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 étale 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.
We investigate special structures due to automorphisms in isogeny graphs of principally polarized abelian varieties, and abelian surfaces in particular. We give theoretical and experimental results on the spectral and statistical properties of (2, 2)-isogeny graphs of superspecial abelian surfaces, including stationary distributions for random walks, bounds on eigenvalues and diameters, and a proof of the connectivity of the Jacobian subgraph of the (2, 2)-isogeny graph. Our results improve our understanding of the performance and security of some recently-proposed cryptosystems, and are also …
We define three different isogeny graphs of principally polarized superspecial abelian varieties, prove foundational results on them, and explain their role in number theory and geometry. This is background to joint work with Yevgeny Zaytman on properties of these isogeny graphs for dimension $g>1$, especially the result that these graphs are connected, but not in general Ramanujan.
We consider the structures formed by isogenies of abelian varieties with polarizations that are not necessarily principal, specifically with the $[\ell]$-polarizations we have previously defined. Our primary interest is in superspecial abelian varieties, where the isogenies are related to quaternionic hermitian forms. We first consider isogeny graphs. We show that these $[\ell]$-isogeny graphs are a generalized Brandt graph and construct them entirely in terms of definite quaternion algebras. We prove that they are connected and give examples to show that the regular graphs obtained are someti…
We study for each fixed integer $g \ge 2$, for all primes $\ell$ and $p$ with $\ell \neq p$, finite regular directed graphs associated with the set of equivalence classes of $\ell$-marked principally polarized superspecial abelian varieties of dimension $g$ in characteristic $p$, and show that the adjacency matrices have real eigenvalues with spectral gaps independent of $p$. This implies a rapid mixing property of natural random walks on the family of isogeny graphs beyond the elliptic curve case and suggests a potential construction of the Charles-Goren-Lauter type cryptographic hash functio…
On counting certain abelian varieties over finite fields
本文从分类学的角度对阿贝尔簇进行了研究。在一系列论文中，目前的作者和T.- C. Yang得到了有限域上超特殊交换曲面个数的显式公式。本文给出了实Weil数为$\sqrt{q}$的简单交换曲面的等距类的大小的一个显式公式。这为我们将超特殊阿贝尔曲面的显式计算推广到超奇异阿贝尔曲面的显式计算奠定了基础。第二部分是在一般情况下引入具有附加结构的阿贝尔簇的亏格和理想复形的概念。目的是将余文中关于具有附加结构的阿贝尔簇的结果推广到相似类上，为进一步研究几何定义质量与算术定义质量之间的联系建立更多的结果.
There is a well known theorem by Deuring which gives a criterion for when the reduction of an elliptic curve with complex multiplication (CM) by the ring of integers of an imaginary quadratic field has ordinary or supersingular reduction. We generalise this and a similar theorem by Goren in dimension 2, and classify the $p$-torsion group scheme of the reduction of 3-dimensional abelian varieties with CM by the ring of integers of a cyclic sextic CM field. We also prove a theorem in arbitrary dimension $g$ that distinguishes ordinary and superspecial reduction for abelian varieties with CM by a…
Ekedahl-Oort strata with a superspecial factor in unitary Shimura varieties
Washington College（华盛顿学院）；Carleton College（卡尔顿学院）；Brooklyn College and The Graduate Center, City University of New York（纽约市立大学布鲁克林学院及研究生院）；Korteweg-de Vries Institute for Mathematics, University of Amsterdam（阿姆斯特丹大学科特维格-德弗里斯数学研究所）；University of Arizona（亚利桑那大学）
Let $\ell$ be a rational prime. Previously, abelian $\ell$-towers of multigraphs were introduced which are analogous to $\Z_{\ell}$-extensions of number fields. It was shown that for towers of bouquets, the growth of the $\ell$-part of the number of spanning trees behaves in a predictable manner (analogous to a well-known theorem of Iwasawa for $\Z_{\ell}$-extensions of number fields). In this paper, we extend this result to abelian $\ell$-towers over an arbitrary connected multigraph (not necessarily simple and not necessarily regular). In order to carry this out, we employ integer-valued pol…
Let $A$ be an abelian variety defined over a number field $k$, let $p$ be an odd prime number and let $F/k$ be a cyclic extension of $p$-power degree. Under not-too-stringent hypotheses we give an interpretation of the $p$-component of the relevant case of the equivariant Tamagawa number conjecture in terms of integral congruence relations involving the evaluation on appropriate points of $A$ of the ${\rm Gal}(F/k)$-valued height pairing of Mazur and Tate. We then discuss the numerical computation of this pairing, and in particular obtain the first numerical verifications of this conjecture in…
Let $π$ be a cuspidal automorphic representation of $\operatorname{GL}_2$ over a totally real number field $F$. Let $K$ be a totally imaginary quadratic extension of $F$. We estimate central values of the $\operatorname{GL}_2 \times \operatorname{GL}_2$ Rankin-Selberg $L$-functions associated to $π$ times representations induced from Hecke characters of $K$ which are ramified only at a given prime ideal $\mathfrak{p}$ of $F$. More specifically, we use spectral decompositions of shifted convolution sums and relations to Fourier-Whittaker coefficients of genuine and non-genuine metaplectic forms…
