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.
- M. Bright, D. Testa, R. van Luijk. Geometry and Arithmetic of Surfaces (in preparation).
- T.D Browning, D.R. Heath-Brown. Integral points on cubic hypersurfaces (Analytic number theory, Cambridge University Press, 2009), 75--90.
- T.D Browning, D.R. Heath-Brown. Rational points on quartic hypersurfaces. J. Reine angew. Math., 629 (2009), 37--88.
- T.D. Browning, D.R. Heath-Brown. Quadratic polynomials represented by norm forms (to appear in Geometric and Functional Analysis).
- N. Bruin, E.V. Flynn, D. Testa. Descent via C_3 x C_3 Isogeny on Abelian Surfaces (in preparation).
- H.M. Bui, D.R. Heath-Brown. A note on the fourth moment of Dirichlet L-functions. Acta Arith., 141 (2010), 335--344.
- E.V. Flynn. Homogeneous Spaces and Degree 4 del Pezzo Surfaces. Manuscripta Math. 129 (2009), 369--380.
- E.V. Flynn, J. Wunderle. Cycles of Covers; Monatsh. Math. 157 (2009), 217--232.
- E.V. Flynn, D. Testa, R. van Luijk. Two-coverings of Jacobians of curves of genus two. Proc. London Math. Soc., 104 (2012), 387--429.
- E.V. Flynn, D. Testa. Finite Weil restriction of curves (in preparation).
- D. Harari, T. Szamuely, with an appendix by E.V. Flynn. Galois Sections for Abelianized Fundamental Groups. Math. Annalen 344 (2009), 779--80.
- D.R. Heath-Brown. Convexity bounds for L-functions. Acta Arith., 136 (2009), 391--395.
- D.R. Heath-Brown. Sums and differences of three k-th powers. J. Number Theory, 129 (2009), 1579--1594.
- D.R. Heath-Brown. Zeros of forms in many variables. Colloquio de Giorgi, 2 (2009), 17--26.
- D.R. Heath-Brown. Zeros of Systems of p-adic Quadratic Forms. Compos. Math., 146 (2010), 271--287.
- D.R. Heath-Brown. Zeros of p-adic forms. Proc. London Math. Soc., 100 (2010), 560--584.
- D.R. Heath-Brown. Pair correlation for fractional parts of $\alpha n^2$. Math. Proc. Camb. Phil. Soc., 148 (2010), 385--407.
- D.R. Heath-Brown. Fractional moments of Dirichlet L-functions. Acta Arith., 145 (2010), 397--409.
- D.R. Heath-Brown. Bounds for the Cubic Weyl Sum. J. Math. Sciences, 171 (2010), 813--823.
- D.R. Heath-Brown. Artin's Conjecture on zeros of $p$-adic Forms. Proceedings of the International Congress of Mathematicians (2010), 249--257.
- D.R. Heath-Brown, D. Testa. Counting Rational Points on Cubic Curves. Science China Math., 53 (2010), 2173--2572.
- D.R. Heath-Brown. A Note on the Chevalley--Warning Theorems. Uspekhi Matematicheskikh Nauk, 66 (2011), 223--232.
- D.R. Heath-Brown, Lillian B. Pierce. Counting rational points on smooth cyclic covers. J. Number Theory, 132 (2012), 1741--1757.
- D.R. Heath-Brown. Powerfree Values of Polynomials (to appear in Quart. J. Math. Oxford).
- D.R. Heath-Brown. Square-free values of $n^2+1$ (to appear in Acta Arith.).
- D.R. Heath-Brown. p-adic Zeros of Systems of Quadratic Forms (in preparation).
- D.R. Heath-Brown. Burgess's bounds for character sums (in preparation).
- A. Laface, D. Testa. Nef and semiample divisors on rational surfaces (to appear in Proceedings of the workshop "Torsors: theory and applications").
- M. Marietti, D. Testa. Conical and Spherical Graphs (to appear in European J. Combinatorics).
- M. Pacini, D. Testa. Recovering plane curves of low degree from their inflection lines and inflection points (to appear in Israel J. Math.).
- C. Salgado, D. Testa, T. Várilly-Alvarado. Survey on the unirationality of del Pezzo surfaces (in preparation).
- M. Stoll, D. Testa. The surface parametrizing cuboids (in preparation).
- D. Testa, A. Várilly-Alvarado, M. Velasco. Cox rings of degree one del Pezzo surfaces. Algebra and Number Theory, 3 (2009), 729--761.
- D. Testa, A. Várilly-Alvarado, M. Velasco. Big rational surfaces. Math. Ann., 351 (2011), 95--107.
- D. Testa. Reducible plane sections of Fermat surfaces (in preparation).
- D. Testa. The surface of cuboids and Siegel modular threefolds (in preparation).

### 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