Isogeny graphs on superspecial abelian varieties: Eigenvalues and Connection to Bruhat-Tits buildings

To go into further explanation we need to fix some notation and the details are left to the relevant sections. Let $p$ be a prime and $g$ be a positive integer. Fix an algebraically closed field $\overline{\mathbb{F}}_{p}$ of the finite field $\mathbb{F}_{p}={\mathbb{Z}}/p{\mathbb{Z}}$. Let $SS_{g}(p)$ be the set of isomorphism classes of all principally polarized superspecial abelian varieties over $\overline{\mathbb{F}}_{p}$ which are of dimension $g$. We denote such an abelian variety $A$ endowed with the principal polarization $\mathcal{L}$ which is an ample line bundle $\mathcal{L}$ on $A$ with the trivial Euler-Poincaré characteristic. For a principally polarized superspecial abelian variety $(A,\mathcal{L})$ we write $[(A,\mathcal{L})]$ for the class of $(A,\mathcal{L})$ in $SS_{g}(p)$. It is known that $SS_{g}(p)$ is a finite set and more precisely that

for all large enough $p$ and for some positive constants $C_{1}(g)$ and $C_{2}(g)$ depending only on $g$ (it follows from the mass formula (1.2) in p.1419 of [Yu12]).

Fix a representative $(A_{0},\mathcal{L}_{0})$ in a class of $SS_{g}(p)$ and a prime $\ell\neq p$. For each $(A,\mathcal{L})$ in a class of $SS_{g}(p)$, there exists an isogeny $\phi_{A}:A_{0}\longrightarrow A$ of $\ell$-power degree such that ${\rm Ker}(\phi_{A})$ is a maximal totally isotropic subspace of $A[\ell^{n}]$ for some $n\geq 0$ (it follows from Theorem 2.7 in Section 2.5 in this paper and Theorem 34 of [JZ21]). We call $\phi_{A}$ an $\ell$-marking of $(A,\mathcal{L})$ from $(A_{0},\mathcal{L}_{0})$. Two $\ell$-markings of $(A,\mathcal{L})$ from $(A_{0},\mathcal{L}_{0})$ differ by only an element in

where $\dagger$ stands for the Rosati involution associated to $\mathcal{L}_{0}$ (see Proposition 2.3).

We define

where $[(A,\mathcal{L})]\in SS_{g}(p)$ and $\phi_{A}$ is an $\ell$-marking from $(A_{0},\mathcal{L}_{0})$. Here two objects $(A_{1},\mathcal{L}_{1},\phi_{A_{1}})$ and $(A_{2},\mathcal{L}_{2},\phi_{A_{2}})$ are said to be equivalent if there exists an isomorphism $f:(A_{1},\mathcal{L}_{1})\longrightarrow(A_{2},\mathcal{L}_{2})$ such that $f\circ\phi_{A_{1}}$ and $\phi_{A_{2}}$ differ by only an element of $\Gamma(A_{0})^{\dagger}$ in which case we write $(A_{1},\mathcal{L}_{1},\phi_{A_{1}})\sim(A_{2},\mathcal{L}_{2},\phi_{A_{2}})$. We write $[(A,\mathcal{L},\phi_{A})]\in SS_{g}(p,\ell,A_{0},\mathcal{L}_{0})$ for the class of $(A,\mathcal{L},\phi_{A})$ where $(A,\mathcal{L})$ is a principally polarized superspecial abelian variety with an $\ell$-marking $\phi_{A}$ from $(A_{0},\mathcal{L}_{0})$. Let $C$ be a maximal totally isotropic subgroup (or a Lagragian subspace in other words) of $A[\ell]$. Then the quotient $A_{C}=A/C$ yields an object, say $(A_{C},\mathcal{L}_{C})$ in a class in $SS_{g}(p)$ and the natural surjection $f_{C}:A\longrightarrow A_{C}$ is called an $(\ell)^{g}$-isogeny (see Proposition 2.1 and Definition 2.3). Any $(\ell)^{g}$-isogeny between two objects in $SS_{g}(p)$ arises in this way. We remark that the number of maximal totally isotropic subgroups $A[\ell]$ is

for each $A$.

