Supersingular abelian varieties over finite fields
Hui Zhu
Let $\rm A$ be a supersingular abelian variety defined over a finite field $\mathbf{k}$. We give an approximate description of the structure of the group ${\rm A}({\mathbf k})$ of $\mathbf k$-rational points of $\rm A$ in terms of the characteristic polynomial $f$ of the Frobenius endomorphism of $\rm A$ relative to $\mathbf k$. Write $f=\prod g_i^{e_i}$ for distinct monic irreducible polynomials $g_i$ and positive integers $e_i$, we show that there is a group homomorphism $\varphi: {\rm A}({\mathbf k}) \longrightarrow \prod ({\mathbf Z}/g_i(1){\mathbf Z})^{e_i}$ that is "almost" an isomorphism in the sense that the size of the kernel and the cokernel of $\varphi$ are bounded by an explicit function of ${\rm dim\ A}$.