeneracy hypotheses suitably. In my talk, I give some historical background and motivation
for this problem and present an outline of the proof, which follows the Pila-Zannier strategy.

Vesselin Dimitrov
On some new results in the Mordell and Bogomolov problems
Abstract :I will report on two recent developments around these classical topics, and discuss theirconnection.
(1) The algebraic sets of a theoretically minimum Arakelov height in an arithmetic variety arestable under taking an intersection. This result turns out to yield a new solution of the Bogomolov
conjecture, by means of a reduction to the torsion case (Manin-Mumford).
(2) A generalization of Bombieri and De Diego’s quantitative estimates on the number of Krational points with large canonical height on a curve C/K of a fixed genus g > 1 and a given
Mordell-Weil rank over K of its Jacobian. The point of our generalization is to introduce a newparameter that quantifies the meaning of ”large height,” so as to make possible the applicationof a new height lower bound of Gao and Habegger to the old problem of uniformly estimating#C(K). I will discuss this result and its relation to the previous, as well as to the work of Davidand Philippon (from their paper Minorations des hauteurs normalisées des sous-variétés des puissances
des courbes elliptiques, IMRS, 2007).
This is a joint work with Ziyang Gao and Philipp Habegger.

Ziyang Gao
Around a height inequality
Abstract : With Philipp Habegger we have recently proved a height inequality for 1-parameterfamilies of abelian varieties. It has several applications, for example it can prove the Geometric
Bogomolov Conjecture over characteristic 0. In this talk, I will start by explaining its application
towards counting the rational points of curves. Then I’ll explain some generalization of thismethod to an arbitrary family of curves. I’ll focus on how the mixed Ax-Schanuel for universal
abelian varieties, extension of a recent result of Mok-Pila-Tsimerman, applies to this problem.
This is work in progress, joint with Vesselin Dimitrov and Philipp Habegger.

Gareth Jones
Counting for functions on the disc
Abstract : I will discuss a counting result for algebraic points of bounded height and bounded
degree on the graph of a function on the disc. Under the assumption of exponential decay along
the interval, and assuming that our function is bounded, we give a power-log-bound. Our boundis also polynomial in the degree of the points. We exploit this to give a dynamical application.
This is all joint work with Gareth Boxall and Harry Schmidt.
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Samuel Le FournQuadratic Chabauty and L-functions
Abstract : The Chabauty method has long stood as the most efficient tool to effectively determine
rational points on algebraic curves. In the context of Serre’s uniformity problem, its realization
for modular curves has been decisive to prove there are no nontrivial rational points on familiesof modular curves whose jacobian admits a “rank zero quotient”. Thanks to Kolyvagin-Logachev
theorems, this can be obtained through nonvanishing of L-functions of modular forms. In thistalk, I will present a work in progress with Dogra and Siksek, proving (again with L-functions)the existence of a “rank=dimension” quotient of jacobians of modular curves from two families,
and how it allows to apply the new “quadratic Chabauty” method devised by Balakrishnan,
Besser, Dogra and Müller.

Davide Lombardo
Endomorphism rings of Jacobians : from theory to practice and back again
Abstract : Over the last 200 years mathematicians have developed a very rich theory of abelianvarieties ; however, certain kinds of explicit calculations have remained out of reach of our computational power until quite recently, when theoretical and technological advances have led toa renewed interest in the computational side of the theory. In this talk I will discuss one of thefundamental algorithmic problems one would like to solve, namely that of determining the endomorphism ring of the Jacobian of an explicitly given curve over a number field ; in particular, I
will describe a method to provably compute the endomorphism ring of a genus 2 Jacobian, anddiscuss its applicability in practical situations. If time permits, I will also describe the problems
one faces when trying to extend the result to Jacobians of curves of higher genus ; this leadsback to interesting theoretical questions, which take the form of a peculiar kind of local-global
principle for endomorphisms.

Álvaro Lozano-Robledo
Recent progress in the classification of torsion subgroups of elliptic curves
Abstract : This talk will be a survey of recent results and methods used in the classification oftorsion subgroups of elliptic curves over finite and infinite extensions of the rationals, and overfunction fields.

