ARITHABVAR · Arithmetic of Abelian Varieties

Arithmetic of Abelian Varieties

Project context and objectives Abelian varieties are objects of central importance in number theory and algebraic geometry and are widely used in public-key cryptography and coding theory. The main aim of the project is a deeper understanding of endomorphism rings of abelian varieties, which classify the 'symmetries' of the abelian varieties. There are four objectives: 1. provide algorithms for computing endomorphism rings of abelian varieties; 2. provide a database of abelian varieties and their endomorphism rings; 3. provide improved bounds for the rank of the Cartier operator on Kummer covers of the projective line and on Artin-Schreier curves in a positive characteristic; 4. extend the method of Smart and Siksek to suitable quotients of Jacobians by their automorphism groups. Work performed The researcher has worked on each of the four objectives. The work was performed for objective 1 in collaboration with Profs. Rachel Pries (Colorado State University) and Yuri Zarhin (Pennsylvania State University) and resulted in two papers. For objective 2, the work was carried out in collaboration with the scientist-in-charge (Samir Siksek, Warwick) and Dr Sander Dahmen (Utrecht) and resulted in an online database. The researcher worked on his own for objective 3, which resulted in one publication. The researcher's work on objective 4 was carried out in collaboration with the scientist-in-charge. Main results The project has substantially improved our understanding of the following: - the dependence of the endomorphism ring of a Jacobian of a hyperelliptic curve on the Galois group of the hyperelliptic polynomial; - the interaction between the endomorphism ring of an abelian variety and its torsion subgroup in a positive characteristic; - the ranks of Cartier operators on Kummer covers of the projective line, their a-numbers and the relations to the ramification data and the characteristic of the base field; - methods for computing Belyi covers of the projective line. Project website: http://homepages.warwick.ac.uk/~masjaf/.

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