Journal of the Ramanujan Mathematical Society

Volume 29, Issue 1, March 2014 pp. 31–62.

Simple geometrically split abelian surfaces over finite fields

Authors: Kuo-Ming James Chou and Ernst Kani
Author institution: Department of Mathematics and Statistics, Queen's University Kingston, Ontario, Canada K7L 3N6

Summary: In this paper we study simple abelian surfaces A over a finite field Fq which are not simple (i.e., which are split) over the algebraic closure Fq of Fq. After presenting a classification theorem of such surfaces, we discuss various existence theorems for these surfaces. Some of these results are closely linked to the structure of the Weil restrictions of certain elliptic curves E/Fqn, as is explained in Section 4. In the last section we apply our results to the study of the Jacobians Ju,v of a family of genus 2 curves Cu,v which were first studied by Legendre in 1832 and more recently by Satoh in 2009 in connection with Public Key Cryptography. As a result, we can refine and simplify Satoh's method for constructing cryptographically safe genus 2 curves.

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