2.1 Superspecial abelian varieties

Let p be a prime number and $k=\overline {\mathbb {F}}_p$ . Let A be an abelian variety over k of dimension $g>0$ , and we denote by $\widehat {A}=\mathrm {Pic}^0(A)$ the dual abelian variety (cf. Section 9 of [Reference MilneMil86]). 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 [Reference Li and OortLO98] 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 $g\ge 2$ is isomorphic to $E^g_0$ . (Here, the assumption $g\ge 2$ is essential, and indeed, this is not true for $g=1$ . See also Theorem 4.1 in Chapter V of [Reference SilvermanSil09].) Throughout this section, we fix a supersingular elliptic curve $E_0$ .

2.2 Principal polarizations

Let A be an abelian variety over $k=\overline {\mathbb {F}}_p$ . A polarization is a class of the Néron–Severi group $\mathrm {NS}(A):=\mathrm {Pic}(A)/\mathrm {Pic}^0(A)$ 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 [Reference MilneMil86] since $k=\overline {\mathbb {F}}_p$ .

For each ample line bundle $\mathcal {L}$ , we define an isogeny $\phi _{\mathcal {L}}:A\longrightarrow \widehat {A},\ x\mapsto t^\ast _x(\mathcal {L})\otimes \mathcal {L}^{-1}$ where $t_x$ stands for the translation by x and we denote by $t^\ast _x$ its pullback.

We study the difference of two $\ell $ -markings. Let us keep the notation in Definition 2.1. By using the principal polarization $\mathcal {L}_1$ , we define the Rosati-involution $\dagger $ on $\mathrm {End}(A_1)$ by

(2.2) $$ \begin{align} f^\dagger=\phi^{-1}_{\mathcal{L}_1}\circ \widehat{f}\circ \phi_{\mathcal{L}_1},\ f\in \mathrm{End}(A_1). \end{align} $$

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

Proof For f, let $\widetilde {f}:A_2\longrightarrow A_1$ be an $(\ell )^g$ -isogeny in Proposition 2.2. Put $\psi _1=\widetilde {f}\circ h\in \mathrm {End}(A_1)$ . Then we have, by definition,

$$ \begin{align*}\psi_1\circ \psi^{\dagger}_1=(\widetilde{f}\circ h) \circ (\phi^{-1}_{\mathcal{L}_1}\circ \widehat{h}\circ \widehat{\widetilde{f}} \circ \phi_{\mathcal{L}_1}).\end{align*} $$

By [Reference Jordan and ZaytmanJZ21, Theorem 34] and Definition 2.1 that $\widehat {\widetilde {f}}\circ \phi _{\mathcal {L}_1}\circ \widetilde {f}= \phi _{\mathcal {L}^{\otimes \ell ^{m}}_2}=\ell ^m \phi _{\mathcal {L}_2}$ and $\widehat {h}\circ \phi _{\mathcal {L}_2}\circ h=\phi _{\mathcal {L}^{\otimes \ell ^{m'}}_1}=\ell ^{m'} \phi _{\mathcal {L}_1}$ for some integers $m',m\ge 0$ . This yields

$$ \begin{align*}\psi_1\circ \psi^{\dagger}_1=\ell^m \widetilde{f}\circ \phi^{-1}_{\mathcal{L}_2} \circ \widehat{\widetilde{f}} \circ \phi_{\mathcal{L}_1}=\ell^{m+m'}\mathrm{id}_{A_1}.\end{align*} $$

Further, $f\circ \psi =(f\circ \widetilde {f})\circ h=\ell ^m h$ . Therefore, we may put $\psi =\ell ^{-m}\psi _1$ as desired.

2.3 Class number of the principal genus for quaternion Hermitian lattices

In this subsection, we refer Section 3.2 of [Reference IbukiyamaIbu20] for the facts and the notation. Let p be a prime number, and let n be a positive integer. Let B be the definite quaternion algebra ramified only at p and $\infty $ . Let us fix a maximal order $\mathcal {O}$ of B.

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:

$$ \begin{align*}\underline{G}_n:(Rings)\longrightarrow (Sets),\ R\mapsto\underline{G}_n(R)&:= \{\gamma\in M_n(\mathcal{O}\otimes_{\mathbb Z} R)\ |\ \gamma\cdot {}^t\overline{\gamma}\\ &=\nu(\gamma)I_n\ \text{for some }\nu(\gamma)\in R^\times \},\end{align*} $$

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 $G^1_n:=\mathrm {Ker}(\nu )$ 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 $\mathbb {A}_{\mathbb Q}$ be the ring of adeles of ${\mathbb Q}$ , and let $\mathbb {A}_f$ be the finite part of $\mathbb {A}_{\mathbb Q}$ . For an $\mathcal {O}$ -lattice L of $B^n$ 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 $G_n(\mathbb {A}_f)$ .

2.4 Ibukiyama–Katsura–Oort–Serre’s result in terms of adelic language

Let us fix a prime p and put $k=\overline {\mathbb {F}}_p$ . 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 $g\ge 2$ . According to [Reference Ibukiyama, Katsura and OortIKO86], we describe $SS_g(p)$ in terms of adelic language. Let us first recall the main result in [Reference Ibukiyama, Katsura and OortIKO86].

We denote by $Z_{G_g}\simeq GL_1$ the center of $G_g=GUSp_g$ . Recall the open compact subgroup $K(\mathcal {O}^g)=\displaystyle \prod _pK_p(\mathcal {O}^g)$ . For each prime $\ell \neq p$ , put $K(\mathcal {O}^g)^{(\ell )}=\displaystyle \prod _{p\neq \ell }K_p(\mathcal {O}^g)$ . Clearly, $K(\mathcal {O}^g)=K(\mathcal {O}^g)^{(\ell )}\times G_g({\mathbb Z}_\ell )$ . We identify $B_\ell =B\otimes _{{\mathbb Q}}{\mathbb Q}_\ell $ (resp. $\mathcal {O}_\ell =\mathcal {O}\otimes _{{\mathbb Z}}{\mathbb Z}_\ell $ ) with $M_2({\mathbb Q}_\ell )$ (resp. $M_2({\mathbb Z}_\ell )$ ). Under this identification, we have $G_g(R)=GSp_g(R)$ for $R={\mathbb Z}_\ell $ or ${\mathbb Q}_\ell $ (cf. Lemma 4 of [Reference GhitzaGhi04]). Therefore, for any subring M of ${\mathbb Q}_\ell $ , $G_g(M)$ is naturally identified with a subgroup of $G_g({\mathbb Q}_\ell )=GSp_g({\mathbb Q}_\ell )$ under the inclusion $M\subset {\mathbb Q}_\ell $ .

Proof For any algebraic closed field F, $G^1_g(F)=USp_g(F)=Sp_g(F)$ . Since $Sp_g$ is simply connected as a group scheme over ${\mathbb Z}$ , so is $G^1_g=USp_g$ . Let $\mathbb {A}^{(\ell )}_f$ be the finite adeles of ${\mathbb Q}$ outside $\ell $ . By the strong approximation theorem (cf. Theorem 7.12, p.427 in Section 7.4 of [Reference Platonov and RapinchukPR94]) for $G^1_g$ with respect to $S=\{\infty ,\ell \}$ and using the exact sequence $1\longrightarrow G^1_g\longrightarrow G_g\stackrel {\nu }{\longrightarrow } GL_1\longrightarrow 1,$ we have a decomposition

(2.3) $$ \begin{align} G_g(\mathbb{A}_f)=G_g(\mathbb{A}^{(\ell)}_f)\times G_g({\mathbb Q}_\ell)=G_g({\mathbb Q})(K(\mathcal{O}^g)^{(\ell)}\times G_g({\mathbb Q}_\ell)). \end{align} $$

Combining Theorem 2.4 with (2.3), we have

(2.4) $$ \begin{align} SS_g(p)&\simeq K(\mathcal{O}^g)\backslash G_g(\mathbb{A}_f)/G_g({\mathbb Q}) \nonumber \\ &\simeq G_g({\mathbb Q})\backslash G_g(\mathbb{A}_f)/K(\mathcal{O}^g)\\ &=G_g({\mathbb Q})\backslash (G_g({\mathbb Q})(K(\mathcal{O}^g)^{(\ell)}\times G_g({\mathbb Q}_\ell)))/K(\mathcal{O}^g) \nonumber \\ &=G_g({\mathbb Z}[1/\ell])\backslash GSp_g({\mathbb Q}_\ell)/ GSp_g({\mathbb Z}_\ell) \nonumber \\ &=G_g({\mathbb Z}[1/\ell])\backslash GSp_g({\mathbb Q}_\ell)/Z_{GSp_g}({\mathbb Q}_\ell) GSp_g({\mathbb Z}_\ell). \nonumber \end{align} $$

We complete the proof.

2.5 Another formulation due to Jordan–Zaytman

Let $\ell \neq p$ be a prime. Both of $SS_g(p)$ and the Bruhat–Tits building $GSp_g({\mathbb Q}_\ell )/Z_{GSp_g}({\mathbb Q}_\ell ) GSp_g({\mathbb Z}_\ell )$ endowed with Hecke theory at $\ell $ . However, it is not transparent to see the compatibility of Hecke actions on both sides under the one-to-one correspondence (2.4). To overcome this, due to Jordan and Zaytman [Reference Jordan and ZaytmanJZ21], we use another formulation of $SS_g(p)$ and its connection to $SS_g(p,\ell ,A_0,\mathcal {L}_0)$ by using $\ell $ -adic Tate modules.

