Fast and Frobenius: Rational Isogeny Evaluation over Finite Fields

Isogeny Class and Frobenius Root Statistics for Abelian Varieties over Finite Fields

Let I(g, q, N) be the number of isogeny classes of g-dimensional abelian varieties over a finite field Fq having a fixed number N of Fq-rational points. We describe the asymptotic (for q →∞) distribution of I(g, q, N) over possible values of N . We also prove an analogue of the Sato—Tate conjecture for isogeny classes of g-dimensional abelian varieties. 2000 Math. Subj. Class. Primary: 11G25, 1...

On isogeny classes of Edwards curves over finite fields

We count the number of isogeny classes of Edwards curves over finite fields, answering a question recently posed by Rezaeian and Shparlinski. We also show that each isogeny class contains a complete Edwards curve, and that an Edwards curve is isogenous to an original Edwards curve over IFq if and only if its group order is divisible by 8 if q ≡ −1 (mod 4), and 16 if q ≡ 1 (mod 4). Furthermore, ...

Tate’s Isogeny Theorem for Abelian Varieties over Finite Fields

Isogeny classes of Hilbert-Blumenthal abelian varieties over finite fields

This paper gives an explicit formula for the size of the isogeny class of a Hilbert-Blumenthal abelian variety over a finite field. More precisely, let OL be the ring of integers in a totally real field dimension g over Q, let N0 and N be relatively prime square-free integers, and let k be a finite field of characteristic relatively prime to both N0N and disc(L,Q). Finally, let (X/k, ι, α) be a...

Jacobians in isogeny classes of abelian surfaces over finite fields

We give a complete answer to the question of which polynomials occur as the characteristic polynomials of Frobenius for genus-2 curves over finite fields.