1 Introduction

The Yagita invariant \({p^\circ }(G)\) of a discrete group G is an invariant that generalizes the period of the p-local Tate–Farrell cohomology of G, in the following sense: it is a numerical invariant defined for any G that is equal to the period when the p-local cohomology of G is periodic. Yagita considered finite groups [6], and Thomas extended the definition to groups of finite vcd [5]. In [3] the definition was extended to arbitrary groups and \({p^\circ }(G)\) was computed for \(G={\mathrm{GL}}(n,{\mathcal {O}})\) for \({\mathcal {O}}\) any integrally closed subring of \({\mathbb {C}}\) and for sufficiently large n (depending on \({\mathcal {O}}\)).

In [2], one of us computed the Yagita invariant for \({\mathrm{Sp}}(2(p+1),{\mathbb {Z}})\). Computations from [3] were used to provide an upper bound, and computations with finite subgroups and with mapping class groups were used to provide a lower bound [4]. The action of the mapping class group of a surface upon the first homology of the surface gives a natural symplectic representation of the mapping class group of a genus \(p+1\) surface inside \({\mathrm{Sp}}(2(p+1),{\mathbb {Z}})\). In the current paper, we compute \({p^\circ }({\mathrm{Sp}}(2n,{\mathcal {O}}))\) for each \(n\ge p-1\) for each \({\mathcal {O}}\) for which \({p^\circ }({\mathrm{GL}}(n,{\mathcal {O}}))\) was computed in [3]. By using a greater range of finite subgroups, we avoid having to consider mapping class groups.

Throughout the paper, we fix a prime p. Before stating our main result, we recall the definitions of the symplectic group \({\mathrm{Sp}}(2n,R)\) over a ring R, and of the Yagita invariant \({p^\circ }(G)\), which depends on the prime p as well as on the group G. The group \({\mathrm{Sp}}(2n,R)\) is the collection of invertible \(2n\times 2n\) matrices M over R such that

$$\begin{aligned} M^\mathrm {T}JM = J,\,\,\,\hbox {where}\,\,\, J:=\left( \begin{array}{l@{\quad }l} 0&{}I_n\\ -I_n&{}0\end{array}\right) . \end{aligned}$$

Here, \(M^\mathrm {T}\) denotes the transpose of the matrix M, and as usual \(I_n\) denotes the \(n\times n\) identity matrix. Equivalently, \(M\in {\mathrm{Sp}}(2n,R)\) if M defines an isometry of the antisymmetric bilinear form on \(R^{2n}\) defined by \(\langle x,y\rangle :=x^\mathrm {T}Jy\). If C is cyclic of order p, then the group cohomology ring \(H^*(C;{\mathbb {Z}})\) has the form

$$\begin{aligned} H^*(C;{\mathbb {Z}})\cong {\mathbb {Z}}[x]/(px),\,\,\,\, x\in H^2(C;{\mathbb {Z}}). \end{aligned}$$

If C is a cyclic subgroup of G of order p, define n(C) a positive integer or infinity to be the supremum of the integers n such that the image of \(H^*(G;{\mathbb {Z}})\rightarrow H^*(C;{\mathbb {Z}})\) is contained in the subring \({\mathbb {Z}}[x^n]\). Now, define

$$\begin{aligned} {p^\circ }(G):={\text {lcm}}\{2n(C)\,\,:\,\, C\le G,\,\,\, |C|=p\}. \end{aligned}$$

It is easy to see that if \(H\le G,\) then \({p^\circ }(H)\) divides \({p^\circ }(G)\) [3, Prop. 1].

2 Results

In the following theorem statement and throughout the paper, we let \(\zeta _p\) be a primitive pth root of 1 in \({\mathbb {C}}\) and we let \({\mathcal {O}}\) denote a subring of \({\mathbb {C}}\) with \(F\subseteq {\mathbb {C}}\) as its field of fractions. We assume that either \(\zeta _p\in {\mathcal {O}}\) or that \({\mathcal {O}}\) is integrally closed in \({\mathbb {C}}\). We define \(l:=|F[\zeta _p]:F|\), the degree of \(F[\zeta _p]\) as an extension of F. For \(t\in {\mathbb {R}}\) with \(t\ge 1\), we define \(\psi (t)\) to be the largest integer power of p less than or equal to t.

Theorem 1

With notation as above, for each \(n\ge p-1\), the Yagita invariant \({p^\circ }({\mathrm{Sp}}(2n,{\mathcal {O}}))\) is equal to \(2(p-1)\psi (2n/l)\) for l even and equal to \(2(p-1)\psi (n/l)\) for l odd.

By the main result of [3], the above is equivalent to the statement that \({p^\circ }({\mathrm{Sp}}(2n,{\mathcal {O}}))={p^\circ }({\mathrm{GL}}(2n,{\mathcal {O}}))\) when l is even and \({p^\circ }({\mathrm{Sp}}(2n,{\mathcal {O}}))={p^\circ }({\mathrm{GL}}(n,{\mathcal {O}}))\) when l is odd. By definition \({\mathrm{Sp}}(2n,{\mathcal {O}})\) is a subgroup of \({\mathrm{GL}}(2n,{\mathcal {O}})\) and there is an inclusion \({\mathrm{GL}}(n,{\mathcal {O}})\rightarrow {\mathrm{Sp}}(2n,{\mathcal {O}})\) defined by

$$\begin{aligned} A\mapsto \left( \begin{array}{l@{\quad }l} A&{}0\\ 0 &{}(A^\mathrm {T})^{-1}\end{array}\right) , \end{aligned}$$

and so for any n, \({p^\circ }({\mathrm{GL}}(n,{\mathcal {O}}))\) divides \({p^\circ }({\mathrm{Sp}}(2n,{\mathcal {O}}))\), which in turn divides \({p^\circ }({\mathrm{GL}}(2n,{\mathcal {O}}))\).