Pick $(A,\mathcal {L})$ from a class in $SS_g(p)$ . For a positive integer n, let $A[\ell ^n]:=\{P\in A(\overline {\mathbb {F}}_p)\ |\ \ell ^n P=0_A\}\simeq ({\mathbb Z}/\ell ^n {\mathbb Z})^{\oplus 2g}$ and put $A[\ell ^\infty ]=\displaystyle \bigcup _{n\ge 1}A[\ell ^n]$ . We denote by $T_\ell (A)$ the $\ell $ -adic Tate module and by $V_\ell (A):=T_\ell (A)\otimes _{{\mathbb Z}_\ell }{\mathbb Q}_\ell $ the $\ell $ -adic rational Tate module (cf. Section 18 of Chapter IV of [Reference MumfordMum70]). Let us define the coefficient ring $R_V$ to be ${\mathbb Z}/\ell ^n{\mathbb Z}$ if $V=A[\ell ^n]$ , ${\mathbb Z}_\ell $ if $V=T_\ell (A)$ , and ${\mathbb Q}_\ell $ if $V=V_\ell (A)$ . The principal polarization $\phi _{\mathcal {L}}:A\stackrel {\sim }{\longrightarrow }\widehat {A}$ yields $V\simeq V^\ast =\mathrm {Hom}_{R_V}(V,R_V)$ , and it induces a non-degenerate alternating pairing $\langle \ast ,\ast \rangle :V\times V\longrightarrow R_V$ . Let C be a maximal isotropic subgroup of $A[\ell ^n]$ for some $n\ge 1$ . Consider the exact sequence

$$\begin{align*}0\longrightarrow T_\ell(A)\stackrel{\subset}{\longrightarrow} V_\ell(A)\stackrel{\pi}{\longrightarrow}V_\ell(A)/T_\ell(A)\simeq A[\ell^\infty] \longrightarrow 0. \end{align*}$$

Then, $T_C:=\pi ^{-1}(C)$ is a lattice of $V_\ell (A)$ . The quotient $A_C:=A/C$ is also a superspecial abelian variety and the line bundle $\mathcal {L}$ is uniquely descend to a principal polarization $\mathcal {L}_C$ on $A_C$ by Corollary of Theorem 2 in Section 23 of Chapter IV of [Reference MumfordMum70] (see also Proposition 11.25 of [Reference Edixhoven, van der Geer and MoonenEGM] for the uniqueness). Therefore, $T_C\simeq T_\ell (A_C)$ has a symplectic ${\mathbb Z}_\ell $ -basis $\{f_{C,i}\}_{i=1}^{2g}\subset {\mathbb Q}^{2g}_\ell $ which means the matrix $P_C:=(f_{C,1},\ldots ,f_{C,2g})\in M_{2g}({\mathbb Q}_\ell )$ belongs to $GSp_g({\mathbb Q}_\ell )$ . Another choice of a symplectic ${\mathbb Z}_\ell $ -basis of $T_C$ yields $P_C\gamma $ for some $\gamma \in GSp_g({\mathbb Z}_\ell )$ . For each $h\in \mathrm {End}(A)\otimes _{\mathbb Z} {\mathbb Z}[1/\ell ]$ which is invertible (hence h is an isogeny of degree a power of $\ell $ ), we see easily that $P_{h(C)}=h^\ast P_C$ where $h^\ast $ is the endomorphism of $V_\ell (A)$ induced from h. In fact, by the functorial property of the pairing (see page 228 of [Reference MumfordMum70]). We identify $G_g({\mathbb Z}[1/\ell ])$ with

(2.5) $$ \begin{align} \Gamma(A)^\dagger:=\{f\in (\mathrm{End}(A)\otimes_{\mathbb Z} {\mathbb Z}[1/\ell])^\times\ |\ f\circ f^\dagger= f^\dagger\circ f\in {\mathbb Z}[1/\ell]^\times \mathrm{id}_A\} \end{align} $$

under the natural inclusion $(\mathrm {End}(A)\otimes _{\mathbb Z} {\mathbb Z}[1/\ell ])^\times \hookrightarrow \mathrm {Aut}((V_\ell (A),\langle \ast ,\ast \rangle ))=GSp_g({\mathbb Q}_\ell )$ .

Fix $(A,\mathcal {L})$ in a class of $SS_g(p)$ . We introduce the following sets which play an important role in the construction of the isogeny graphs:

(2.6) $$ \begin{align} \mathrm{Iso}_{\ell^\infty}(A,\mathcal{L}):=\{ [(A_C,\mathcal{L}_C)]\in SS_g(p) \ |\ n\ge 1,\ C\subset A[\ell^n]:\text{a maximal isotropic subgroup}\} \end{align} $$

and

(2.7) $$ \begin{align} SS_g(p,\ell,A,\mathcal{L}):=\{ [(B,\mathcal{M},\phi_B)] \ |\ [(B,\mathcal{M})]\in SS_g(p)\}, \end{align} $$

where $\phi _B:A\longrightarrow B$ is an $\ell $ -marking and $ [(B,\mathcal {M},\phi _B)] $ stands for the equivalent class of $(B,\mathcal {M},\phi _B)$ . Here, such 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 in $\Gamma (A_1)^\dagger $ . By definition, the natural map from $SS_g(p,\ell ,A,\mathcal {L})$ to $\mathrm {Iso}_{\ell ^\infty }(A,\mathcal {L})$ is surjective, while $\mathrm {Iso}_{\ell ^\infty }(A)$ is included in $SS_g(p)$ . With the above observation, we have obtained a map

(2.8) $$ \begin{align} \mathrm{Iso}_{\ell^\infty}(A,\mathcal{L})\longrightarrow G_g({\mathbb Z}[1/\ell])\backslash GSp_g({\mathbb Q}_\ell)/GSp_g({\mathbb Z}_\ell),\ [(A_C,\mathcal{L}_C)] \mapsto G_g({\mathbb Z}[1/\ell])P_C GSp_g({\mathbb Z}_\ell). \end{align} $$

We then show a slightly modified version of Jordan–Zaytman’s theorem, Theorem 46 of [Reference Jordan and ZaytmanJZ21] in conjunction with $SS_g(p,\ell ,A,\mathcal {L})$ .

Theorem 2.6 Fix $(A,\mathcal {L})$ in a class of $SS_g(p)$ . Keep the notation being as above. It holds that $\mathrm {Iso}_{\ell ^\infty }(A,\mathcal {L})=SS_g(p)$ and the map (2.8) induces a bijection

$$ \begin{align*}\mathrm{Iso}_{\ell^\infty}(A,\mathcal{L}) \stackrel{\sim}{\longrightarrow} G_g({\mathbb Z}[1/\ell])\backslash GSp_g({\mathbb Q}_\ell)/GSp_g({\mathbb Z}_\ell)=G_g({\mathbb Z}[1/\ell])\backslash GSp_g({\mathbb Q}_\ell)/Z_{GSp_g}({\mathbb Q}_\ell) GSp_g({\mathbb Z}_\ell).\end{align*} $$

Further, the natural map $SS_g(p,\ell ,A,\mathcal {L})\longrightarrow \mathrm {Iso}_{\ell ^\infty }(A,\mathcal {L})$ is also bijective.

Proof Surjectivity of (2.8) follows in reverse from the construction by using Corollary of Theorem 2 in Section 23 of Chapter IV of [Reference MumfordMum70] to guarantee the existence of a principal polarization. By Proposition 2.5 and $\mathrm {Iso}_{\ell ^\infty }(A,\mathcal {L})\subset SS_g(p)$ , we have

$$ \begin{align*}|SS_g(p)|=|G_g({\mathbb Z}[1/\ell])\backslash GSp_g({\mathbb Q}_\ell)/Z_{GSp_g}({\mathbb Q}_\ell) GSp_g({\mathbb Z}_\ell)|\le |\mathrm{Iso}_{\ell^\infty}(A,\mathcal{L})| \le |SS_g(p)|,\end{align*} $$

and it yields first two claims. With a natural surjection $SS_g(p,\ell ,A, \mathcal {L})\longrightarrow \mathrm {Iso}_{\ell ^\infty }(A,\mathcal {L})$ and (2.8), we have a surjective map

$$ \begin{align*}SS_g(p,\ell,A,\mathcal{L}) \longrightarrow G_g({\mathbb Z}[1/\ell])\backslash GSp_g({\mathbb Q}_\ell)/Z_{GSp_g}({\mathbb Q}_\ell) GSp_g({\mathbb Z}_\ell).\end{align*} $$

However, by construction and the identification $(\mathrm {End}(A)\otimes _{\mathbb Z} {\mathbb Z}[1/\ell ])^\times = G_g({\mathbb Z}[1/\ell ])$ , two objects of $SS_g(p,\ell ,A,\mathcal {L})$ which go to one element in the target differ by only $\ell $ -markings. Therefore, the above map is bijective. Hence, $SS_g(p,\ell ,A,\mathcal {L})\stackrel {\sim }{\longrightarrow }\mathrm {Iso}_{\ell ^\infty }(A,\mathcal {L}) =SS_g(p)$ .

Note that the factor $Z_{GSp_g}({\mathbb Q}_\ell )\simeq {\mathbb Q}^\times _\ell $ is intentionally inserted in front of $GSp_g({\mathbb Z}_\ell )$ as explained in the proof of Proposition 2.5.

As a byproduct, we have the following.

