Simple Geometrically Split Abelian Surfaces Over Finite Fields

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 F_q of F_q. 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/F_qⁿ, as is explained in Section 4. In the last section we apply our results to the study of the Jacobians J_u,v of a family of genus 2 curves C_u,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.