Algebraic and Analytic Methods in Number Theory

EPSRC grant number EP/F060661/1 

Prof. E. Victor Flynn - Principal Investigator

Prof. Roger Heath-Brown - Co-Investigator

Dr. Damiano Testa - Postdoctoral research associate

Frank Gounelas - DPhil student

Project description

This project brings together three areas of mathematics: Algebraic Number Theory, Analytic Number Theory and Algebraic Geometry. The goal is to use geometric methods to understand and solve problems of an arithmetic nature. The relationship to Algebraic Number Theory is quite direct: algebraic manipulations of formulas typically have geometric interpretations that often make it possible to generalize and clarify the initial arguments. Problems in Analytic Number Theory frequently give rise to questions with a highly geometric flavor. Moreover we are particularly interested in Manin's conjecture on the density of rational points on algebraic varieties, which lies naturally on the border between Algebraic Geometry and Analytic Number Theory.

A large body of research in Algebraic and Analytic Number Theory makes use of elliptic curves. We expect that substantial benefits and innovations will arise from the interplay between the broad areas described above. Examples include recent explicit techniques on Abelian varieties (such as Jacobians of higher genus curves) and analytic questions about number fields, distributions of rational points, and average rank. Our main aim in this project is to investigate questions where tools from these areas can be used. For covering techniques on higher genus curves, we plan investigate the average rank of the Jacobians of the covering curves, to investigate rate of growth of torsion on Abelian varieties, to take existing applications of elliptic curves which have provided analytic results about class numbers, and generalise them to higher genus.

Publications and Preprints

• A. Arnth-Jensen, E.V. Flynn; Nontrivial Sha in the Jacobian of an Infinite Family of Curves of Genus 2; J. Théor. Nombres Bordeaux 21 (2009), 1--13
• T.D. Browning, D.R. Heath-Brown; Quadratic polynomials represented by norm forms, submitted September 2011
• E.V. Flynn; Homogeneous Spaces and Degree 4 del Pezzo Surfaces; Manuscripta Math. 129 (2009), 369--380
• E.V. Flynn, D. Testa, R. van Luijk; Two-coverings of Jacobians of curves of genus two (to appear in Proceedings of the London Mathematical Society)
• E.V. Flynn, J. Wunderle; Cycles of Covers; Monatsh. Math. 157 (2009), 217--232
• D. Harari, T. Szamuely, with an appendix by E.V. Flynn; Galois Sections for Abelianized Fundamental Groups; Math. Annalen 344 (2009), 779--800
• D.R. Heath-Brown; Zeros of Systems of p-adic Quadratic Forms; Compos. Math. 146 (2010), no. 2, 271--287
• D.R. Heath-Brown; Zeros of p-adic forms; Proc. Lond. Math. Soc. (3) 100 (2010), no. 2, 560--584
• D.R. Heath-Brown; A Note on the Chevalley--Warning Theorems, Uspekhi Matematicheskikh Nauk, 66 (2011), 223--232
• D.R. Heath-Brown; Artin's Conjecture for $p$-adic Forms, Proceedings of ICM 2010, to appear
• D.R. Heath-Brown; Powerfree Values of Polynomials, Quart. J. Math. Oxford Ser. (2), to appear
• D.R. Heath-Brown; Square-free values of $n^2+1$, submitted October 2010
• D.R. Heath-Brown, Lillian B. Pierce; Counting rational points on smooth cyclic covers, submitted  September 2011
• D.R. Heath-Brown, D. Testa; Counting Rational Points on Cubic Curves, Science China Math, 53 (2010), 2173-2572
• A. Laface, D. Testa; Nef and semiample divisors on rational surfaces (to appear in Proceedings of the workshop "Torsors: theory and applications", Edinburgh, 10-14 January 2011)
• M. Pacini, D. Testa; Recovering plane curves of low degree  from their inflection lines and inflection points
• D. Testa, A. Várilly-Alvarado, M. Velasco; Big rational surfaces (Math. Annalen, Volume 351, Number 1, 95-107)
• D. Testa, A. Várilly-Alvarado, M. Velasco; Cox rings of degree one del Pezzo surfaces (Algebra and Number Theory, Vol. 3 (2009), No. 7, 729--761)
• M. Bright, D. Testa, R. van Luijk; Geometry and Arithmetic of Surfaces (in preparation)
• M. Stoll, D. Testa; The surface parametrizing cuboids

Attendance at Seminars, Conferences and Workshops related to the grant

• Algebra and Geometry seminar - University of Bristol, November 12, 2008
• Arithmetic of K3 surfaces - Banff, Canada, November 30-December 5, 2008
• Classical Algebraic Geometry Today - MSRI, USA, January 26-30, 2009
• Number Theory seminar - University of Oxford, February 5, 2009
• Algebraic Geometry seminar - University of Cambridge, February 11, 2009
• Number Theory seminar - University of Cambridge, February 17, 2009
• Meeting M. Bright and R. van Luijk at Warwick University, June 13-19, 2009
• Arithmetic and algebraic geometry of higher-dimensional varieties - University of Bristol, August 31-September 4, 2009
• Séminaire "Variétés rationnelles" - Ecole normale supérieure, Paris, November 6, 2009
• Spaces of rational curves and their interactions with diophantine problems - Columbia University, June 1-5, 2009
• Explicit Methods in Number Theory - Mathematisches Forschungsinstitut Oberwolfach, Germany, July 12-18, 2009
• Joint BMC/BAMC Meeting of 2010 - Edinburgh, 6-9 April, 2010
• Arithmetic of Surfaces - Lorentz Center, 25-29 October, 2010
• V Iberoamerican Congress on Geometry, Pucón, Chile, December 10-13, 2010
• Workshop "Torsors: theory and applications", Edinburgh, 10-14 January, 2011
• Moduli spaces and moduli stacks, Columbia University, New York, May 23-27, 2011
• Sino-French Summer Institute 2011 in Arithmetic Geometry, Chern Institute of Mathematics, Tianjin, China, July 10-15, 2011
• Toric Varieties in Cortona, summer graduate workshop, Cortona, Italy, July 18-29, 2011