We are now ready to define the ($\ell$-marked) $(\ell)^{g}$-isogeny graph ${\mathcal{G}}^{SS}_{g}(\ell,p)$ for $SS_{g}(p,\ell,A_{0},\mathcal{L}_{0})$ is defined as a directed graph where

• •

  the set of vertices $V({\mathcal{G}}^{SS}_{g}(\ell,p))$ is $SS_{g}(p,\ell,A_{0},\mathcal{L}_{0})$ and

• •

  the set of directed edges between two vertices $v_{1}$ and $v_{2}$ is the set of equivalence classes of $(\ell)^{g}$-isogenies between corresponding principally polarized superspecial abelian varieties commuting with marking isogenies representing $v_{1}$ and $v_{2}$. In other words, if $v_{1}$ and $v_{2}$ correspond to $[(A_{1},\mathcal{L}_{1},\phi_{A_{1}})]$ and $[(A_{2},\mathcal{L}_{2},\phi_{A_{2}})]$ with $\ell$-markings $\phi_{A_{1}}:(A_{0},\mathcal{L}_{0})\longrightarrow(A_{1},\mathcal{L}_{1})$ and $\phi_{A_{2}}:(A_{0},\mathcal{L}_{0})\longrightarrow(A_{2},\mathcal{L}_{2})$ respectively, then an edge from $v_{1}$ to $v_{2}$ is an $(\ell)^{g}$-isogeny $f:(A_{1},\mathcal{L}_{1})\longrightarrow(A_{2},\mathcal{L}_{2})$ such that two markings $f\circ\phi_{A_{1}}$ and $\phi_{A_{2}}$ of $(A_{2},\mathcal{L}_{2})$ from $(A_{0},\mathcal{L}_{0})$ differ by an element of $\Gamma(A_{0})^{\dagger}$.

Our graph is regular since it has $N_{g}(\ell)$-outgoing edges from each vertex, possibly loops and multiple edges from one to another. The associated random walk operator for ${\mathcal{G}}^{SS}_{g}(\ell,p)$ is self-adjoint with respect to a weighted inner product by the inverse of the order of the reduced automorphism group (see Section 5.2). Our first main result is the following:

We defined the normalized Laplacian $\Delta$ on a regular directed multigraph ${\mathcal{G}}$ of degree $d$ by $\Delta=1-(1/d)M$ for the adjacency matrix $M$ of ${\mathcal{G}}$. Note that $\Delta$ has the simple smallest eigenvalue $0$ provided that the graph is strongly connected, i.e., there exists a directed edge path from any vertex to any other vertex. In Theorem 1.1, we actually have an explicit lower bound for $\lambda_{2}$: For every integer $g\geq 2$, for all primes $\ell$ and $p$ with $p\neq\ell$,

(Corollary 5.5 in Section 5.4). In the course of the proof of Theorem 1.1, we relate ${\mathcal{G}}^{SS}_{g}(\ell,p)$ to a finite quotient $\Gamma\backslash\mathcal{S}_{g}$ (see Subsection 3.3) of the special 1-complex $\mathcal{S}_{g}$ defined in terms of the Bruhat-Tits building for $PGSp_{g}({\mathbb{Q}}_{\ell})$ (see Theorem 2.7 and Section 4.5). We then move on $\mathcal{S}_{g}$ to prove the desired property by using Kazhdan’s Property (T) of $PGSp_{g}({\mathbb{Q}}_{\ell})$ for $g\geq 2$.

In [JZ21], Jordan and Zaytman introduced a big isogeny graph $Gr_{g}(\ell,p)$ based on $SS_{g}(p)$. We will show in Section 2 and Section 3 that there exist natural identifications

which induce natural isomorphisms as graphs between three objects:

1. (1)

   $Gr_{g}(\ell,p)$,

2. (2)

   ${\mathcal{G}}^{SS}_{g}(\ell,p)$, and

3. (3)

   the regular directed graph defined by $\Gamma\backslash\mathcal{S}_{g}$.

It follows from this that the adjacency matrices of the above three graphs agree with each other. Therefore, the structure of Jordan-Zaytman’s graph $Gr_{g}(\ell,p)$ is revealed by our main theorem:

This result implies the rapid mixing property of a lazy version of the walk; see [FS21b, Theorem 4.9].

