On the parity of supersingular Weil polynomials
  Let q be an odd power of a prime p and let A/Fq be a supersingular abelian variety of dimension g. We show that if p>2g+1, then the characteristic polynomial of the q-Frobenius is an even polynomial. This generalizes the well-known result on the vanishing of the trace of the p-Frobenius when p>3 for supersingular elliptic curves over Fp. 
David Ayotte, Antonio Lei, Jean-Christophe Rondy-Turcotte
math.NT