This article presents an algorithm to compute Hilbert modular forms on the quadratic field Q( √ 5). It also provides a list of all modular abelian varieties defined over Q( √ 5), with prime conductor of norm less than 100 (up to Q-isogeny).

Let X be a p-divisible group with constant Newton polygon over a normal Noetherian scheme S. We prove that there exists an isogeny X → Y such that Y admits a slope filtration. In case S is regular this was proved by N. Katz for dim S = 1 and by T. Zink for dim S ≥ 1. 2000 Mathematics Subject Classification: 14L05, 14F30

The l modular polynomial, φl(x, y), parameterizes pairs of elliptic curves with an isogeny of degree l between them. Modular polynomials provide the defining equations for modular curves, and are useful in many different aspects of computational number theory and cryptography. For example, computations with modular polynomials have been used to speed elliptic curve point-counting algorithms ([B...

We give parametrisation of curves C of genus 2 with a maximal isotropic (Z/3) in J [3], where J is the Jacobian variety of C, and develop the theory required to perform descent via (3, 3)-isogeny. We apply this to several examples, where it can shown that non-reducible Jacobians have nontrivial 3-part of the Tate-Shafarevich group.

Let X be a p-divisible group with constant Newton polygon over a normal noetherian scheme S. We prove that there exists an isogeny to X → Y such that Y admits a slope filtration. In case S is regular this was proved by N.Katz for dim S = 1 and by T.Zink for dim S ≥ 1.

On the twisted Fermat cubic, an elliptic divisibility sequence arises as the sequence of denominators of the multiples of a single rational point. We prove that the number of prime terms in the sequence is uniformly bounded. When the rational point is the image of another rational point under a certain 3-isogeny, all terms beyond the first fail to be primes.

We show that there are no non-zero semi-stable abelian varieties over Q( √ 5) with good reduction outside 3 and we show that the only semi-stable abelian varieties over Q with good reduction outside 15 are, up to isogeny over Q, powers of the Jacobian of the modular curve X0(15).

We study symmetric correspondences with completely decomposable minimal equation on smooth projective curves C . The Jacobian of then decomposes correspondingly. For all positive integers g and ℓ , we give series examples genus n ( − 1 ) + satisfying equations degree such that the has at least 2 isogeny components.

The Oriented Supersingular Isogeny Diffie–Hellman is a post-quantum key exchange scheme recently introduced by Colò and Kohel. It based on the group action of an ideal class quadratic imaginary order subset supersingular elliptic curves, in this sense it can be viewed as generalization popular isogeny CSIDH. From algorithmic standpoint, however, OSIDH quite different from In sense, uses groups ...