We discuss some theoretical features for each of our work and previous works due to Pizer and Jordan-Zaytman. Instead of using $Gr_{1}(\ell,p)$, Pizer handled the moduli space of supersingular elliptic curves with non-trivial levels to avoid that non-trivial automorphisms happen (see [Piz98], [Piz90]). Therefore, his graphs are regular undirected graphs so that they are Ramanujan by Eichler’s theorem via Jacquet-Langlands theory. However, if $p\equiv 1$ mod 12, then each vertex of $Gr_{1}(\ell,p)$ does not have non-trivial automorphisms other than $-1$.

Jordan-Zaytman’s graphs $Gr_{g}(\ell,p)$ are useful and fit into the computational implementations (cf. [CDS20], [KT20], [FS21a], [FS21b]) as explained in the next subsection. However, it may be hard to directly obtain the uniform estimation of the eigenvalues of the normalized Laplacian. Our graphs do not, unfortunately, well-behave in the computational aspects. However, there is a natural correspondence between $SS_{g}(p,\ell,A_{0},\mathcal{L}_{0})$ and $\mathcal{S}_{g}$ as explained. A point here is that these two objects have markings from a fixed object while $SS_{g}(p)$ does not have it. However, fortunately, there is a natural correspondence between $SS_{g}(p)$ and $SS_{g}(p,\ell,A_{0},\mathcal{L}_{0})$. Then eventually we can relate $SS_{g}(p)$ with $\mathcal{S}_{g}$ via the intermediate object $SS_{g}(p,\ell,A_{0},\mathcal{L}_{0})$.

It seems interesting to consider the moduli space of principal polarized superspecial abelian varieties with a non-trivial level so that the reduced automorphism group of any object is trivial. This will be discussed somewhere else.

This study is largely motivated by a possible approach to cryptographic hash functions from isogeny graphs. Let us begin with a brief review of the notation of hash functions, which are widely used in computer science. For a general reference on hash functions in cryptographic context, see Chapter 5 of [KL14] for example.

A hash function $H$ is an efficiently computable function taking as input a message of any length and outputting a value of fixed length $s$, i.e. $H:\{0,1\}^{*}\to\{0,1\}^{s}$, where $\{0,1\}^{*}=\cup_{n=1}^{\infty}\{0,1\}^{n}$. A standard condition required for hash functions in cryptography is collision resistance; it is computationally hard for any probabilistic polynomial-time algorithm to find a pair of distinct messages $(m_{1},m_{2})$ such that $H(m_{1})=H(m_{2})$. Collision resistant hash functions have numerous applications in cryptography. For example, such functions are used as components of pseudo-random generators, Hash-based Message Authentication Code, digital signatures and so on.

However, despite its importance, it is hard to construct such a function because to design a collision resistant hash function requires suitable mixing and compressing bit strings of any length. As one can see in [Gol11], [CGL09], there is an approach to design hash functions by employing expander graphs on which random walks mix rapidly. Due to Pizer’s work [Piz90], [Piz98], isogeny graphs of supersingular elliptic curves have attracted attention as a tool for realizing a good expansion property. In this subsection, we explain this research direction and state our research question.

In Section 2, we give two interpretations of $SS_{g}(p)$ according to works of Ibukiyama-Katsura-Oort-Serre and Jordan-Zaytman. The former is helpful to compute the cardinality of $SS_{g}(p)$ while the latter is helpful to make the compatibility of Hecke operators at $\ell$ transparent. As mentioned before, this is a crucial step to apply Property (T) (hence, Theorem 5.4) with our family $\{\mathcal{G}^{SS}_{g}(\ell,p)\}_{p\neq\ell}$. In Section 3, we discuss a comparison between the graph ${\mathcal{G}}_{g}^{SS}(\ell,p)$ and that of Jordan-Zaytman $Gr_{g}(\ell,p)$. In Section 4, we study Bruhat-Tits buildings for symplectic groups. Then, in Section 5, the main result is proved in terms of the terminology in the precedent sections. Finally, in Section 6.1 we give a speculation in view of the theory of automorphic forms.

