论文标题
具有真实Weil数字
Polarized superspecial simple abelian surfaces with real Weil numbers
论文作者
论文摘要
令$ q $是\ mathbb {n} $的prime $ p \的奇怪力量，而$ \ mathrm {ppsp}（\ sqrt {q}）$是一组有限的同构类别，这些类别是主要极化的superspecial superspecial superspecial superspecial superspecial superspecial superspecial superspecial supery class y Mathb ymath $ _q $ _q $ _q $ _q p} $ q $ -Numbers $ \ pm \ sqrt {q} $。我们为$ \ mathrm {ppsp}（\ sqrt {q}）$生成了明确的公式：（i）$ \ mathrm {ppsp}（\ sqrt {q}）$ \ mathrm {ppsp}的基数，即〜 （ii）类型编号公式，即内态数的数量，直到$ \ mathrm {ppsp}（\ sqrt {q}）$的基础abelian表面的同构。对于该等级类别的偏振分类成员的其他集合，也获得了类似的公式，根据它们的极化模块将其分组在一起。 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.
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.
