Hyperelliptic curves over F_2 of every 2-rank without extra automorphisms

Abstract

We prove that for any pair of integers 0\leq r\leq g such that g\geq 3 or r>0, there exists a (hyper)elliptic curve C over F_2 of genus g and 2-rank r whose automorphism group consists of only identity and the (hyper)elliptic involution. As an application, we prove the existence of principally polarized abelian varieties (A,\lambda) over F_2 of dimension g and 2-rank r such that \Aut(A,\lambda)={\pm 1}.