For a set $X$, the cardinality is denoted by $|X|$. Throughout the paper, we use the Landau asymptotic notations: for positive real-valued functions $f(n)$ and $g(n)$ for integers $n$, we denote by $f(n)=o(g(n))$ if $f(n)/g(n)\to 0$ as $n\to\infty$, by $f(n)=O(g(n))$ if there exists a positive constant $C>0$ such that $f(n)\leq Cg(n)$ for all large enough $n$, and $f(n)=\Theta(g(n))$ if we have both $f(n)=O(g(n))$ and $g(n)=O(f(n))$.

Let $n$ be a positive integer and $I_{n}$ the identity matrix of size $n$. Put

We define a functor $G:(Rings)\longrightarrow(Sets)$ from the category of rings to the category of sets by

for each commutative ring $R$ and we call $\nu(M)$ the similitude of $M$. It is well-known that the functor $G$ is represented by a smooth group scheme $GSp_{n}$ over ${\mathbb{Z}}$. The similitude defines a homomorphism $\nu:GSp_{n}\longrightarrow GL_{1}$ as group schemes over ${\mathbb{Z}}$. We define $Sp_{n}:={\rm Ker}(\nu)$ which is called the symplectic group of rank $n$. The similitude splits and in fact it is given by $a\mapsto{\rm diag}(I_{n},aI_{n})$. It follows from this that $G\simeq Sp_{n}\rtimes GL_{1}$. In the sections related to abelian varieties, we put $n=g$ while we keep $n$ in Section 4 through 5.

For any algebraic group $H$ over a field, we denote by $Z_{H}$ the center of $H$.

We would like to thank Professors Yevgeny Zaytman and Bruce Jordan for informing us of issues in the first version of this paper and correcting references; the current version has been revised substantially since then—–we greatly acknowledge them for their comments. We would also like to thank Professor Tsuyoshi Takagi for many helpful discussions and encouragement, and Professor Ken-ichi Kawarabayashi for fostering an ideal environment which made this collaboration possible. Y.A. is supported by JST, ACT-X Grant Number JPMJAX2001, Japan. R.T. is partially supported by JSPS Grant-in-Aid for Scientific Research JP20K03602 and JST, ACT-X Grant Number JPMJAX190J, Japan. T.Y. is partially supported by JSPS KAKENHI Grant Number (B) No.19H01778.

Let $p$ be a prime number and $k$ be an algebraically closed field of a finite field of characteristic $p$. In our purpose, we may put $k=\overline{\mathbb{F}}_{p}$. Let $A$ be an abelian variety over $k$ of dimension $g>0$ and we denote by $\widehat{A}={\rm Pic}^{0}(A)$ the dual abelian variety (cf. Section 9 of [Mil86]). The abelian variety $A$ is said to be superspecial if $A$ is isomorphic to $E^{g}=\overbrace{E\times\cdots\times E}^{g}$ for some supersingular elliptic curve $E$ over $k$ (see Sections 1.6 and 1.7 of [LO98] for another definition in terms of $a$-number). As explained in loc.cit., for any fixed supersingular elliptic curve $E_{0}$ over $k$, every superspecial abelian variety of dimension $g\geq 2$ is isomorphic to $E^{g}_{0}$. (Here the assumption $g\geq 2$ is essential and indeed, this is not true for $g=1$. See also Theorem 4.1 in Chapter V of [Sil09].) Throughout this section, we fix a supersingular elliptic curve $E_{0}$.

Let $A$ be an abelian variety over $k=\overline{\mathbb{F}}_{p}$. A polarization is a class of the Néron-Severi group ${\rm NS}(A):={\rm Pic}(A)/{\rm Pic}^{0}(A)$ which is represented by an ample line bundle on $A$. The definition of polarizations here is different from the usual one but it is equivalent by Remark 13.2 of [Mil86] since $k=\overline{\mathbb{F}}_{p}$.

For each ample line bundle $\mathcal{L}$ we define a homomorphism

where $t_{x}$ stands for the translation by $x$ and we denote by $t^{\ast}_{x}$ its pullback. By APPLICATION 1, p.60 in Section 6 of Chapter II in [Mum70], $\phi_{\mathcal{L}}$ is an isogeny, hence it has a finite kernel since $\mathcal{L}$ is ample. If we write $D$ for an ample divisor on $A$ corresponding to $\mathcal{L}$, namely, $\mathcal{L}\simeq\mathcal{O}(D)$, then by Riemann-Roch theorem in p.150 of loc.cit., we have

where $\chi(\mathcal{L})$ stands for the Euler-Poincare characteristic of $\mathcal{L}$ and $(D^{g})$ is the $g$-fold self-intersection number of $D$. Since $\mathcal{L}$ is ample, $\chi(\mathcal{L})>0$.

