Isogeny orbits in a family of abelian varieties

Isogeny graphs of ordinary abelian varieties

Fix a prime number . Graphs of isogenies of degree a power of are well-understood for elliptic curves, but not for higher-dimensional abelian varieties. We study the case of absolutely simple ordinary abelian varieties over a finite field. We analyse graphs of so-called l-isogenies, resolving that, in arbitrary dimension, their structure is similar, but not identical, to the “volcanoes” occurri...

Isogeny Classes of Abelian Varieties over Function Fields

Let K be a field, K̄ its separable closure, Gal(K) = Gal(K̄/K) the (absolute) Galois group of K. Let X be an abelian variety over K. If n is a positive integer that is not divisible by char(K) then we write Xn for the kernel of multiplication by n in X(Ks). It is well-known [21] that Xn ia a free Z/nZ-module of rank 2dim(X); it is also a Galois submodule in X(K̄). We write K(Xn) for the field of d...

Isogeny Classes of Abelian Varieties with No Principal Polarizations

We provide a simple method of constructing isogeny classes of abelian varieties over certain fields k such that no variety in the isogeny class has a principal polarization. In particular, given a field k, a Galois extension l of k of odd prime degree p, and an elliptic curve E over k that has no complex multiplication over k and that has no k-defined p-isogenies to another elliptic curve, we c...

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...

Tate’s Isogeny Theorem for Abelian Varieties over Finite Fields