2.6 The Hecke operator at $\ell $

Finally, we discuss a relation of the map (2.8) with the Hecke operator at $\ell $ . We refer Section 3 in Chapter VII of [Reference Chai and FaltingsCF90] for general facts and Sections 16–19 of [Reference van der GeerGee08] as a reader’s friendly reference. For each prime $\ell $ different from p and a class $[(A,\mathcal {L},\phi _A)]\in SS_g(p,\ell ,A_0,\mathcal {L}_0)$ , we define the (geometric) Hecke correspondences $T(\ell )^{\mathrm {geo}}_{(A_0,\mathcal {L}_0)}$ at $\ell $ :

(2.9) $$ \begin{align} T(\ell)^{\mathrm{geo}}_{(A_0,\mathcal{L}_0)}([(A,\mathcal{L},\phi_A)]):=\sum_{C\subset A[\ell]\atop \text{maximal isotropic}}[(A_C,\mathcal{L}_C,f_C\circ \phi_A)], \end{align} $$

where $f_C:A\longrightarrow A_C$ is the natural projection. Similarly, we also define the (geometric) Hecke correspondences $T(\ell )^{\mathrm {geo}}$ at $\ell $ on $SS_g(p)$ :

(2.10) $$ \begin{align} T(\ell)^{\mathrm{geo}}([(A,\mathcal{L})]):=\sum_{C\subset A[\ell]\atop \text{maximal isotropic}}[(A_C,\mathcal{L}_C)]. \end{align} $$

Recall $GSp_g({\mathbb Q}_\ell )=GSp({\mathbb Q}^{2g}_\ell ,\ \langle \ast ,\ast \rangle )$ where $\langle \ast ,\ast \rangle $ is the standard symplectic pairing on ${\mathbb Q}^{2g}_\ell \times {\mathbb Q}^{2g}_\ell $ . Put $V={\mathbb Q}^{2g}_\ell $ . As seen before, each element of $GSp_g({\mathbb Q}_\ell )/GSp_g({\mathbb Z}_\ell )$ can be regarded as a lattice L of V such that $\langle \ast ,\ast \rangle _{L\times L}$ gives a ${\mathbb Z}_\ell $ -integral symplectic structure on L. Using this interpretation, each element of $GSp_g({\mathbb Q}_\ell )/Z_{GSp_g}({\mathbb Q}_\ell ) GSp_g({\mathbb Z}_\ell )$ can be regard as a homothety class $[L]$ for such an L. For each L being as above, we define the Hecke correspondence on $GSp_g({\mathbb Q}_\ell )/GSp_g({\mathbb Z}_\ell )$ at $\ell $

(2.11) $$ \begin{align} T(\ell)([L]):=\sum_{L\subset L_1 \subset \ell^{-1}L \atop L_1/L\text{:maximal isotropic}}[L_1], \end{align} $$

where $L_1$ runs over all lattice enjoying $L\subset L_1 \subset \ell ^{-1}L$ as denoted and that $L_1/L$ is a maximal isotropic subgroup of $\ell ^{-1}L/L$ with respect to the symplectic pairing $\langle \ast ,\ast \rangle _{\ell ^{-1}L/L\times \ell ^{-1}L/L}$ . Clearly, the action of $G_g({\mathbb Z}[1/\ell ])$ (given by multiplication from the left) on lattices are equivariant under $T(\ell )$ . Therefore, it also induces a correspondence on $G_g({\mathbb Z}[1/\ell ])\backslash GSp_g({\mathbb Q}_\ell )/Z_{GSp_g}({\mathbb Q}_\ell ) GSp_g({\mathbb Z}_\ell )$ and by abusing notation, we denote it by $T(\ell )$ . For a set X, we write $\mathrm {Div}(X)_{\mathbb Z}:=\bigoplus _{P\in X}{\mathbb Z} P$ . The identification (2.8) with the bijection

(2.12) $$ \begin{align} SS_g(p,\ell,A_0,\mathcal{L}_0)\stackrel{\sim}{\longrightarrow}SS_g(p),\ [(A,\mathcal{L},\phi_A)]\mapsto [(A,\mathcal{L})] \end{align} $$

yields a bijection

(2.13) $$ \begin{align} SS_g(p,\ell,A_0,\mathcal{L}_0)\longrightarrow G_g({\mathbb Z}[1/\ell])\backslash GSp_g({\mathbb Q}_\ell)/GSp_g({\mathbb Z}_\ell). \end{align} $$

Then we have obtained the following.

Theorem 2.8 The following diagram is commutative:

2.7 The Hecke action and automorphisms

In this subsection we describe the behavior of the Hecke action of $T(\ell )$ on the finite set

$$ \begin{align*}G_g({\mathbb Z}[1/\ell])\backslash GSp_g({\mathbb Q}_\ell)/GSp_g({\mathbb Z}_\ell)= G_g({\mathbb Z}[1/\ell])\backslash GSp_g({\mathbb Q}_\ell)/Z_{GSp_g}({\mathbb Q}_\ell)GSp_g({\mathbb Z}_\ell)\end{align*} $$

in terms of automorphism groups of objects in $SS_g(p,\ell ,A_0,\mathcal {L}_0)$ .

Put $\Gamma =G_g({\mathbb Z}[1/\ell ]),\ G=GSp_g({\mathbb Q}_\ell )$ , $Z=Z_{GSp_g}({\mathbb Q}_\ell )$ , and $K=GSp_g({\mathbb Z}_\ell )$ for simplicity. We write

$$ \begin{align*}\Gamma\backslash G/K=\{\Gamma x_1ZK,\ldots, \Gamma x_h ZK \},\ x_1,\ldots x_h\in G,\end{align*} $$

where $h=h_g(p,1)=|\Gamma \backslash G/ZK|$ . For each $i\in \{1,\ldots ,h\}$ , the coset $\Gamma x_i ZK$ is naturally identified with

$$ \begin{align*}\Gamma/\Gamma\cap x_i ZK x^{-1}_i=(\Gamma Z/Z)/((\Gamma\cap x_i ZK x^{-1}_i)Z/Z).\end{align*} $$

Next, we study the image of each element of $\Gamma \backslash G/K=\Gamma \backslash G/ZK$ under the Hecke action of $T(\ell )$ . Since $T(\ell )$ is defined in terms of lattices (see (2.11)), we define another formulation in terms of elements in G. Let $t_\ell :=\mathrm {diag}(\overbrace {1,\ldots ,1}^{g},\overbrace {\ell ,\ldots ,\ell }^{g})\in G$ . We decompose

(2.14) $$ \begin{align} Kt_\ell K=\coprod_{t\in T}g_t K, \end{align} $$

where T is the index set so that $|T|=N_g(\ell )$ . For each $i,j\in \{1,\ldots ,h\}$ , we define

(2.15) $$ \begin{align} m_{ij}:=\{t\in T\ |\ \Gamma x_i g_t ZK=\Gamma x_j ZK\}, \end{align} $$

which is independent of the choice of the representatives $\{g_t\}_{t\in T}$ . Let $W(\ell ):=\{g_t ZK\ |\ t\in T\}$ . Then, for each $i\in \{1,\ldots ,h\}$ , recall $\widetilde {\Gamma }_i=(\Gamma \cap x_i ZK x^{-1}_i)Z/Z$ , and the finite group $x_i^{-1}\widetilde {\Gamma }_i x_i\subset KZ/Z$ acts on $W(\ell )$ from the left by multiplication. The action induces the orbit decomposition

(2.16) $$ \begin{align} W(\ell)=\coprod_{t\in T'}O_{x_i^{-1}\widetilde{\Gamma}_i x_i}(g_tKZ) \end{align} $$

for some subset $T'\subset T$ .

Proof By assumption, $x_j=\gamma x_i g_t z k$ for some $\gamma \in \Gamma ,\ z\in Z$ , and $k\in K$ . For each $\alpha Z\in x_i^{-1}\widetilde {\Gamma }_i x_i=(x^{-1}_i\Gamma x_i\cap K)Z/Z$ , let us consider the element $kg^{-1}_t \alpha g_t k^{-1}Z$ in $G/Z$ . By using $x_j=\gamma x_i g_t z k$ , we see that the element belongs to $x^{-1}_j \Gamma x_j Z/Z$ . Further, if $\alpha Z$ is an element of $\mathrm {Stab}_{x_i^{-1}\widetilde {\Gamma }_i x_i}(g_t KZ)$ , $kg^{-1}_t \alpha g_t k^{-1}Z$ also belongs to K. Therefore, we have a group homomorphism

$$ \begin{align*}\mathrm{Stab}_{x_i^{-1}\widetilde{\Gamma}_i x_i}(g_t KZ)\stackrel{\tiny{\mathrm{the\ conjugation\ by\ }kg^{-1}_t}}{\longrightarrow} (x^{-1}_j\Gamma x_j\cap K)Z/Z\simeq \widetilde{\Gamma}_j.\end{align*} $$

Clearly, this map is injective and we have the claim.

We also study the converse of the correspondence from $\Gamma x_i g_t ZK$ to $\Gamma x_i ZK$ for each $i\in \{1,\ldots ,h\}$ . Clearly, $g^{-1}_tZK\in W(\ell )$ .

Lemma 2.11 For each $i\in \{1,\ldots ,h\}$ and $t\in T'$ , if $ \Gamma x_i g_t ZK=\Gamma x_j ZK$ for some ${j\in \{1,\ldots ,h\}}$ , then $|\mathrm {Stab}_{x_i^{-1}\widetilde {\Gamma }_i x_i}(g_t KZ)|= |\mathrm {Stab}_{x_j^{-1}\widetilde {\Gamma }_j x_j}(g^{-1}_t KZ)|$ . In particular, it holds

$$ \begin{align*}|\widetilde{\Gamma}_j |\cdot |O_{x_i^{-1}\widetilde{\Gamma}_i x_i}(g_tKZ)|= |\widetilde{\Gamma}_i |\cdot |O_{x_j^{-1}\widetilde{\Gamma}_j x_j}(g^{-1}_tKZ)|.\end{align*} $$

Finally, we study the corresponding results in $SS_g(p,\ell ,A_0,\mathcal {L}_0)$ under the identification

(2.17) $$ \begin{align} SS_g(p,\ell,A_0,\mathcal{L}_0)\longrightarrow G_g({\mathbb Z}[1/\ell])\backslash GSp_g({\mathbb Q}_\ell)/GSp_g({\mathbb Z}_\ell) \end{align} $$

given by Theorem 2.6. We write

$$ \begin{align*}SS_g(p,\ell,A_0,\mathcal{L}_0)=\{w_i=[(A_i,\mathcal{L}_i,\phi_{A_i})]\ |\ i=1,\ldots,h\}.\end{align*} $$

Let us fix $i\in \{1,\ldots ,h\}$ , and we denote by $\mathrm {LG}_i(\ell )=\{C_t\}_{t\in T}$ the set of all totally maximal isotropic subspace of $A_i[\ell ]$ with respect to the Weil pairing associated with $\mathcal {L}_i$ . Here, we use the same index T as $W(\ell )$ defined before. Then the group $\mathrm {RA}_i:=\mathrm {Aut}((A_i,\mathcal {L}_i))/\{\pm 1\}$ acts on $\mathrm {LG}(\ell )$ since each element there preserves the polarization. As in (2.16), we also have the decomposition

$$ \begin{align*}\mathrm{LG}(\ell)=\coprod_{t\in T'}O_{\mathrm{RA}_i}(C_t).\end{align*} $$

Suppose $\Gamma x_i ZK$ corresponds to $w_i=[(A_i,\mathcal {L}_i,\phi _{A_i})]$ under (2.17).