We study the difference of two $\ell$-markings. Let us keep the notation in Definition 2.3. By using the principal polarization $\mathcal{L}_{1}$ we define the Rosati-involution $\dagger$ on ${\rm End}(A_{1})$ by

Notice that $\dagger$ is an anti-involution.

In this subsection we refer Section 3.2 of [Ibu20] for the facts and the notation. Let $p$ be a prime number and $n$ be a positive integer. Let $B$ be the definite quaternion algebra ramified only at $p$ and $\infty$ (see Proposition 5.1, p.368 of [Piz80] for an explicit realization). We write $B=\Big{(}\displaystyle\frac{a,b}{{\mathbb{Q}}}\Big{)}={\mathbb{Q}}\cdot 1\oplus{\mathbb{Q}}\cdot i\oplus{\mathbb{Q}}\cdot j\oplus{\mathbb{Q}}\cdot ij$ with $i^{2}=a,\ j^{2}=b,$ and $ij=-ji$. For each $x=x_{0}+x_{1}i+x_{2}j+x_{3}ij$, the conjugation of $x$ is defined by $\overline{x}=x_{0}-x_{1}i-x_{2}j-x_{3}ij$. Then $N(x)=x\overline{x},\ {\rm Tr}(x)=x+\overline{x}$ are called the norm and the trace of $x$ respectively. Let us fix a maximal order $\mathcal{O}$ of $B$ which is also explicitly given in Proposition 5.2 of [Piz80]. The pairing $(\ast,\ast)$ on $B^{n}$ is defined by the following manner:

A submodule $L\subset B^{n}$ is said to be $\mathcal{O}$-lattice if

• •

  $L$ is a ${\mathbb{Z}}$-lattice in $B^{n}$, hence $L\otimes_{\mathbb{Z}}{\mathbb{Q}}=B^{n}$;

• •

  $L$ is a left $\mathcal{O}$-module.

For an $\mathcal{O}$-lattice $L$, we define by $N(L)$ the two sided (fractional) ideal of $B$ generated by $(x,y),\ x,y\in L$. The ideal $N(L)$ is called the norm of $L$.

For a commutative ring $R$, we extend the conjugation on $\mathcal{O}\subset B$ to $\mathcal{O}\otimes_{\mathbb{Z}}R$ by $\overline{x\otimes r}:=\overline{x}\otimes r$ for each $x\in\mathcal{O}$ and $r\in R$. Further, for each $\gamma=(\gamma_{ij})\in M_{n}(\mathcal{O}\otimes_{\mathbb{Z}}R)$ (the set of $n\times n$ matrices over $\mathcal{O}\otimes_{\mathbb{Z}}R$) we define $\overline{\gamma}:=(\overline{\gamma}_{ij})$. We define the algebraic group $G_{n}$ over ${\mathbb{Z}}$ which represents the following functor from the category of rings to the category of sets:

where $I_{n}$ stands for the identity matrix of size $n$. The similitude map $\nu:G_{n}\mapsto GL_{1}$ is defined by $\gamma\mapsto\nu(\gamma)$. Put $G^{1}_{n}:={\rm Ker}(\nu)$ as an algebraic group. The group scheme $G_{n}$ $($resp. $G^{1}_{n})$ over ${\mathbb{Z}}$ is said to be the generalized unitary symplectic group $($unitary symplectic group$)$ and it is symbolically denoted by $G_{n}=GUSp_{n}$ $($resp. $G^{1}_{n}=USp_{n})$. It is easy to see that $G_{n}({\mathbb{R}})$ is compact modulo center and $G^{1}_{n}({\mathbb{R}})$ is, in fact, compact, since $B$ is definite. By definition, $G_{n}$ $($resp. $G^{1}_{n})$ is an inner form of $GSp_{n}$ $($resp. $Sp_{n})$.

