Reinventing Rational Points

Institut Henri Poincaré via YouTube

Start learning Write review

Explore this comprehensive thematic period from Institut Henri Poincaré that brings together leading mathematicians to examine rational points on algebraic varieties, one of mathematics' oldest and most fundamental problems. Delve into the intersection of arithmetic geometry, analytic number theory, and logic as experts present cutting-edge research on Diophantine equations and their rational solutions. Discover how modern geometric properties of algebraic varieties provide new perspectives on classical problems, while analytic techniques reveal the "average" behavior of rational and integral points. Examine major conjectures including Mazur's conjectures on topological closures, the Brauer-Manin obstruction for describing rational point closures in adelic spaces, and the Batyrev-Manin conjectures on growth patterns of bounded height rational points. Learn about revolutionary techniques from additive combinatorics and arithmetic invariant theory that have transformed the field, enabling solutions to longstanding problems in arithmetic geometry. Follow detailed presentations on del Pezzo fibrations, motivic Euler products, homogeneous spaces, toric varieties, abelian varieties, and fundamental group connections. Investigate specialized topics including the Poonen-Voloch conjecture, Mordell-Weil rank jumps, Campana's orbifolds, strong approximation methods, and p-adic approaches to rational points on curves through Poonen's four-part lecture series. Witness the emerging synergy between analytic and geometric approaches, where counting problems inspire geometric innovations and geometric conjectures open new analytical research directions.

Syllabus

Obstruction de Brauer-Manin pour les surfaces (...) - Colliot-Thélène - Workshop 1 - CEB T2 2019
Motivic Euler products - Bilu - Workshop 1 - CEB T2 2019
Sections of del Pezzo fibrations over P1 - Tanimoto - Workshop 1 - CEB T2 2019
Complexes of tori and rational points on homogeneous (...) - Harari - Workshop 1 - CEB T2 2019
On a conjecture of Poonen and Voloch I: Probabilistic models(...) - Sawin - Workshop 1 - CEB T2 2019
On a conjecture of Poonen and Voloch II: Lattice point (...) - Le Boudec - Workshop 1 - CEB T2 2019
Mordell Weil rank jumps and the Hilbert property - Salgado - Workshop 1 - CEB T2 2019
Structure of homogeneous spaces and applications to (...) - Demarche - Workshop 1 - CEB T2 2019
The local-global principle for stacky curves - Poonen - Workshop 1 - CEB T2 2019
Campana’s orbifolds, points of bounded height and fibrations - Smeets - Workshop 1 - CEB T2 2019
Local-Global principles for tori over arithmetic surfaces - Hartmann - Workshop 1 - CEB T2 2019
Serre’s problem for diagonal conics - Sofos - Workshop 1 - CEB T2 2019
Counting rational points of cubic hypersurfaces - Salberger - Workshop 1 - CEB T2 2019
Approximation fine pour les points rationnels sur les (...) - Wittenberg - Workshop 1 - CEB T2 2019
A quantitative version of the fibration method - Loughran - Workshop 1 - CEB T2 2019
Strong approximation for a family of norm varieties - Xu - Workshop 1 - CEB T2 2019
Integral points of bounded height on toric varieties - Chambert-Loir - Workshop 1 - CEB T2 2019
Rational lines on cubic hypersurfaces - Brandes - Workshop 1 - CEB T2 2019
Affine and mod-affine varieties in arithmetic geometry. - Charles - Workshop 2 - CEB T2 2019
Perfect points on abelian varieties in positive characteristic. - Rössler - Workshop 2 - CEB T2 2019
Families of abelian varieties with a common isogeny factor. - Cadoret - Workshop 2 - CEB T2 2019
Homogeneous spaces, algebraic K-theory and cohomological(...) - Izquierdo - Workshop 2 - CEB T2 2019
«Special» manifolds: rational points and entire curves. - Campana - Workshop 2 - CEB T2 2019
Rational points and fundamental groups. - Ellenberg - Workshop 2 - CEB T2 2019
Mazur's program B. - Zureick-Brown - Workshop 2 - CEB T2 2019
2^k-Selmer groups and Goldfeld's conjecture. - Smith - Workshop 2 - CEB T2 2019
Quasi-hyperbolicity via explicit symmetric (...) - Várilly-Alvarado - Workshop 2 - CEB T2 2019
Sous-groupe de Brauer invariant et application - Cao - Workshop 2 - CEB T2 2019
Descent obstructions on constant curves over global (...) - Creutz - Workshop 2 - CEB T2 2019
Tamagawa Numbers of Linear Algebraic Groups over (...) - Rosengarten - Workshop 2 - CEB T2 2019
Minimization and reduction of plane curves - Stoll - Workshop 2 - CEB T2 2019
Squares represented by a product of three ternary (...) - Harpaz - Workshop 2 - CEB T2 2019
Persistence of the Brauer-Manin obstruction under field extension - Viray - Workshop 2 - CEB T2 2019
Low degree points on curves. - Vogt - Workshop 2 - CEB T2 2019
Purity for the Brauer group of singular schemes - Česnavičius - Workshop 2 - CEB T2 2019
Endomorphisms of certain superelliptic jacobians and l-adic (..) - Zarhin - Workshop 2 - CEB T2 2019
Resolution in characteristic 0 using weighted blowing up. - Abramovich - Workshop 2 - CEB T2 2019
[T2 2019] Les mathématiques au fil du temps ; Les points rationnels - Schappacher/Peyre
p-adic approaches to rational points on curves - Poonen - Lecture 1/4 - CEB T2 2019
p-adic approaches to rational points on curves - Poonen - Lecture 2/4 - CEB T2 2019
p-adic approaches to rational points on curves - Poonen - Lecture 3/4 - CEB T2 2019
p-adic approaches to rational points on curves - Poonen - Lecture 4/4 - CEB T2 2019
Presentation of the T2 2019 "Reinventing Rational Points"