Fix an isogeny class of abelian varieties with commutative endomorphism algebra over a finite field. This isogeny class is determined by a Weil polynomial fA without multiple roots. We give a classification of groups of rational points on varieties from this class in terms of Newton polygons of fA(1− t).

Let E be a Q-isogeny class of elliptic curves defined over Q. The isogeny graph associated to is which has vertex for each element and an edge prime degree that maps one another E, with the recorded as label edge. isogeny-torsion where, in addition, we abstract group structure torsion subgroup Q corresponding curve. main result article determination graphs classes correspond infinitely many j-i...

We estimate the fraction of isogeny classes of abelian varieties over a finite field which have a given characteristic polynomial P (T ) modulo l. As an application we find the proportion of isogeny classes of abelian varieties with a rational point of order l.

Isogeny volcanoes are an interesting structure that have had several recent applications. An isogeny volcano is a connected component of a larger graph called a cordillera. In this paper, we further explore properties of how to compute volcanoes given that we have already computed one of a different degree. This allows us to compute volcanoes of composite degree more efficiently than a direct c...

Understanding the dynamics of nonlinear maps is an important but difficult problem, and there are not many methods available. In this paper, we study the dynamics of a simple function, f(x) = x + x−1, on fields of characteristic two and provide explicit information about structure of it. The main idea is to lift it to the dynamics of an isogeny on an elliptic curve and study the dynamics of the...

We investigate the fields of definition up to isogeny of the abelian varieties known as building blocks. These varieties are defined as the Q-varieties admitting real or quaternionic multiplications of the maximal possible degree allowed by their dimensions (cf. Pyle (2004)). The Shimura-Taniyama conjecture predicts that every such variety is isogenous to a non-CM simple factor of a modular Jac...

The remarkable structure and computationally explicit form of isogeny graphs of elliptic curves over a finite field has made them an important tool for computational number theorists and practitioners of elliptic curve cryptography. This expository paper recounts the theory behind these graphs and examines several recently developed algorithms that realize substantial (often dramatic) performan...

gives an equation for a curve that represents the moduli problem [Γ0(p)] for elliptic curves over a perfect field of characteristic p. This moduli problem associates to such an elliptic curve its finite flat subgroup schemes of rank p. A choice of such a subgroup scheme is equivalent to an isogeny from the elliptic curve with a prescribed kernel. The j-invariants of the source and target curves...

We call a simple abelian variety over Fp super-isolated if its (Fp-rational) isogeny class contains no other varieties. The motivation for considering these varieties comes from concerns about isogeny based attacks on the discrete log problem. We heuristically estimate that the number of super-isolated elliptic curves over Fp with prime order and p ≤ N , is roughly Θ̃( √ N). In contrast, we prov...