David Masser
Avoiding Jacobians
Abstract : It is classical that, for example, there is a simple abelian variety of dimension 4 which
is not the jacobian of any curve of genus 4, and it is not hard to see that there is one definedover the field of all algebraic numbers Q. In 2012 Chai and Oort asked if there is a simple
abelian fourfold, defined over Q, which is not even isogenous to any jacobian. In the same year
Tsimerman answered ”yes”. Recently Zannier and I have done this over the rationals Q, and with
”yes, almost all”. In my talk I will explain ”almost all” the concepts involved.
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Ricardo Menares
p-adic equidistribution of CM points and integral properties of singular moduli
Abstract : We establish equidistribution properties of CM points on the moduli space of p-adic
elliptic curves. We obtain a description of the equidistribution properties of p-adic Hecke orbits
as well. A key ingredient is a p-adic version of Linnik’s theorem on the asymptotic distributionof integer points on spheres. A theorem of Habegger states that there are at most finitely many
singular moduli that are units. As an application of our results on p-adic equidistribution, weshow that for every finite set of primes S, there are at most finitely many singular moduli thatare S-units.
This is a joint work with Sebastian Herrero and Juan Rivera-Letelier.

Lucia Mocz
A new Northcott property for Faltings height
Abstract : The Faltings height is a useful invariant for addressing questions in arithmetic geometry. In his celebrated proof of the Mordell and Shafarevich conjectures, Faltings shows theFaltings height satisfies a certain Northcott property, which allows him to deduce his finitenessstatements. In this work we prove a new Northcott property for the Faltings height. Namely weshow, assuming the Colmez Conjecture and the Artin Conjecture, that there are finitely many
CM abelian varieties of a fixed dimension which have bounded Faltings height. The techniquedeveloped uses new tools from integral p-adic Hodge theory to study the variation of Faltingsheight within an isogeny class of CM abelian varieties. In special cases, we are able to use thesetechniques to moreover develop new Colmez-type formulas for the Faltings height.

Martin Orr
Unlikely intersections with Hecke correspondences
Abstract : I will talk about a theorem on unlikely intersections between a curve and Hecke correspondences in Ag × Ag (where Ag is the moduli space of principally polarised abelian varietiesof dimension g). This is inspired by work of Habegger and Pila for the case g = 1.
The proof has two arithmetic steps which I will discuss : (1) a height bound, giving an effective
version of the Siegel property from the reduction theory of arithmetic groups ; (2) a Galois bound,relying on the Masser-Wustholz isogeny theorem.

Pierre Parent
Stable models for modular curves in prime level
Abstract : We describe stable models for modular curves associated with all maximal subgroups
in prime level, including in particular the new case of non-split Cartan curves.
Joint work with Bas Edixhoven.

5Héctor H. Pastén Vásquez
Shimura curves and bounds for the abc conjecture
Abstract : I will explain some new connections between the abc conjecture and modular forms.
In particular, I will outline a proof of a new unconditional estimate for the abc conjecture, whichlies beyond the existing techniques in this context. The proof involves a number of tools suchas Shimura curves, CM points, analytic number theory, and Arakelov geometry. It also requires
some intermediate results of independent interest, such as bounds for the Manin constant beyond
the semi-stable case. If time permits, I will also explain some results towards Szpiro’s conjecture
over totally real number fields which are compatible with the discriminant term appearing in
Vojta’s conjecture for algebraic points of bounded degree.

Jérôme Poineau
Local and global Berkovich spaces
Abstract : We will first give an introduction to p-adic analytic geometry in the sense of Berkovich. We will explain that it provides a convenient setting to study p-adic analogues of classicalquestions such as the inverse Galois problem over Qp (T ) (after Harbater and Serre-Liu) or theexistence of zeroes of quadratic forms over Qp (T ) (after Harbater-Hartmann-Krashen and Mehmeti).
In the second part, we will use a more general definition of Berkovich spaces that allows
arbitrary Banach rings as base rings, e.g. Z endowed with the usual absolute value. Over thelatter, Berkovich spaces look like fibrations that contain complex analytic spaces as well as p-adic
analytic spaces for every prime number p. We will shortly describe those spaces as well as sometypical rings of functions that appear in this setting (Harbater’s convergent arithmetic powerseries). We will finally explain how Berkovich spaces over Z can be used to parametrize certain
natural families such as elliptic curves over arbitrary local fields, archimedean or not (in whichcase, one recovers only those with bad reduction, i.e. Tate curves).

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