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Register a new userPolarized superspecial simple abelian surfaces with real Weil numbers
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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.
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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.
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.
For an algebraic number $alpha$ and $gammain mathbb{R}$, $h(alpha)$ be the (logarithmic) Weil height, and $h_gamma(alpha)=(mathrm{deg}alpha)^gamma h(alpha)$ be the $gamma$-weighted (logarithmic) Weil height of $alpha$. Let $f:overline{mathbb{Q}}to [0,infty)$ be a function on the algebraic numbers $overline{mathbb{Q}}$, and let $Ssubset overline{mathbb{Q}}$. The Northcott number $mathcal{N}_f(S)$ of $S$, with respect to $f$, is the infimum of all $Xgeq 0$ such that ${alpha in S; f(alpha)< X}$ is infinite. This paper studies the set of Northcott numbers $mathcal{N}_f(mathcal{O})$ for subrings of $overline{mathbb{Q}}$ for the house, the Weil height, and the $gamma$-weighted Weil height. We show: (1) Every $tgeq 1$ is the Northcott number of a ring of integers of a field w.r.t. the house. (2) For each $tgeq 0$ there exists a field with Northcott number in $ [t,2t]$ w.r.t. the Weil height $h(cdot)$. (3) For all $0leq gammaleq 1$ and $gamma<gamma$ there exists a field $K$ with $mathcal{N}_{h_{gamma}}(K)=0$ and $mathcal{N}_{h_gamma}(K)=infty$. For $(1)$ we provide examples that satisfy an analogue of Julia Robinons property (JR), examples that satisfy an analogue of Vidaux and Videlas isolation property, and examples that satisfy neither of those. Item $(2)$ concerns a question raised by Vidaux and Videla due to its direct link with decidability theory via the Julia Robinson number. Item (3) is a strong generalisation of the known fact that there are fields that satisfy the Lehmer conjecture but which are not Bogomolov in the sense of Bombieri and Zannier.
It was discovered some years ago that there exist non-integer real numbers $q>1$ for which only one sequence $(c_i)$ of integers $c_i in [0,q)$ satisfies the equality $sum_{i=1}^infty c_iq^{-i}=1$. The set of such univoque numbers has a rich topological structure, and its study revealed a number of unexpected connections with measure theory, fractals, ergodic theory and Diophantine approximation. In this paper we consider for each fixed $q>1$ the set $mathcal{U}_q$ of real numbers $x$ having a unique representation of the form $sum_{i=1}^infty c_iq^{-i}=x$ with integers $c_i$ belonging to $[0,q)$. We carry out a detailed topological study of these sets. For instance, we characterize their closures, and we determine those bases $q$ for which $mathcal{U}_q$ is closed or even a Cantor set. We also study the set $mathcal{U}_q$ consisting of all sequences $(c_i)$ of integers $c_i in [0,q)$ such that $sum_{i=1}^{infty} c_i q^{-i} in mathcal{U}_q$. We determine the numbers $r >1$ for which the map $q mapsto mathcal{U}_q$ (defined on $(1, infty)$) is constant in a neighborhood of $r$ and the numbers $q >1$ for which $mathcal{U}_q$ is a subshift or a subshift of finite type.
We consider an abelian variety defined over a number field. We give conditional bounds for the order of its Tate-Shafarevich group, as well as conditional bounds for the Neron-Tate height of generators of its Mordell-Weil group. The bounds are implied by strong but nowadays classical conjectures, such as the Birch and Swinnerton-Dyer conjecture and the functional equation of the L-series. In particular, we improve and generalise a result by D. Goldfeld and L. Szpiro on the order of the Tate-Shafarevich group, and extends a conjecture of S. Lang on the canonical height of a system of generators of the free part of the Mordell-Weil group. The method is an extension of the algorithm proposed by Yu. Manin for finding a basis for the non-torsion rational points of an elliptic curve defined over the rationals.
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