We remark that the fourth claim of (2) in the above proposition was proved in Lemma 3.2 of [Reference Florit and SmithFS22].

3.1 Jordan–Zaytman’s big isogeny graph

We basically follow the notation in Sections 7.1 and 5.3 of [Reference Jordan and ZaytmanJZ21]. The $(\ell )^g$ -isogeny (big) graph $Gr_g(\ell ,p)$ due to Jordan–Zaytman for $SS_g(p)$ is defined as a directed (regular) graph where:

• • the set of vertices $V(Gr_g(\ell ,p))$ is $SS_g(p)$ and

• • the set of directed edges between two vertices $v_1=[(A_1,\mathcal {L}_1)]$ and $v_2=[(A_2,\mathcal {L}_2)]$ is the set of equivalence classes of $(\ell )^g$ -isogenies between $(A_1,\mathcal {L}_1)$ and $(A_2,\mathcal {L}_2)$ . Here, two isogenies $f,h:(A_1,\mathcal {L}_1)\longrightarrow (A_2,\mathcal {L}_2)$ are said to be equivalent if there exist automorphisms $\phi \in \mathrm {Aut}(A_1,\mathcal {L}_1)$ and $\psi \in \mathrm {Aut}(A_2,\mathcal {L}_2)$ such that $\psi \circ h=f\circ \phi $ .

The case when $g=1$ is nothing but Pizer’s graph $G(1,p;\ell )$ handled in [Reference PizerPiz90].

3.2 The ( $\ell $ -marked) $(\ell )^g$ -isogeny graph

Similarly, 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 (regular) 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 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 $v_2$ under the identification. 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 $f:(A_1,\mathcal {L}_1)\longrightarrow (A_2,\mathcal {L}_2)$ .

3.3 The graph defined by the special 1-complex

Put $\Gamma =G_g({\mathbb Z}[1/\ell ]),\ G=GSp_g({\mathbb Q}_\ell )$ , $Z=Z_{GSp_g}({\mathbb Q}_\ell )$ , and $K=GSp_g({\mathbb Z}_\ell )$ for simplicity. We consider the graph associated with the quotient $\Gamma \backslash \mathcal {S}_g$ where $\Gamma =G_g({\mathbb Z}[1/\ell ])$ and $\mathcal {S}_g=GSp_g({\mathbb Q}_\ell )/Z_{GSp_g}({\mathbb Q}_\ell )GSp_g({\mathbb Z}_\ell )$ .

Two elements $v_1=\Gamma g_1 ZK$ and $v_2=\Gamma g_2 ZK$ in $\Gamma \backslash G/ZK$ said to be adjacent if ${v_2=\Gamma g_1 g_t ZK}$ for some $t \in T$ where $\{g_t\}_{t\in T}$ is defined in (2.14).

The graph in question, say $\mathrm {BTQ}^1_g(\ell ,p)$ , is a directed (regular) graph where:

• • the set of vertices $V(\mathrm {BTQ}^1_g(\ell ,p))$ is $\Gamma \backslash G/ZK$ and

• • the set of directed edges between two vertices $v_1=\Gamma g_1 ZK$ and $v_2=\Gamma g_2 ZK$ is defined by the adjacency condition in the above sense. Namely, an edge from $v_1$ from $v_2$ is $g_t$ with $t\in T$ such that $v_2=\Gamma g_1 g_t ZK$ .

3.4 Comparison theorem

Let us keep the notation in this section. We define

$$ \begin{align*}\mathrm{RA}(v):= \left\{\begin{array}{@{}ll} \mathrm{RA}(A,\mathcal{L}), & \text{if }v=[(A,\mathcal{L})]\in SS_g(p)\text{ or }v=[(A,\mathcal{L},\phi_A)]\in SS_g(p,\ell,A_0,\mathcal{L}_0),\\ (\Gamma\cap x ZK x^{-1})Z/Z, & \text{if }v=\Gamma x ZK\text{ in the case of }\mathrm{BTQ}^1_g(\ell,p). \end{array} \right.\end{align*} $$

Further, we also define

$$ \begin{align*}\mathrm{Ker}(e):= \left\{\begin{array}{@{}ll} \mathrm{Ker}(f), & \text{if }e\text{ is a class of }(\ell)^g\text{-isogeny }f\text{ in the case of }SS_g(p)\text{ or }SS_g(p,\ell,A_0,\mathcal{L}_0),\\ g_t, & \text{if }e\text{ is an edge defined by }g_t,\ t\in T\text{ in the case of }\mathrm{BTQ}^1_g(\ell,p). \end{array} \right.\end{align*} $$

We will prove the following comparison theorem which plays an important role in our study.

Theorem 3.1 The identifications (2.12) and (2.13) induce the following graph isomorphisms:

$$ \begin{align*}Gr_g(\ell,p)\stackrel{(2.12)\atop \sim}{\longleftarrow} \mathcal{G}^{SS}_{g}(\ell,p) \stackrel{(2.13)\atop\sim}{\longrightarrow}\mathrm{BTQ}^1_g(\ell,p).\end{align*} $$

Further, the following properties are preserved under the isomorphisms:

• • The Hecke action of $T(\ell )^{\mathrm {geo}},\ T(\ell )^{\mathrm {geo}}_{(A_0,\mathcal {L}_0)}$ , or $T(\ell )$ on each set of the vertices defines $N_g(\ell )$ -neighbors of a given vertex.

• • Each edge e from $v_1$ to $v_2$ has an opposite $\widehat {e}$ such that

  $$ \begin{align*}|\mathrm{RA}(v_2)|\cdot |O_{\mathrm{RA}(v_1)}(\mathrm{Ker}(e))|=|\mathrm{RA}(v_1)|\cdot |O_{\mathrm{RA}(v_2)}(\mathrm{Ker}(\widehat{e}))|.\end{align*} $$

We remark that Theorem 2.8 is insufficient to prove the above corollary, while Theorem 3.1 tells us more finer information for the relation of the reduced automorphisms and the multiplicity of each edge.

4.1 Symplectic groups revisited for the buildings

Let F be a non-archimedean local field of characteristic different from $2$ , and let $\mathit {O}$ be the ring of integers. We fix a uniformizer $\varpi $ and identify the residue field $\mathit {O}/\varpi \mathit {O}$ with a finite field ${\mathbb F}_q$ of order q. Further, we denote by $F^\times $ and $\mathit {O}^\times $ the multiplicative groups in F and $\mathit {O}$ , respectively. Let $\mathrm {ord}_\varpi $ be a discrete valuation in F, normalized so that $\mathrm {ord}_\varpi (F^\times )={\mathbb Z}$ . For example, we consider the $\ell $ -adic field ${\mathbb Q}_\ell $ for a prime $\ell $ with the ring of integers ${\mathbb Z}_\ell $ , where $\ell $ is a uniformizer and the residue field is ${\mathbb F}_\ell ={\mathbb Z}/\ell {\mathbb Z}$ .

For a positive integer n, let $V:=F^{2n}$ be the symplectic space over F equipped with the standard symplectic pairing $\langle \ast , \ast \rangle $ defined by $\langle v, w\rangle =\,{}^{t}v J_n w$ for $v, w \in F^{2n}$ . For V, there exists a basis $\{v_1, \dots , v_n, w_1, \dots , w_n\}$ such that $\langle v_i, w_j\rangle =\delta _{i j}$ and $\langle v_i, v_j\rangle =\langle w_i, w_j\rangle =0$ for all $i, j=1, \dots , n$ , where $\delta _{ij}$ equals $1$ if $i=j$ and $0$ if $i \neq j$ , and we call it a symplectic basis of $(V, \langle \ast , \ast \rangle )$ . Each choice of a symplectic basis yields an isomorphism between the isometry group and $Sp_n(F)$ .

Note that the following elements are in $GSp_n(F)$ :

$$\begin{align*}t_\lambda:=\mathrm{diag}(1, \dots, 1, \lambda, \dots, \lambda)= \left( \begin{array}{cc} I_n & 0\\ 0 & \lambda I_n \end{array} \right) \quad \text{for }\lambda \in F^\times. \end{align*}$$

In the subsequent sections, we consider the projectivized groups: let $PSp_n(F)$ and $PGSp_n(F)$ be the groups $Sp_n(F)$ and $GSp_n(F)$ modulo the centers, respectively. If we naturally identify $PSp_n(F)$ with a normal subgroup of $PGSp_n(F)$ , then the quotient group $PGSp_n(F)/PSp_n(F)$ is isomorphic to $({\mathbb Z}/2{\mathbb Z})\times \mathit {O}^\times $ , which is generated by the images of $t_\lambda =\mathrm {diag}(1, \dots , 1, \lambda , \dots , \lambda )$ for $\lambda \in \varpi \mathit { O}^\times $ . Similarly, letting $PSp_n(\mathit {O})$ and $PGSp_n(\mathit {O})$ be the groups $Sp_n(\mathit {O})$ and $GSp_n(\mathit {O})$ modulo the centers, respectively, we consider $PSp_n(\mathit {O})$ as a subgroup in $PGSp_n(\mathit {O})$ .

