Rational and integral points on higher-dimensional varieties -- AIM, May 2014

Open problems

Elliptic curves

• Statistics in families of elliptic curves: rank, size of Selmber groups
• Consequences of the Sato-Tate conjecture
• Lang-Trotter conjecture, on average

M. Bhargava, A positive proportion of plane cubics fail the Hasse principle

M. Bhargava, A positive proportion of elliptic curves over Qhave rank one

M. Bhargava, Pencils of quadrics and the arithmetic of hyperelliptic curves

M. Bhargava, Most hyperelliptic curves over Q have no rational points

A. Cojocaru, D. Grant, N. Jones, One-parameter families of elliptic curves over Q with maximal Galois representations

A. Cojocaru, C. David, Frobenius fields for elliptic curves

Universal torsors

• Explicit equations for universal torsors on Del Pezzo surfaces
• General properties of Cox rings

U. Derenthal, On the Cox ring of Del Pezzo surfaces

U. Derenthal, Y. Tschinkel, Universal torsors over Del Pezzo surfaces and rational points

M. Stillman, D. Testa, M. Velasco, Computation of the universal torsor of a smooth Del Pezzo surface of degree 4, (letter to B. Hassett)

V. Batyrev, O.N. Popov, The Cox ring of a Del Pezzo surface, Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), 85--103, Progr. Math., 226, Birkhäuser Boston, Boston, MA, 2004.

B. Hassett, Y. Tschinkel, Universal torsors and Cox rings, Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), 149--173, Progr. Math., 226, Birkhäuser Boston, Boston, MA, 2004.

E. Peyre, Torseurs universels et methode du cercle, Rational points on algebraic varieties, 221--274, Progr. Math., 199, Birkhäuser, Basel, 2001

P. Salberger, Tamagawa measures on universal torsors and points of bounded height on Fano varieties. Nombre et repartition de points de hauteur bornee (Paris, 1996), Asterisque No. 251 (1998), 91--258

E. Peyre, Terme principal de la fonction zeta des hauteurs et torseurs universels, Nombre et repartition de points de hauteur bornee (Paris, 1996), Asterisque No. 251 (1998), 259--298

Brauer-Manin obstructions

• Effectivity of Brauer-Manin obstructions
• Geometric approaches to the construction of Azumaya algebras on K3 surfaces
• Evaluation of local invariants
• Examples of surfaces with interesting obstructions to Hasse principle and weak approximation
• Kummer surfaces

D. Harari, F. Voloch, The Brauer-Manin obstruction for integral points on curves

B. Hassett, A. Kresch, Y. Tschinkel, Effective computation of Picard groups and Brauer-Manin obstruction of degree two K3 surfaces over number fields

B. Hassett, A. Varilly-Alvarado, Failure of the Hasse principle on general K3 surfaces

J-L. Colliot-Thelene, D. Coray, J.-J. Sansuc, Descente et principe de Hasse pour certaines varietes rationnelles, J. für die reine und ang. Math. 320 (1980), 150-191.

M. Bright, The Brauer–Manin obstruction on a general diagonal quartic surface, Acta Arithmetica 147, (2011), 291-302.

A. Varilly-Alvarado, B. Viray, Failure of the Hasse principle for Enriques surfaces, Advances in Mathematics 226 (2011), 4884-4901.

A. Varilly-Alvarado, B. Viray, Arithmetic of del Pezzo surfaces of degree 4 and vertical Brauer groups, Advances in Mathematics 255 (2014), 153-181

K3 and Enriques surfaces

• Potential density of rational points
• Rational curves
• Picard ranks and specializations of Picard numbers
• Automorphisms
• Moduli spaces
• Applications of the Kuga-Satake construction
• Punctual Hilbert schemes

M. Bhargava, Orbit Parametrizations for K3 Surfaces

D. Harari, A. Skorobogatov, Non-abelian descent and the arithmetic of Enriques surfaces

M. Artebani, A. Laface, D. Testa, On Büchi's K3 surface

B. Hassett, A. Varilly-Alvarado, Failure of the Hasse principle on general K3 surfaces

Counting points of bounded height

D. Loughran, On the number of varieties in a family which contain a rational point

Chr. Frei, M. Pieropan, O-minimality on twisted universal torsors and Manin's conjecture over number fields

U. Derenthal, Chr. Frei, Counting imaginary quadratic points via universal torsors

E. Peyre, Counting points on varieties using universal torsors, Arithmetic of higher-dimensional algebraic varieties, Progress in math 226 (2003), 61-81

Higher dimensions

T. Browning, R. Heath-Brown, Forms in many variables and differing degrees

T. Browning, D. Loughran, Varieties with too many rational points

J. Bruedern, T. Wooley, The Hasse principle for systems of diagonal cubic forms

T. Browning, P. Vishe, Cubic hypersurfaces and a version of the circle method for number fields

D. Harari, J.-L. Colliot-Thélène Approximation forte en famille

D. Harari, Le defaut d’approximation forte pour les groupes algebriques commutatifs

D. Harari, A. Skorobogatov, Descent theory for open varieties