Let's fix a finite field F and consider abelian varieties of dimension g over F. Can we say how many isogeny classes there are? Is it even clear that there's more than one isogeny class? For g=1, and some fixed F with characteristic not 2 or 3, we could probably just write down all the Weierstrass equations and count isogeny classes by brute force, but is there a cleaner way to do it?



Can we count isogeny classes of abelian varieties?
Let q be the order of your finite field. Then the category of abelian varieties over $\mathbb{F}\_q$ up to isogeny is semisimple - any object is isogenous to a product of simple ones in an essentially unique way, so this reduces your question to one about simple objects.


For simple abelian varieties over $\mathbb{F}\_q$, there is the Tate-Honda classification which states that the isogeny classes are in bijective correspondence with Weil $q$-integers (algebraic numbers that have absolute value $q^{1/2}$ in all complex embeddings) up to Galois conjugacy.


I learned this from Milne's "[Points on Shimura varieties mod $p$](http://www.jmilne.org/math/articles/1979a.pdf)" (one of the articles from the Corvallis proceedings that Kevin mentioned), which has a nice and fairly elementary discussion in section 5.