4.2 Bruhat–Tits building: the construction

Let $(V, \langle \ast , \ast \rangle )$ be a symplectic space over F of dimension $2n$ . We define a lattice $\Lambda $ in V as a free $\mathit {O}$ -module of rank $2n$ . Note that if $\Lambda $ is a lattice, then $\Lambda /\varpi \Lambda $ is a vector space over ${\mathbb F}_q$ of dimension $2n$ . We say that a lattice $\Lambda $ is primitive if $\langle \Lambda , \Lambda \rangle \subseteq \mathit {O}$ where $\langle \Lambda , \Lambda \rangle :=\{\langle v, w\rangle \mid v, w \in \Lambda \}$ , and $\langle \ast , \ast \rangle $ induces a non-degenerate alternating form on $\Lambda /\varpi \Lambda $ over ${\mathbb F}_q$ .

Let $\Lambda _i$ for $i=1, 2$ be lattices in V, and we say that they are homothetic if ${\Lambda _1 =\alpha \Lambda _2}$ for some $\alpha \in F^\times $ . This defines an equivalence relation in the set of lattices in V. We denote the homothety class of a lattice $\Lambda $ by $[\Lambda ]$ . Let us define the set ${\mathbb L}_n$ of homothety classes $[\Lambda ]$ of lattices such that there exist a representative $\Lambda $ of $[\Lambda ]$ and a primitive lattice $\Lambda _0$ satisfying that $\varpi \Lambda _0 \subseteq \Lambda \subseteq \Lambda _0$ and $\langle \Lambda , \Lambda \rangle \subseteq \varpi \mathit {O}$ . By the definition, if $[\Lambda ] \in {\mathbb L}_n$ , then a representative $\Lambda $ yields a subspace $\Lambda /\varpi \Lambda _0$ of $\Lambda _0/\varpi \Lambda _0$ with some primitive lattice $\Lambda _0$ such that it is totally isotropic, i.e., the induced non-degenerate alternating form $\langle \ast , \ast \rangle $ vanishes on $\Lambda /\varpi \Lambda _0$ in $\Lambda _0/\varpi \Lambda _0$ . Further, we define the incidence relation in ${\mathbb L}_n$ and denote by $[\Lambda _1] \sim [\Lambda _2]$ for two distinct homothety classes if there exist representatives $\Lambda _i$ of $[\Lambda _i]$ for $i=1, 2$ and a primitive lattice $\Lambda _0$ such that $\varpi \Lambda _0 \subseteq \Lambda _i \subseteq \Lambda _0$ for $i=1, 2$ , and either $\Lambda _1 \subseteq \Lambda _2$ or $\Lambda _2 \subseteq \Lambda _1$ holds.

The Bruhat–Tits building ${\mathcal B}_n$ (in short, building) for the group $PGSp_n(F)$ (or $Sp_n(F)$ ) is the clique complex whose set of vertices $\mathrm {Ver}({\mathcal B}_n)$ is ${\mathbb L}_n$ , i.e., $\sigma \subset \mathrm {Ver}({\mathcal B}_n)$ defines a simplex if any distinct vertices in $\sigma $ are incident. The building ${\mathcal B}_n$ is a simplicial complex of dimension n; note that each chamber (i.e., a simplex of maximal dimension) $[\Lambda _0], [\Lambda _1], \dots , [\Lambda _n]$ corresponds to a sequence of lattices $\Lambda _0 \subseteq \Lambda _1 \subseteq \cdots \subseteq \Lambda _n \subseteq \varpi ^{-1}\Lambda _0$ , where $\varpi ^{-1}\Lambda _0$ is primitive, such that

$$\begin{align*}\{0\} \subseteq \Lambda_1/\Lambda_0 \subseteq \Lambda_2/\Lambda_0 \subseteq \cdots \subseteq \Lambda_n/\Lambda_0 \subseteq \varpi^{-1}\Lambda_0/\Lambda_0 \end{align*}$$

forms a complete flag of a maximal totally isotropic subspace $\Lambda _n/\Lambda _0$ in $\varpi ^{-1}\Lambda _0/\Lambda _0$ over ${\mathbb F}_q$ .

The group $Sp_n(F)$ acts on ${\mathcal B}_n$ as simplicial automorphisms: let us fix a symplectic basis $\{v_1, \dots , v_n, w_1, \dots , w_n\}$ of $(V, \langle \ast , \ast \rangle )$ , which we identify with the standard symplectic space over F. Then the action is defined by $[\Lambda ] \mapsto [M \Lambda ]$ for $[\Lambda ] \in \mathrm {Ver}({\mathcal B}_n)$ and $M \in Sp_n(F)$ , and this action is simplicial since it preserves the incidence relation. Moreover, this yields the action of the projectivized group $PSp_n(F)$ on ${\mathcal B}_n$ .

We define the label (or, color) on the set of vertices $\mathrm {Ver}({\mathcal B}_n)$ . For any lattice $\Lambda $ , there exists some $\gamma \in GL_{2n}(F)$ such that $\gamma u_1, \dots , \gamma w_n$ form an $\mathit {O}$ -basis of $\Lambda $ . Let

$$\begin{align*}\mathbf{lab}_n[\Lambda]:=\mathrm{ord}_\varpi(\det \gamma)\quad \mod 2n. \end{align*}$$

Note that this depends only on the homothety class of $\Lambda $ since $\det (\alpha \gamma )=\alpha ^{2n}\det (\gamma )$ for $\alpha \in F^\times $ and for $\gamma \in GL_{2n}(F)$ , and $\det \gamma \in \mathit {O}^\times $ for $\gamma \in GL_{2n}(\mathit {O})$ . Hence, the function $\mathbf {lab}_n:\mathrm {Ver}({\mathcal B}_n) \to {\mathbb Z}/2n{\mathbb Z}$ is well defined and we call $\mathbf {lab}_n[\Lambda ]$ the label of a vertex $[\Lambda ] \in \mathrm {Ver}({\mathcal B}_n)$ . For example, let us consider a sequence of lattices $\Lambda _0, \dots , \Lambda _n$ , where

(4.1) $$ \begin{align} \Lambda_k:=\mathit{O} u_1\oplus \cdots \oplus \mathit{O} u_k \oplus \mathit{O} \varpi u_{k+1}\oplus \cdots \oplus \mathit{O} \varpi w_1\oplus \cdots \oplus \mathit{O} \varpi w_n \quad \text{for }0 \le k <n, \end{align} $$

and $\Lambda _n:=\mathit {O} u_1\oplus \cdots \oplus \mathit {O} u_n \oplus \mathit {O} \varpi w_1\oplus \cdots \oplus \mathit {O} \varpi w_n$ . Then $\Lambda _0 \subseteq \cdots \subseteq \Lambda _n \subseteq \varpi ^{-1}\Lambda _0$ and $\varpi ^{-1}\Lambda _0$ is primitive, and since the chain $\Lambda _1/\Lambda _0 \subseteq \cdots \subseteq \Lambda _n/\Lambda _0$ forms a maximal totally isotropic flag in $\varpi ^{-1}\Lambda _0/\Lambda _0$ over ${\mathbb F}_q$ , the corresponding homothety classes $[\Lambda _0], \dots , [\Lambda _n]$ define a chamber in ${\mathcal B}_n$ . In this case, we have that $\mathbf {lab}_n[\Lambda _k]=2n-k \mod 2n$ for $0 \le k \le n$ . We call the chamber determined by $[\Lambda _0], \dots , [\Lambda _n]$ the fundamental chamber ${\mathcal C}_0$ . Here, we note that $\mathbf {lab}_n$ misses the values $1, \dots , n-1$ in ${\mathbb Z}/2n {\mathbb Z}$ . It is known that $Sp_n(F)$ acts transitively on the set of chambers [Reference GarrettGar97, Section 20.5], i.e., every chamber is of the form $\gamma {\mathcal C}_0$ for $\gamma \in Sp_n(F)$ . By definition, the action of $Sp_n(F)$ preserves the labels on $\mathrm {Ver}({\mathcal B}_n)$ . It thus implies that the action is not vertex-transitive for any $n \ge 1$ .

4.3 Apartments

Let us introduce a system of apartments in the building ${\mathcal B}_n$ . See [Reference GarrettGar97, Chapter 20] and [Reference ShemanskeShe07] for basics and more details. A frame is an unordered n-tuple: $\{\lambda _1^1, \lambda _1^2\}$ , $\dots $ , $\{\lambda _n^1, \lambda _n^2\}$ , such that each $\{\lambda _i^1, \lambda _i^2\}$ is an unordered pair of lines which span a two-dimensional symplectic subspace with the induced alternating form for $i=1, \dots , n$ , and

$$\begin{align*}V=V_1\oplus \cdots \oplus V_n \quad \text{where }V_i:=\lambda_i^1\oplus\lambda_i^2\text{ and }V_i \bot V_j\text{ if }i \neq j, \end{align*}$$

i.e., $\langle v, v'\rangle =0$ for all $v \in V_i$ and all $v' \in V_j$ if $i \neq j$ . An apartment defined by a frame $\{\lambda _i^1, \lambda _i^2\}$ for $i=1, \dots , n$ is a maximal subcomplex of ${\mathcal B}_n$ on the set of vertices $[\Lambda ]$ such that

$$\begin{align*}\Lambda=\bigoplus_{i=1}^n \left(M_i^1\oplus M_i^2\right) \quad \text{where }M_i^j\text{ is a rank one free }\mathit{O}\text{-module in }\lambda_i^j\text{ for }j=1, 2, \end{align*}$$

for some (equivalently, every) representative $\Lambda $ in the homothety class. We define a system of apartments as a maximal set of apartments.

