Ribet bimodules and principally polarized superspecial abelian varieties with quaternion action

Abstract

In an influential paper [K. Ribet, Bimodules and abelian surfaces, in Algebraic number theory, 359-407, Adv. Stud. Pure Math., Vol. 17, 1989] on the bad reduction of Shimura curves, Ribet studies certain superspecial abelian surfaces over ${\overline{\mathbb{F}}}_p$ with quaternion multiplication by a maximal order $\mathcal{O}$ in an indefinite quaternion $\mathbb{Q}$-algebra ramified at $p$. In particular, he classifies the $p$-divisible groups of such $\mathcal{O}$-abelian surfaces by classifying $({\mathcal{O}}_p,{\mathcal{O}}_p)$-bimodules $L_p$ that are free over ${\mathbb{Z}}_p$ (i.e.bilattices) under an additional admissible assumption. In this paper, we generalize Ribet's result by removing the admissible assumption and producing a complete classification of $({\mathcal{O}}_p,{\mathcal{O}}_p)$-bilattices $L_p$. Equip the right order ${\mathcal{O}}_p$ with the canonical involution, and suppose additionally that the left order ${\mathcal{O}}_p$ is equipped with an orthogonal involution $\ast$. We derive the necessary and sufficient condition for the existence of a perfect quaternion hermitian form $⟨,{⟩}_p:L_p\times L_p\to {\mathcal{O}}_p$ on the right ${\mathcal{O}}_p$-lattice $L_p$ inducing the given involution $\ast$ on the left order ${\mathcal{O}}_p$, and give a complete classification of such self-dual quaternion hermitian $({\mathcal{O}}_p,\ast ,{\mathcal{O}}_p)$-bilattices $(L_p,⟨,{⟩}_p)$. Globally, we apply these classification results to the study of the existence of principal polarizations on superspecial abelian varieties over ${\overline{\mathbb{F}}}_p$ equipped with $\mathcal{O}$-action.

50 pages, comments welcome!