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Register a new userOn superspecial abelian surfaces over finite fields III
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In the paper [On superspecial abelian surfaces over finite fields II. J. Math. Soc. Japan, 72(1):303--331, 2020], Tse-Chung Yang and the first two current authors computed explicitly the number $lvert mathrm{SSp}_2(mathbb{F}_q)rvert$ of isomorphism classes of superspecial abelian surfaces over an arbitrary finite field $mathbb{F}_q$ of even degree over the prime field $mathbb{F}_p$. There it was assumed that certain commutative $mathbb{Z}_p$-orders satisfy an etale condition that excludes the primes $p=2, 3, 5$. We treat these remaining primes in the present paper, where the computations are more involved because of the ramifications. This completes the calculation of $lvert mathrm{SSp}_2(mathbb{F}_q)rvert$ in the even degree case. The odd degree case was previous treated by Tse-Chung Yang and the first two current authors in [On superspecial abelian surfaces over finite fields. Doc. Math., 21:1607--1643, 2016]. Along the proof of our main theorem, we give the classification of lattices over local quaternion Bass orders, which is a new input to our previous works.
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Let $q$ be an odd power of a prime $pin mathbb{N}$, 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 three 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 members of $mathrm{PPSP}(sqrt{q})$; (iii) the refined class number formula with respect to each finite group $mathbf{G}$, i.e. the number of elements of $mathrm{PPSP}(sqrt{q})$ whose automorphism group coincides with $mathbf{G}$. Similar formulas are obtained for other polarized superspecial members of this isogeny class using 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.
In this paper we determine the number of endomorphism rings of superspecial abelian surfaces over a field $mathbb{F}_q$ of odd degree over $mathbb{F}_p$ in the isogeny class corresponding to the Weil $q$-number $pmsqrt{q}$. This extends earlier works of T.-C. Yang and the present authors on the isomorphism classes of these abelian surfaces, and also generalizes the classical formula of Deuring for the number of endomorphism rings of supersingular elliptic curves. Our method is to explore the relationship between the type and class numbers of the quaternion orders concerned. We study the Picard group action of the center of an arbitrary $mathbb{Z}$-order in a totally definite quaternion algebra on the ideal class set of said order, and derive an orbit number formula for this action. This allows us to prove an integrality assertion of Vigneras [Enseign. Math. (2), 1975] as follows. Let $F$ be a totally real field of even degree over $mathbb{Q}$, and $D$ be the (unique up to isomorphism) totally definite quaternion $F$-algebra unramified at all finite places of $F$. Then the quotient $h(D)/h(F)$ of the class numbers is an integer.
Let $[X,lambda]$ be a principally polarized abelian variety over a finite field with commutative endomorphism ring; further suppose that either $X$ is ordinary or the field is prime. Motivated by an equidistribution heuristic, we introduce a factor $ u_v([X,lambda])$ for each place $v$ of $mathbb Q$, and show that the product of these factors essentially computes the size of the isogeny class of $[X,lambda]$. The derivation of this mass formula depends on a formula of Kottwitz and on analysis of measures on the group of symplectic similitudes, and in particular does not rely on a calculation of class numbers.
We study the dynamics of maps arising from the composition of two non-commuting involution on a K3 surface. These maps are a particular example of reversible maps, i.e., maps with a time reversing symmetry. The combinatorics of the cycle distribution of two non-commuting involutions on a finite phase space was studied by Roberts and Vivaldi. We show that the dynamical systems of these K3 surfaces satisfy the hypotheses of their results, providing a description of the cycle distribution of the rational points over finite fields. Furthermore, we extend the involutions to include the case where there are degenerate fibers and prove a description of the cycle distribution in this more general situation.
Let Fq be a finite field with q=8 or q at least 16. Let S be a smooth cubic surface defined over Fq containing at least one rational line. We use a pigeonhole principle to prove that all the rational points on S are generated via tangent and secant operations from a single point.
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