Rational Points on Higher-Dimensional Varieties

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Rational Points on Higher-Dimensional Varieties

 25 - 29 Apr 2022

ICMS, Bayes Centre, Edinburgh


  • Daniel Loughran, University of Bath
  • Rachel Newton, King's College London
  • Efthymios Sofos, University of Glasgow


The topic of rational points on varieties over the rational numbers is the modern perspective on the theory of Diophantine equations.

There is a good (partially conjectural) understanding now of the situation for algebraic curves. The proof of the Mordell conjecture for curves of genus at least 2 by Faltings is one of the crowning achievements in the area, and much of the work on elliptic curves is driven by the Birch-Swinnerton-Dyer conjecture. Recent highlights include the work of Bhargava and his collaborators on average ranks of elliptic curves. The situation in higher dimensions is much murkier however.

The aim of the meeting was to bring together leading experts and early career researchers to make progress on understanding rational points on surfaces and higher dimensional varieties. Traditionally there have been two separate communities working in the area, using tools from analytic number theory and algebraic geometry, respectively. Spectacular progress has been made in recent times by managing to bridge these communities, with a particular highlight being applications of Green-Tao-Ziegler's work on primes in arithmetic progressions to the fibration method. The emphasis in the meeting was on building upon this bridge and further inspiring collaboration between the analytic and geometric communities.

Specific topics covered included the following:

  1. Schinzel's Hypothesis with probability 1

  2. Campana points

  3. Purity of strong approximation

  4. Rational points in families

  5. Brauer--Manin obstruction

Recorded Talks


Registration and Refreshments
Welcome and Housekeeping Efthymios Sofos (University of Glasgow), Simon Kershaw (ICMS)
Jennifer Berg, Bucknell University Rational points on constant subvarieties of abelian varieties
Lunch Break at ICMS Lunch served at ICMS
Ashvin Swaminathan, Princeton University On the distribution of 2-Selmer groups of hyperelliptic Jacobians
Refreshment Break
Tim Santens, KU Leuven Quartic diagonal surfaces with a Brauer-Manin obstruction
Florian Wilsch, Institute of Science and Technology Austria Integral points of bounded height on a certain toric variety
Welcome Reception Welcome Reception at ICMS until 18.30
Marta Pieropan, Utrecht University Campana points on Fano varieties
Refreshment Break
Sam Streeter, University of Bristol Semi-integral points and quadrics
Rosa Winter, King's College London Concurrent exceptional curves and torsion points on del Pezzo surfaces of degree one
Lunch Break at ICMS Lunch served at ICMS
Margherita Pagano, Leiden university Brauer-Manin obstruction at primes with good reduction
Refreshment Break
Alec Shute, Institute of Science and Technology Austria Polynomials represented by norm forms via the beta sieve
End of Day
Leonhard Hochfilzer, Goettingen University Diagonal cubic hypersurfaces over F_q(t).
Julian Demeio, Max Planck Institute, Bonn On the distribution of rational points on ramified covers of abelian varieties
Refreshment Break
Tim Browning, Institute of Science and Technology Austria Manin-Peyre for some quadric bundle threefolds
Lunch Break at ICMS Packed Lunch available for participants to either eat at ICMS, or take with them
End of day Free afternoon for participants
Margaret Bilu, Université de Bordeaux / IST Austria A motivic circle method
Refreshment Break
Alexei Skorobogatov, Imperial College London Reduction of Kummer surfaces modulo 2
Lunch Break at ICMS Lunch served at ICMS
Julian Lyczak, Institute of Science and Technology Austria The Loughran-Smeets conjecture over a more general base
Refreshment Break
Nick Rome, University of Michigan The number of diagonal planar conics with a point
Kevin Destagnol, Université Paris Saclay (Orsay) The Loughran--Smeets conjecture for Châtelet type varieties
End of talks for the day
Workshop Dinner Workshop Dinner at ICMS until 21.00
Jean-Louis Colliot-Thélène, Université Paris-Saclay, Centre d'Orsay Arithmetic of intersections of two quadrics, after nearly 40 years
Refreshment Break
Roger Heath-Brown, University of Oxford Rational points on DP5's with a conic fibration
Closing Remarks
Lunch at ICMS Packed Lunch available for participants to either eat at ICMS, or take with them
Workshop Ends