Polarized superspecial simple abelian surfaces with real Weil numbers

Abstract

Let $q$ be an odd power of a prime $p\in \mathbb{N}$, and $\mathrm{PPSP}(\sqrt{q})$ be the finite set of isomorphism classes of principally polarized superspecial abelian surfaces in the simple isogeny class over ${\mathbb{F}}_q$ corresponding to the real Weil $q$-numbers $\pm \sqrt{q}$. We produce explicit formulas for $\mathrm{PPSP}(\sqrt{q})$ of the following kinds: (i) the class number formula, i.e.~the cardinality of $\mathrm{PPSP}(\sqrt{q})$; (ii) the type number formula, i.e. the number of endomorphism rings up to isomorphism of the underlying abelian surfaces of $\mathrm{PPSP}(\sqrt{q})$. Similar formulas are obtained for other collections of polarized superspecial members of this isogeny class grouped together according to their polarization modules. We observe several surprising identities involving the arithmetic genus of certain Hilbert modular surface on one side and the class number or type number of $(P,P_{+})$-polarized superspecial abelian surfaces in this isogeny class on the other side.

52 pages. The present paper is reorganized based on the revision of the previous two papers arXiv:2207.12736v1 and arXiv:1909.11858v4. Main results remain the same

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Cited by in corpus (1)

• Hecke eigensystems of automorphic forms (mod $p$) of Hodge type and algebraic modular forms