Following [Reference ShemanskeShe07], we fix a symplectic basis $\{u_1, \dots , u_n, w_1, \dots , w_n\}$ of V and a uniformizer $\varpi $ in F and lighten the notation: we denote a lattice

$$\begin{align*}\Lambda=\mathit{O} \varpi^{a_1}u_1\oplus \cdots \oplus \mathit{O} \varpi^{a_n}u_n \oplus \mathit{O} \varpi^{b_1}w_1\oplus \cdots \oplus \mathit{O} \varpi^{b_n}w_n \quad \text{for }a_i, b_i \in {\mathbb Z}, i=1, \dots, n, \end{align*}$$

by $\Lambda =(a_1, \dots , a_n; b_1, \dots , b_n)$ , and the homothety class by $[\Lambda ]=[a_1, \dots , a_n; b_1, \dots , b_n]$ . For $\Lambda $ , we have $\langle \Lambda , \Lambda \rangle \subset \mathit {O}$ if and only if $\langle \varpi ^{a_i}u_i, \varpi ^{b_i}w_i\rangle =\varpi ^{a_i+b_i} \in \mathit {O}$ for all $i=1, \dots , n$ . This is equivalent to that $a_i+b_i\ge 0$ for all $i=1, \dots , n$ , in which case, $\Lambda /\varpi \Lambda $ is a non-degenerate alternating space with the induced form over the residue field $\mathit {O}/\varpi \mathit {O}$ if and only if $a_i+b_i=0$ for all $i=1, \dots , n$ .

For the fixed basis, let $\lambda _i^1:=F u_i$ and $\lambda _i^2:=F w_i$ for $i=1, \dots , n$ . The frame $\{\lambda _i^1, \lambda _i^2\}_{i=1, \dots , n}$ determines an apartment $\Sigma _0$ in the building ${\mathcal B}_n$ for $Sp_n(F)$ . We call $\Sigma _0$ the fundamental apartment. The chain of lattice $\Lambda _0 \subseteq \cdots \subseteq \Lambda _n$ in (4.1) defines a chamber ${\mathcal C}_0$ in $\Sigma _0$ containing $[\Lambda _0]$ . The rest of chambers in the apartment $\Sigma _0$ are obtained by the action of the affine Weyl group (of type $\widetilde C_n$ ) attached to the building.

Example 4.1 If $n=2$ , then we have eight chambers containing vertex $[\Lambda _0]=[1,1;1,1]$ in a fixed apartment, where the fundamental chamber ${\mathcal C}_0$ is defined by the chain

$$\begin{align*}\Lambda_0=(1,1;1,1) \subset (0,1;1,1) \subset (0,0;1,1) \subset (0,0;0,0)=\varpi^{-1}\Lambda_0. \end{align*}$$

4.4 Special vertices and the special $1$ -complex

For any lattice $\Lambda $ in a symplectic space $(V, \langle \ast , \ast \rangle )$ , let us define the dual by $\Lambda ^\ast :=\{v \in V \mid \langle v, w\rangle \in \mathit {O} \ \text {for all }w \in \Lambda \}$ . Note that $\Lambda ^\ast $ is also a lattice in V. For every $\alpha \in F^\times $ , we have that $(\alpha \Lambda )^\ast =\alpha ^{-1}\Lambda ^\ast $ , whence the homothety class $[\Lambda ^\ast ]$ depends only on $[\Lambda ]$ . Let us call a vertex $[\Lambda ]$ in the building ${\mathcal B}_n$ self-dual if $[\Lambda ^\ast ]=[\Lambda ]$ . Below we characterize self-dual vertices in terms of labels – it is essentially proved in [Reference ShemanskeShe07, Proposition 3.1]; so we omit the proof.

For $[\Lambda ] \in \mathrm {Ver}({\mathcal B}_n)$ , let us call $[\Lambda ]$ a special vertex if $[\Lambda ^\ast ]=[\Lambda ]$ . We define the special 1-complex ${\mathcal S}_n$ as a one-dimensional subcomplex of ${\mathcal B}_n$ based on the set of special vertices

$$\begin{align*}\mathrm{Ver}({\mathcal S}_n):=\big\{[\Lambda] \in \mathrm{Ver}({\mathcal B}_n) \mid [\Lambda^\ast]=[\Lambda]\big\}, \end{align*}$$

and $1$ -simplices (edges) are defined between two incident vertices in ${\mathcal B}_n$ (cf. Section 4.1): for $[\Lambda _1]$ , $[\Lambda _2]$ in $\mathrm {Ver}({\mathcal S}_n)$ , we have $[\Lambda _1] \sim [\Lambda _2]$ if and only if there exist representatives $\Lambda _1$ and $\Lambda _2$ from $[\Lambda _1]$ and $[\Lambda _2]$ , respectively, such that either $\varpi ^{-1}\Lambda _1$ is primitive and $\Lambda _1 \subseteq \Lambda _2 \subseteq \varpi ^{-1}\Lambda _1$ , or the analogous relation where the roles of $\Lambda _1$ and $\Lambda _2$ are interchanged holds. Note that since special vertices are those that are self-dual, if $\varpi ^{-1}\Lambda _1$ is primitive, then $\Lambda _2/\Lambda _1$ is a maximal totally isotropic subspace of $\varpi ^{-1}\Lambda _1/\Lambda _1$ over ${\mathbb F}_q$ . Lemma 4.2 shows that $[\Lambda ] \in \mathrm {Ver}({\mathcal S}_n)$ if and only if $\mathbf {lab}_n[\Lambda ]=0$ or $n\ \ \mod 2n$ . The following proposition has been shown in [Reference ShemanskeShe07, Proposition 3.6].

We note that $GSp_n(F)$ does not act on ${\mathcal B}_n$ through the linear transformation of lattices. Indeed, a vertex of label $2n-1\ \ \mod 2n$ in the fundamental chamber ${\mathcal C}_0$ is sent by $t_\varpi \in GSp_n(F)$ to a vertex of label $n-1\ \ \mod 2n$ , which does not belong to $\mathrm {Ver}({\mathcal B}_n)$ . However, restricted on ${\mathcal S}_n$ , the group $GSp_n(F)$ acts on ${\mathcal S}_n$ . Moreover, the action of $GSp_n(F)$ on ${\mathcal S}_n$ is vertex-transitive since for $t_\varpi =\mathrm {diag}(1, \dots , 1, \varpi , \dots , \varpi )$ in $GSp_n(F)$ , we have that

$$\begin{align*}t_\varpi[\Lambda_0]=[\Lambda_n] \quad \text{where } t_\varpi=\left( \begin{array}{cc} I_n & 0\\ 0 & \varpi I_n \end{array} \right)\ \text{and }[\Lambda_0], [\Lambda_n] \in {\mathcal C}_0. \end{align*}$$

Note that $t_\varpi $ permutes the labels on $\mathrm {Ver}({\mathcal S}_n)$ . This defines the action of $PGSp_n(F)$ on ${\mathcal S}_n$ .

5.1 Property (T)

Let G be a topological group, and let $(\pi , {\mathcal H})$ be a unitary representation of G, where we assume that any Hilbert space ${\mathcal H}$ is complex. For any compact subset Q in G, let

$$\begin{align*}\kappa(G, Q, \pi):=\inf\Big\{\max_{s \in Q}\|\pi(s)\varphi-\varphi\| \mid \varphi \in {\mathcal H}, \ \|\varphi\|=1\Big\}, \end{align*}$$

and further let $\kappa (G, Q):=\inf \kappa (G, Q, \pi )$ , where the above infimum is taken over all equivalence classes of unitary representations $(\pi , {\mathcal H})$ without nonzero invariant vectors. We call $\kappa (G, Q)$ the optimal Kazhdan constant for the pair $(G, Q)$ . We say that G has Property (T) if there exists a compact set Q in G such that $\kappa (G, Q)>0$ . It is known that for a local field F, if $n \ge 2$ , then $Sp_n(F)$ has Property (T), while if $n=1$ , then $Sp_1(F)=SL_2(F)$ and it fails to have Property (T) [Reference Bekka, de la Harpe and ValetteBHV08, Theorem 1.5.3 and Example 1.7.4].

For any $n \ge 2$ , $PSp_n(F)$ has Property (T) since $Sp_n(F)$ does [Reference Bekka, de la Harpe and ValetteBHV08, Theorem 1.3.4]. Similarly, for any $n \ge 2$ , the group $PGSp_n(F)$ has Property (T) since $PGSp_n(F)/PSp_n(F)$ admits a finite invariant Borel regular measure (see Section 4.1 and [Reference Bekka, de la Harpe and ValetteBHV08, Theorem 1.7.1]). (We note that for any $n \ge 1$ , the group $GSp_n(F)$ does not have Property (T) because it admits a surjective homomorphism onto ${\mathbb Z}$ [Reference Bekka, de la Harpe and ValetteBHV08, Corollary 1.3.5].)

We say that a subset Q of G is generating if the sub-semigroup generated by Q coincides with G. If G has Property (T) and Q is an arbitrary compact generating set of G (provided that it exists), then $\kappa (G, Q)>0$ [Reference Bekka, de la Harpe and ValetteBHV08, Proposition 1.3.2]. We will construct an appropriate compact generating set in the following.

5.2 A random walk operator

In this section, fix an integer $n \ge 1$ . Recall that $K=PGSp_n(\mathit {O})$ , and letting $o:=[\Lambda _0]$ , we identify K with the stabilizer of o in $PGSp_n(F)$ . Let $a:=[t_\varpi ] \in PGSp_n(F)$ , and let us choose $\xi _i \in PSp_n(F) (\subset PGSp_n(F))$ for $i=0, 1, \dots , n+1$ such that $\xi _0:=\mathrm {id}$ and for $i=1. \dots , n+1$ each $\xi _i$ projects onto the reflection $s_i$ in the affine Weyl group acting on the fundamental apartment $\Sigma _0$ .

