- English

# 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.