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Séance spéciale du colloque "Non-Archimedean and Tropical Geometry"

Pour plus d'informations sur le colloque "Non-Archimedean and Tropical Geometry"

École polytechnique  – Salle de conférences du Centre de Mathématiques Laurent Schwartz

 

14h00-15h00 : Vlerë Mehmeti (Université Paris-Saclay)

"A Hasse Principle on Berkovich Analytic Curves"

Abstract: Patching techniques, under various forms and inspired from results in complex analysis, have in the past been used as an approach to the inverse Galois problem. Recently, these techniques have become a very important tool in the study of local-global principles. I will explain how patching can be adapted to Berkovich analytic curves. Working in this setting, one can then obtain several local-global principles, all of which are applicable to quadratic forms.

 

15h00-16h00 : Erwan Brugallé (Université de Nantes)

"Euler characteristic and signature of real semi-stable degenerations"

Abstract: It is interesting to compare the Euler characteristic of the real part of a real algebraic variety to the signature of its complex part. For example, a theorem by Itenberg and Bertrand states that both quantities are equal for "primitive T-Hypersurfaces". After defining these latter, I will give a motivic proof of this theorem via the motivic nearby fiber of a real semi-stable degeneration. This proof extends in particular the original statement by Itenberg and Bertrand to non-singular tropical varieties.

 

16h00-16h30 : Pause café

 

16h30-17h30 : Jérôme Poineau (Université de Caen Normandie)

 "Analytic dynamics over Z and torsion points of elliptic curves"

Abstract: Let Y be a Berkovich space over Z. Recall that such a space naturally contains non-Archimedean parts (such as usual p-adic Berkovich spaces) and Archimedean parts (such as complex analytic spaces). Denote by X the relative projective line over Y. For each point y in Y, let µ_y be a measure defined on the fiber X_y (which is an analytic projective line over the complete residue field associated to y). Inspired by the work of Favre on endomorphisms on hybrid Berkovich spaces, we prove general continuity results for families of measures of the form (µ_y)_{y∈Y} coming from dynamical systems on X. Following a strategy by DeMarco-Krieger-Ye, we then deduce new cases of a conjecture of Bogomolov-Fu-Tschinkel on uniform bounds on the number of common images on ℙ^1 of torsion points of two elliptic curves.