Let us define a subset $\Omega :=\{k \xi _i a k', k (\xi _i a)^{-1}k' \mid k, k' \in K, i=0, \dots , n+1\}$ in $PGSp_n(F)$ , where we simply write

$$\begin{align*}\Omega= K \Omega_0 K, \quad \text{where }\Omega_0:=\{\xi_0 a, \dots, \xi_{n+1}a, (\xi_0 a)^{-1}, \dots, (\xi_{n+1}a)^{-1}\}. \end{align*}$$

Note that $\Omega $ is compact and symmetric, i.e., $x \in \Omega $ if and only if $x^{-1} \in \Omega $ . Let $\nu $ be a Haar measure on K normalized so that $\nu (K)=1$ . Let us define the probability measure $\mu $ on $PGSp_n(F)$ as the distribution of $k \zeta k'$ where $k, k'$ , and $\zeta $ are independent and $k, k'$ are distributed according to $\nu $ and $\zeta $ is uniformly distributed on $\{\xi _i a, (\xi _i a)^{-1} \mid i=0, \dots , n+1\}$ . In other words,

$$\begin{align*}\mu=\nu\ast \mathrm{Unif}_{\Omega_0}\ast \nu, \quad \text{where }\mathrm{Unif}_{\Omega_0}:=\frac{1}{2(n+2)}\sum_{i=0}^{n+1}\left(\delta_{\xi_i a}+\delta_{(\xi_i a)^{-1}}\right), \end{align*}$$

and $\delta _x$ denotes the Dirac distribution at x. We write the convolution $\mu _1 \ast \mu _2$ for two probability measures $\mu _1, \mu _2$ on a group G. Note that the support of $\mu $ is $\Omega $ . If we define the probability measure $\check \mu $ on $PGSp_n(F)$ as the distribution of $x^{-1}$ where x has the law $\mu $ , then the definition of $\mu $ implies that $\check \mu =\mu $ .

For simplicity of notation, let $G:=PGSp_n(F)$ in the following discussion. Recall that $\mathrm {Ver}({\mathcal S}_n)=Go$ (cf. Section 4.3). Let us denote by $\ell ^2({\mathcal S}_n)$ the Hilbert space of square-summable complex-valued functions on $\mathrm {Ver}({\mathcal S}_n)$ equipped with the inner product $\langle \varphi , \psi \rangle :=\sum _{v \in \mathrm {Ver}({\mathcal S}_n)}\varphi (v)\overline {\psi (v)}$ for $\varphi , \psi \in \ell ^2({\mathcal S}_n)$ . Let us define an operator ${\mathcal A}_\mu : \ell ^2({\mathcal S}_n) \to \ell ^2({\mathcal S}_n)$ by

$$\begin{align*}{\mathcal A}_\mu \varphi(\xi o)=\int_{G}\varphi(\xi \gamma o)d\mu(\gamma) \qquad \text{for }\xi \in G. \end{align*}$$

Note that ${\mathcal A}_\mu $ is well defined by the definition of $\mu $ since $\mathrm {Ver}({\mathcal S}_n)=G o$ and K is the stabilizer of o. Lemma 5.1(2) shows that ${\mathcal A}_\mu $ is the normalized adjacency operator on ${\mathcal S}_n$ . Since $\check \mu =\mu $ , the operator ${\mathcal A}_\mu $ is self-adjoint on $\ell ^2({\mathcal S}_n)$ . Similarly, if we define ${\mathcal A}_{\mu ^{\ast t}}: \ell ^2({\mathcal S}_n) \to \ell ^2({\mathcal S}_n)$ in the same way with respect to the tth convolution power $\mu ^{\ast t}$ of $\mu $ , then we have that by induction ${\mathcal A}_{\mu }^t={\mathcal A}_{\mu ^{\ast t}}$ for all positive integer $t \ge 1$ .

Let us consider any closed subgroup $\Gamma $ of G such that $\Gamma $ acts on ${\mathcal S}_n$ from left with a compact quotient space $\Gamma \backslash {\mathcal S}_n$ , where the action is given by $(\gamma , v) \mapsto \gamma v$ for $\gamma \in \Gamma $ and $v \in {\mathcal S}_n$ . Since $\Gamma $ acts on ${\mathcal S}_n$ by simplicial automorphisms (as $PGSp_n(F)$ does), the quotient $\Gamma \backslash {\mathcal S}_n$ naturally admits a finite (unoriented) graph structure induced from ${\mathcal S}_n$ . Let us denote the finite graph by the same symbol $\Gamma \backslash {\mathcal S}_n$ . Note that since ${\mathcal S}_n$ is connected by Proposition 4.3, the graph $\Gamma \backslash {\mathcal S}_n$ is connected for any such $\Gamma $ . Here, however, we do not assume that $\Gamma $ is torsion-free; thus, the graph $\Gamma \backslash {\mathcal S}_n$ may have loops and not regular. Although ${\mathcal S}_n$ admits a bipartite graph structure, $\Gamma \backslash {\mathcal S}_n$ is not necessarily bipartite unless $\Gamma $ factors through $PSp_n(F)$ .

For each $v \in {\mathcal S}_n$ , let $\Gamma _v:=\{\gamma \in \Gamma \mid \gamma v=v\}$ . Note that $\Gamma _v$ is finite and $|\Gamma _v|$ is independent of the choice of representatives for $v \in \Gamma \backslash {\mathcal S}_n$ . Let us define $\ell ^2(\Gamma \backslash {\mathcal S}_n)$ the space of complex-valued functions on $\Gamma \backslash {\mathcal S}_n$ equipped with the inner product defined by

$$\begin{align*}\langle \varphi, \psi\rangle:=\sum_{v \in \Gamma \backslash {\mathcal S}_n} \varphi(v)\overline{\psi(v)}\frac{1}{|\Gamma_{v}|} \quad \text{for }\varphi, \psi \in \ell^2(\Gamma \backslash {\mathcal S}_n). \end{align*}$$

The group $\Gamma $ acts on $\ell ^2({\mathcal S}_n)$ by $\varphi \mapsto \varphi \circ \gamma ^{-1}$ for $\gamma \in \Gamma $ and $\varphi \in \ell ^2({\mathcal S}_n)$ , and since this $\Gamma $ -action and ${\mathcal A}_\mu $ on $\ell ^2({\mathcal S}_n)$ commute, the following operator ${\mathcal A}_{\Gamma , \mu ^{\ast t}}$ on $\ell ^2(\Gamma \backslash {\mathcal S}_n)$ is well defined for all positive integer t:

$$\begin{align*}{\mathcal A}_{\Gamma, \mu^{\ast t}}\varphi(\Gamma \xi o)=\int_{G}\varphi(\Gamma \xi \gamma o)d\mu^{\ast t}(\gamma) \qquad \text{for }\Gamma \xi o \in \Gamma\backslash {\mathcal S}_n\text{ and }\varphi \in \ell^2(\Gamma\backslash {\mathcal S}_n). \end{align*}$$

We have that ${\mathcal A}_{\Gamma , \mu }^t={\mathcal A}_{\Gamma , \mu ^{\ast t}}$ for all integer $t \ge 1$ , and ${\mathcal A}_{\Gamma , \mu }$ is self-adjoint, i.e., $\langle {\mathcal A}_{\Gamma , \mu }\varphi , \psi \rangle =\langle \varphi , {\mathcal A}_{\Gamma , \mu }\psi \rangle $ for $\varphi , \psi \in \ell ^2(\Gamma \backslash {\mathcal S}_n)$ . In other words, the operator ${\mathcal A}_{\Gamma , \mu }$ defines a Markov chain on $\Gamma \backslash {\mathcal S}_n$ reversible with respect to the measure $1/|\Gamma _v|$ for each vertex v.

5.3 Spectral gap

We normalize the Haar measure on G in such a way that K has the unit mass. Let $L^2(\Gamma \backslash G)$ denote the complex $L^2$ -space with respect to the (right) Haar measure for which each double coset $\Gamma \xi K$ has the mass $1/|\xi ^{-1}\Gamma \xi \cap K|$ . Note that the mass coincides with $1/|\Gamma _{\xi o}|$ since $\Gamma _{\xi o}=\Gamma \cap \xi K \xi ^{-1}$ . We consider $L^2(\Gamma \backslash G)^K$ the subspace of K-fixed vectors in $L^2(\Gamma \backslash G)$ and naturally identify it with $\ell ^2(\Gamma \backslash {\mathcal S}_n)$ (including the inner product). Let us define the unitary representation $\pi $ of G on $L^2(\Gamma \backslash G)$ by

$$\begin{align*}\pi(\gamma)\varphi(\Gamma \xi)=\varphi(\Gamma \xi\gamma) \quad \text{for }\varphi \in L^2(\Gamma \backslash G)\text{ and }\xi, \gamma \in G. \end{align*}$$

Note that $\varphi \in L^2(\Gamma \backslash G)^K$ if and only if $\pi (k)\varphi =\varphi $ for all $k \in K$ .

Let

$$\begin{align*}T_\Gamma(\gamma) \varphi(\Gamma \xi):=\int_K\varphi(\Gamma \xi k \gamma)\,d\nu(k) \quad \text{for }\varphi \in L^2(\Gamma \backslash G)\text{ and }\gamma \in G, \end{align*}$$

where we recall that $\nu $ is the normalized Haar measure on K.

