ARITHABVAR · Arithmetic of Abelian Varieties

The subject of abelian varieties is of central importance in number theory and algebraic geometry. In this proposal, we aim for a deeper understanding of endomorphism rings of abelian varieties (these classify the 'symmetries' of the abelian varieties). We will tackle the following four problems: (i) Give algorithms for computing endomorphism rings of abelian varieties ( specified as Jacobians of curves or as modular abelian varieties). (ii) Give a database of abelian varieties and their endomorphism rings. (iii) Give improved bounds for the rank of the Cartier operator on Kummer covers of the projective line and on Artin-Schreier curves in positive characteristic. (iv) Extend the method of Smart and Siksek (a variant of the Diffie-Hellman protocol in cryptography) to suitable quotients of Jacobians by their automorphism groups. The subject of endomorphism rings of abelian varieties is an area of strength for the US, but not for Europe. The incoming fellow, Dr Arsen Elkin, has made outstanding contributions to this subject and the project aims to redress the imbalance between the US and Europe. Moreover, Warwick has outstanding algebraic geometry and number theory groups, but the vital subjects of abelian varieties and arithmetic geometry in positive characteristic are missing from Warwick. The project will introduce and help establish these subjects to Warwick. The project will link Dr. Elkin and his distinguished American collaborators with researchers at Barcelona, Bayreuth, Bristol, Cambridge, Leiden, Leuven, Oxford, Royal Holloway and Warwick.