Let us keep introducing some notation. For two $\mathcal{O}$-lattices $L_{1},L_{2}$ of $B^{n}$, they are said to be globally equivalent (locally equivalent at a rational prime $p$) if $L_{1}=L_{2}\gamma$ for some $\gamma\in G_{n}({\mathbb{Q}})$ ($L_{1}\otimes_{\mathbb{Z}}{\mathbb{Z}}_{p}=(L_{2}\otimes_{\mathbb{Z}}{\mathbb{Z}}_{p})\gamma$ for some $\gamma\in G_{n}({\mathbb{Q}}_{p})$). We also say $L_{1}$ and $L_{2}$ belong to the same genus if $L_{1}$ is locally equivalent to $L_{2}$ for each rational prime $p$. For each $\mathcal{O}$-lattice $L$, we denote by $\mathcal{L}(L)$ the set of all $\mathcal{O}$-lattices $L_{1}$ such that $L_{1}$ and $L$ belong to the same genus. The set $\mathcal{L}(L)$ is called a genus and we denote by $\mathcal{L}(L)/\sim$ the set of globally equivalent classes of $\mathcal{L}(L)$.

Let $\mathbb{A}_{\mathbb{Q}}$ be the ring of adeles of ${\mathbb{Q}}$ and $\mathbb{A}_{f}$ be the finite part of $\mathbb{A}_{\mathbb{Q}}$. For an $\mathcal{O}$-lattice $L$ and each rational prime $p$, put $K_{p}(L):=\{\gamma_{p}\in G_{n}({\mathbb{Q}}_{p})\ |\ (L\otimes_{\mathbb{Z}}{\mathbb{Z}}_{p})\gamma_{p}=L\otimes_{\mathbb{Z}}{\mathbb{Z}}_{p}\}$ which is an open compact subgroup of $G_{n}({\mathbb{Q}}_{p})$. Then $K(L):=\displaystyle\prod_{p}K_{p}(L)$ makes up an open compact subgroup of $G_{n}(\mathbb{A}_{f})$. For each element $\gamma=(\gamma_{p})_{p}$ of $G_{n}(\mathbb{A}_{\mathbb{Q}})$ and an $\mathcal{O}$-lattice $L$, put

and it is easy to see that $L\gamma$ is also an $\mathcal{O}$-lattice which is locally equivalent to $L$ at each prime $p$. Hence we have

where $G({\mathbb{Q}})$ is diagonally embedded in $G_{n}(\mathbb{A}_{f})$ as $h\mapsto(h)_{p}$. As for the computation of the class number of the principal genus, the case of $n=1$ is due to Eichler [Eic38] (see also Theorem 1.12, p.346 of [Piz80]) and the case of $n=2$ is handled by Hashimoto-Ibukiyama [HI80].

Let us fix a prime $p$ and put $k=\overline{\mathbb{F}}_{p}$. For each positive integer, we denote by $SS_{g}(p)$ the set of all isomorphism classes of principally polarized abelian variety over $k$ of dimension $g$. Henceforth we assume $g\geq 2$. According to [IKO86] we describe $SS_{g}(p)$ in terms of adelic language.

Let $E_{0}$ be a supersingular elliptic curve over $k$. It is well-known that $B:={\rm End}(E_{0})\otimes_{\mathbb{Z}}{\mathbb{Q}}$ is the definite quaternion algebra ramified only at $p$ and $\infty$ with a maximal order $\mathcal{O}={\rm End}(E_{0})$. For each prime $q$, put $\mathcal{O}_{q}:=\mathcal{O}\otimes_{\mathbb{Z}}{\mathbb{Z}}_{q}$. Put $A_{0}=E^{g}_{0}$ and define a divisor on $A_{0}$ by

where $0_{E_{0}}$ stands for the origin of $E_{0}$. By using suitable parallel transformations, it is easy to see that $(D^{g})=g!$. It follows from (2.1) that $\mathcal{L}_{0}:=\mathcal{O}(D)$ is a principal polarization. Let us fix the principally polarized abelian variety $(A_{0},\mathcal{L}_{0})$. Pick another principally polarized abelian variety $(A,\mathcal{L})$. As explained before, $A$ is isomorphic to $A_{0}$ and by pulling back $\mathcal{L}$ to $A_{0}$, there is one to one correspondence between $SS_{g}(p)$ and the set $PP_{g}(A_{0})$ of isomorphism classes of principal polarizations on $A_{0}$. Therefore, we have