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Rational points and algebraic cycles

1 July 2012 - 31 December 2012, EPFL, Lausanne, Switzerland

Overview

The study of rational solutions to polynomial equations is ancient, but recently there have been advances, thanks to ideas coming from unexpected directions within mathematics. This semester-long program at the CIB will study rational points on varieties, along with related problems in algebraic geometry over non-algebraically closed fields, in particular those concerning algebraic cycles.

Topics

We expect the following to be investigated during the course of the semester:

• Cohomological obstructions to rational points (e.g., Brauer-Manin obstruction, descent obstruction)
• Arithmetic applications of étale homotopy theory
• Nonabelian analogues of Chabauty's method
• Fundamental groups and anabelian conjectures
• Motivic conjectures
• Arithmetic aspects of Chow groups
• Diophantine subsets of rational points
• Variation of ranks and Selmer and Shafarevich-Tate groups in families of abelian varieties

Activities

All events take place at EPFL in Lausanne unless otherwise specified.

Workshops

• Cohomological methods in arithmetic geometry, September 10-14, 2012 at U. Zürich
• Arithmetic of abelian varieties in families, November 12-16, 2012 (register) (financial support)

Minicourses

• Arithmetic applications of étale homotopy theory, by Yonatan Harpaz and Tomer Schlank, Mondays and Wednesdays from July 2 to August 29, 2012, during 14:00-14:45 and 15:15-16:00 each day in room AAC006 (website) (alternative lecture notes)
• Arithmetic of del Pezzo and K3 surfaces, by Anthony Várilly-Alvarado, Tuesdays from July 10 to August 21, 2012, during 10:00-11:00 and 11:30-12:30 each day in room AAC006 (website) (lecture notes)
• The motivic fundamental group, by Marc Levine, October 1-5, 2012, during 10:00-10:45 and 11:00-11:45 each day in room AAC006 (website) (abstract)

Bernoulli lectures

These are distinguished lectures for a general mathematical audience.

• Gerd Faltings, Tuesday, September 18, 2012, 16:15 in room SV1717A: "Finiteness for rational points on curves: three methods"
• Jean-Louis Colliot-Thélène, Thursday, November 8, 2012, 16:15 in room SV1717A: "Solutions entières des équations diophantiennes" (slides)
• Raman Parimala, Thursday, December 13, 2012, 17:15 in room CM 5: "Invariants of quadratic forms"

RatPAC seminar

Thursdays (unless otherwise specified), 11:15-12:30, in room AAC006

• July 5: Yuri Zarhin (Penn State), Polynomials in one variable and ranks of certain tangent maps
• July 12: Yuri Tschinkel (New York), Introduction to anabelian geometry
• July 19: Felipe Voloch (Austin), Anabelian geometry and descent obstructions on moduli spaces
• July 26: René Pannekoek (Leiden), On p-adic density of rational points on Kummer surfaces
• August 2: Tatiana Bandman (Bar-Ilan University), Word maps on PSL(2,F_q) and SL_2(F_q)
• August 9: Jennifer Park (MIT), A universal first-order formula for the ring of integers inside a number field
• August 16: Kestutis Cesnavicius (MIT), The p-parity conjecture for elliptic curves with a p-isogeny
• August 23: Altan Erdogan (Koc University, Istanbul), A universal formula for the canonical lifting of an ordinary elliptic curve
• August 30: Martin Bright (Beirut), Bad reduction of the Brauer-Manin obstruction
• September 6: Damiano Testa (Warwick), The Büchi K3 surface and its rational points
• September 13: No seminar (due to the workshop in Zürich)
• September 20: Bjorn Poonen (MIT), Modeling the distribution of Selmer groups, Shafarevich-Tate groups, and ranks of elliptic curves
• September 27: Alexei Skorobogatov (London), Pencils of conics and quadrics with many degenerate fibres
• October 11: Philippe Michel (EPFL), Linnik's ergodic method and representations of binary forms by quaternary forms
• October 18: Ronald van Luijk (Leiden), Density of rational points on Del Pezzo surfaces of degree one
• October 25: Anthony Várilly-Alvarado (Rice), On the unirationality of del Pezzo surfaces of degree 2
• November 1: Peter Jossen (Orsay), Periods over complex function fields
• November 8: Bianca Viray (Brown), On singular moduli for arbitrary discriminants
• November 15: No seminar (due to workshop)
• November 22: David Harari (Orsay), The Brauer-Manin obstruction in positive characteristic
• November 29: François Charles (Rennes), Frobenius distribution for pairs of elliptic curves and exceptional isogenies
• December 6: Yonatan Harpaz (Jerusalem), The Brauer-Manin obstruction for singular curves and applications to surfaces
• December 13: Alena Pirutka (Strasbourg): Unramified cohomology and Chow groups
• WEDNESDAY December 19 at 11:15: Maciej Ulas (Kraków): Polynomial solutions of the Diophantine equation a(x^p-y^q)=b(z^r-w^s), with 1/p+1/q+1/r+1/s=1 and related results
• WEDNESDAY December 19 at 14:00: Ekaterina Amerik (Orsay): On the dynamical Brauer-Manin problem
• December 20 at 11:15: Damaris Schindler (Bristol): Norms as products of polynomials
• December 20 at 14:00: Cyril Demarche (Jussieu): Local-global principles for multinorm equations

Movie

July 12, 2012, 14:00-15:15, in room MX F1: Colors of math, a documentary by Ekaterina Eremenko (CIB guest Yuri Tschinkel was involved in the production of this film) (trailer) (Moscow film festival)

Organizing Committee

Hélène Esnault (U. Duisburg-Essen), Andrew Kresch (U. Zürich), Bjorn Poonen (MIT), Alexei Skorobogatov (Imperial College London).

If you are interested in participating, please contact the organizers.