We give an algorithm to compute $$(\ell ,\ell )$$ -isogenies from the Jacobians of genus three hyperelliptic curves non-hyperelliptic over a finite field characteristic different 2 in time $$\tilde{O}(\ell ^3)$$ , where $$\ell $$ is odd prime which coprime characteristic. An important application reduce discrete logarithm problem Jacobian curve corresponding curve.

There exists a weight 1 Katz-Siegel modular form ξ1 ∧ ξ2 ∧ ξ3 for Abelian schemes of relative dimension 3; moreover the following statements hold. d Let k be a field of characteristic different to 2. Let (A, a) be a principally polarised Abelian threefold with indecomposable polarisation a; then if χ18(A, a) 6= 0, we have χ18(A, a) (ξ1 ∧ ξ2 ∧ ξ3) ∈ k∗4 if and only if (A, a) is a k rational Jaco...

As usual, we write Z,Q,Fp,C for the ring of integers, the field of rational numbers, the finite field with p elements and the field of complex numbers respectively. If Z is a smooth algebraic variety over an algebraically closed field then we write Ω(Z) for the space of differentials of the first kind on Z. If Z is an abelian variety then we write End(Z) for its ring of (absolute) endomorphisms...

In [17] the author proved that in characteristic 0 the jacobian J(C) = J(Cf ) of a hyperelliptic curve C = Cf : y 2 = f(x) has only trivial endomorphisms over an algebraic closure Ka of the ground field K if the Galois group Gal(f) of the irreducible polynomial f ∈ K[x] is “very big”. Namely, if n = deg(f) ≥ 5 and Gal(f) is either the symmetric group Sn or the alternating group An then the ring...