Lemma 5.2 For every $n \ge 1$ , and for all $\varphi \in L^2(\Gamma \backslash G)^K$ , we have that

$$ \begin{align*} {\mathcal A}_{\Gamma, \mu}\varphi=\frac{1}{2(n+2)}\sum_{\gamma \in \Omega_0}T_\Gamma(\gamma)\varphi. \end{align*} $$

Moreover, for all $\gamma \in \Gamma $ and for all $\varphi _1, \varphi _2 \in L^2(\Gamma \backslash G)^K$ , we have that $\langle T_\Gamma (\gamma )\varphi _1, \varphi _2\rangle =\langle \pi (\gamma )\varphi _1, \varphi _2\rangle $ .

Proof Let us show the first claim. Recalling that $\mu =\nu \ast \mathrm {Unif}_{\Omega _0}\ast \nu $ , for ${\varphi \in L^2(\Gamma \backslash G)^K}$ and $\xi , \gamma \in G$ , we have that

$$ \begin{align*} {\mathcal A}_{\Gamma, \mu}\varphi(\Gamma \xi) &=\int_G \varphi(\Gamma \xi\gamma)\,d\mu(\gamma) =\int_{K \times \Omega_0 \times K}\varphi(\Gamma \xi k_1 \gamma k_2)\,d\nu(k_1)d\mathrm{Unif}_{\Omega_0}(\gamma)d\nu(k_2)\\ &=\frac{1}{2(n+2)}\sum_{\gamma \in \Omega_0}\int_K \varphi(\Gamma \xi k\gamma)\,d\nu(k) =\frac{1}{2(n+2)}\sum_{\gamma \in \Omega_0}T_\Gamma(\gamma)\varphi(\Gamma \xi), \end{align*} $$

where the third equality follows since $\varphi $ is a K-fixed vector and the last identity follows from the definition of $T_\Gamma (\gamma )$ . Hence, the first claim holds. The second claim follows from a formal computation based on the right-invariance of the Haar measure on $\Gamma \backslash G$ , so we omit the details.

Let us denote by $\ell _0^2(\Gamma \backslash {\mathcal S}_n)$ the orthogonal complement to the space of constant functions in $\ell ^2(\Gamma \backslash {\mathcal S}_n)$ . Note that ${\mathcal A}_{\Gamma , \mu }$ acts on $\ell _0^2(\Gamma \backslash {\mathcal S}_n)$ since the operator is self-adjoint. Given the right representation $(\pi , L^2(\Gamma \backslash G))$ , letting $L_0^2(\Gamma \backslash G)$ be the orthogonal complement to constant functions in $L^2(\Gamma \backslash G)$ , we define $(\pi _0, L_0^2(\Gamma \backslash G))$ by restricting $\pi $ to $L_0^2(\Gamma \backslash G)$ . The space $\ell _0^2(\Gamma \backslash {\mathcal S}_n)$ is identified with the space of K-fixed vectors in $L_0^2(\Gamma \backslash G)$ under the identification between $\ell ^2(\Gamma \backslash {\mathcal S}_n)$ and $L^2(\Gamma \backslash G)^K$ . It is crucial that $\pi _0$ has no nonzero invariant vector.

Proposition 5.3 For every $n \ge 1$ , let $\Gamma $ be a closed subgroup of $G=PGSp_n(F)$ such that $\Gamma \backslash {\mathcal S}_n$ is finite. For all $\varphi \in \ell _0^2(\Gamma \backslash {\mathcal S}_n)$ with $\|\varphi \|=1$ , we have that

$$ \begin{align*} \langle (I-{\mathcal A}_{\Gamma, \mu})\varphi, \varphi\rangle \ge \frac{1}{4(n+2)}\kappa(G, \Omega)^2, \end{align*} $$

where $\kappa (G, \Omega )$ is the optimal Kazhdan constant for the pair $(G, \Omega )$ .

Proof For $\varphi \in \ell _0^2(\Gamma \backslash {\mathcal S}_n)$ , it follows that $ \langle (I-{\mathcal A}_{\Gamma , \mu })\varphi , \varphi \rangle $ equals

$$ \begin{align*} \langle \varphi, \varphi \rangle -\frac{1}{2(n+2)}\sum_{\gamma \in \Omega_0}\langle \pi(\gamma)\varphi, \varphi\rangle =\frac{1}{4(n+2)}\sum_{\gamma \in \Omega_0}\|\varphi-\pi_0(\gamma)\varphi\|^2, \end{align*} $$

where identifying $\varphi $ with a K-fixed vector, we have used Lemma 5.2, and the equality follows since $\pi _0$ is the restriction of $\pi $ and

$$\begin{align*}\|\varphi-\pi_0(\gamma)\varphi\|^2=\langle \varphi, \varphi\rangle-\langle \pi_0(\gamma)\varphi, \varphi\rangle-\langle \pi_0(\gamma^{-1})\varphi, \varphi \rangle+\langle \pi_0(\gamma)\varphi, \pi_0(\gamma)\varphi\rangle, \end{align*}$$

and $\pi _0(\gamma )$ is unitary, and furthermore $\gamma \in \Omega _0$ if and only if $\gamma ^{-1} \in \Omega _0$ . Moreover, we have that

$$ \begin{align*} \sum_{\gamma \in \Omega_0}\|\varphi-\pi_0(\gamma)\varphi\|^2 \ge \max_{\gamma \in \Omega_0}\|\varphi-\pi_0(\gamma)\varphi\|^2 =\max_{\gamma \in \Omega}\|\varphi-\pi_0(\gamma)\varphi\|^2, \end{align*} $$

which follows from the first claim of Lemma 5.1(2) and since $\varphi $ is a K-fixed vector and $\pi _0$ is a unitary representation. Therefore, we obtain

$$\begin{align*}\langle (I-{\mathcal A}_{\Gamma, \mu})\varphi, \varphi\rangle \ge \frac{1}{4(n+2)}\max_{\gamma \in \Omega}\|\varphi-\pi_0(\gamma)\varphi\|^2. \end{align*}$$

Since $\pi _0$ has no nonzero invariant vector, we conclude the claim.

The proof of Theorem 1.1 now follows from Theorem 5.4 with $\Gamma $ applied to $G_g({\mathbb Z}[1/\ell ])$ modulo the center and Corollary 3.2.

5.4 An explicit lower bound for the spectral gap

Appealing to the results by Oh [Reference OhOh02], we obtain explicit lower bounds for the second smallest eigenvalues of Laplacians on the graphs ${\mathcal G}^{SS}_{g}(\ell ,p)$ for $g \ge 2$ .

Corollary 5.5 For every integer $g \ge 2$ , for all primes $\ell $ and p with $p \neq \ell $ ,

$$\begin{align*}\lambda_2\left({\mathcal G}^{SS}_{g}(\ell,p)\right) \ge \frac{1}{4(g+2)}\left(\frac{\ell-1}{2(\ell-1)+3\sqrt{2\ell (\ell+1)}}\right)^2. \end{align*}$$

Proof We keep the notations in the preceding subsections and put $n=g$ . Let $F:={\mathbb Q}_\ell $ . Note that $\Omega ^2$ contains K and $a^2$ . The definition of the optimal Kazhdan constant shows that

$$\begin{align*}\kappa\left(G, \Omega^2\right) \ge \kappa\left(PSp_n({\mathbb Q}_\ell), \Omega^2\cap PSp_n({\mathbb Q}_\ell)\right). \end{align*}$$

Furthermore, the right-hand side is at least $\kappa \left (Sp_n({\mathbb Q}_\ell ), \Omega _\ast \right )$ , where

$$\begin{align*}\Omega_\ast:=\{Sp_n({\mathbb Z}_\ell), s\} \quad \text{and} \quad s:=\mathrm{diag}(\ell^{-1}, \dots, \ell^{-1}; \ell, \dots, \ell). \end{align*}$$

Applying [Reference OhOh02, Theorem 8.4] to $Sp_n({\mathbb Q}_\ell )$ for $n\ge 2$ with a maximal strongly orthogonal system $\mathrm {L}$ in the case of $C_n (n \ge 2)$ [Reference OhOh02, Appendix], we have that

$$\begin{align*}\kappa\left(Sp_n({\mathbb Q}_\ell), \Omega_\ast\right) \ge \chi_{\mathrm{L}}(s)=\frac{\sqrt{2(1-\xi_{\mathrm{L}}(s))}}{\sqrt{2(1-\xi_{\mathrm{L}}(s))}+3}, \end{align*}$$

where

$$\begin{align*}\xi_{\mathrm{L}}(s) \le \frac{2(\ell-1)+(\ell+1)}{\ell(\ell+1)}=\frac{3\ell-1}{\ell(\ell+1)}. \end{align*}$$

Hence, we have for all $n \ge 2$ and all prime $\ell $ ,

$$\begin{align*}\kappa\left(Sp_n({\mathbb Q}_\ell), \Omega_\ast\right) \ge \frac{\sqrt{2}(\ell-1)}{\sqrt{2}(\ell-1)+3\sqrt{\ell(\ell+1)}}, \end{align*}$$

and since $\kappa (G, \Omega ) \ge (1/2)\kappa (G, \Omega ^2)$ , we obtain

$$\begin{align*}\kappa(G, \Omega) \ge \frac{\ell-1}{2(\ell-1)+3\sqrt{2\ell (\ell+1)}}. \end{align*}$$

Combining the above inequality with $c_n=(4(n+2))^{-1}\kappa (G, \Omega )^2$ in the proof of Theorem 5.4, we conclude that for all $n \ge 2$ and all prime $\ell $ ,

$$\begin{align*}\lambda_2(\Delta_{\Gamma, \mu}) \ge \frac{1}{4(n+2)}\left(\frac{\ell-1}{2(\ell-1)+3\sqrt{2\ell (\ell+1)}}\right)^2. \end{align*}$$

Applying to the case when $\Gamma $ is $G_g({\mathbb Z}[1/\ell ])$ modulo the center together with Corollary 3.2 yields the claim.