Before we start, we recall two standard facts concerning symplectic matrices that will be used in the proof of Corollary 3: if M is in the symplectic group, then \(\det (M)=1\) and M is conjugate to the inverse of its transpose \((M^{-1})^\mathrm {T}=(M^\mathrm {T})^{-1}\). We shall use the notation \({\mathbb {F}}_p^\times \) to denote the multiplicative group of units in the field \({\mathbb {F}}_p\).

Proposition 2

Let f(X) be a polynomial over the field \({\mathbb {F}}_p\) and suppose that 0 is not a root of f,  but that f factors as a product of linear polynomials over \({\mathbb {F}}_p\). If there is a polynomial g and an integer n so that \(f(X)=g(X^n)\), then n has the form \(n=mp^q\) for some m dividing \(p-1\) and some integer \(q\ge 0\). If p is odd and for each \(i\in {\mathbb {F}}_p^\times \), the multiplicity of i as a root of f is equal to that of \(-i\), then m is even.

Proof

The only part of this that is not contained in [3, Prop. 6] is the final statement. Since \((1-iX)(1+iX)=1-i^2X^2\) is a polynomial in \(X^2\), the final statement follows. For the benefit of the reader, we sketch the rest of the proof. If \(n=mp^q\) where p does not divide m, then \(g(X^n)=g(X^m)^{p^q}\), so we may assume that \(q=0\). If \(g(Y)=0\) has roots \(y_i\), then the roots of \(g(X^m)=0\) are the roots of \(y_i-X^m=0\). Since p does not divide m, these polynomials have no repeated roots; since their roots are assumed to lie in \({\mathbb {F}}_p\) it is now easy to show that m divides \(p-1\). \(\square \)

Corollary 3

With notation as in Theorem 1, let G be a subgroup of \({\mathrm{Sp}}(2n,F)\). Then the Yagita invariant \({p^\circ }(G)\) divides the number given for \({p^\circ }({\mathrm{Sp}}(2n,{\mathcal {O}}))\) in the statement of Theorem 1.

Proof

As in [3, Cor. 7], for each \(C\le G\) of order p, we use the total Chern class to give an upper bound for the number n(C) occurring in the definition of \({p^\circ }(G)\). If C is cyclic of order p, then C has p distinct irreducible complex representations, each one dimensional. If we write \(H^*(C;{\mathbb {Z}})={\mathbb {Z}}[x]/(px)\), then the total Chern classes of these representations are \(1+ix\) for each \(i\in {\mathbb {F}}_p\), where \(i=0\) corresponds to the trivial representation. The total Chern class of a direct sum of representations is the product of the total Chern classes, and so when viewed as a polynomial in \({\mathbb {F}}_p[x]=H^*(C;{\mathbb {Z}})\otimes {\mathbb {F}}_p\), the total Chern class of any faithful representation \(\rho :C\rightarrow {\mathrm{GL}}(2n,{\mathbb {C}})\) is a non-constant polynomial of degree at most 2n all of whose roots lie in \({\mathbb {F}}_p^\times \). Now, let F be a subfield of \({\mathbb {C}}\) with \(l=|F[\zeta _p]:F|\) as in the statement. The group C has \((p-1)/l\) non-trivial irreducible representations over F, each of dimension l, and the total Chern classes of these representations have the form \(1-ix^l\), where i ranges over the \((p-1)/l\) distinct lth roots of unity in \({\mathbb {F}}_p\). In particular, the total Chern class of any representation \(\rho :C\rightarrow {\mathrm{GL}}(2n,F)\le {\mathrm{GL}}(2n,{\mathbb {C}})\) is a polynomial in \(x^l\) whose x-degree is at most 2n. If \(\rho \) has image contained in \({\mathrm{Sp}}(2n,{\mathbb {C}})\), then it factors as \(\rho = \iota \circ {\widetilde{\rho }}\) with \({\widetilde{\rho }}:C\rightarrow {\mathrm{Sp}}(2n,{\mathbb {C}})\) and \(\iota \) is the inclusion of \({\mathrm{Sp}}(2n,{\mathbb {C}})\) in \({\mathrm{GL}}(2n,{\mathbb {C}})\). In this case, the matrix representing a generator for C is conjugate to the transpose of its own inverse; in particular, it follows that the multiplicities of the irreducible complex representations of C with total Chern classes \(1+ix\) and \(1-ix\) must be equal for each i. Hence in this case, if p is odd, the total Chern class of the representation \(\rho =\iota \circ {\widetilde{\rho }}\) is a polynomial in \(x^2\). If \(p=2\) (which implies that \(l=1\)), then the total Chern class of any representation \(\rho :C\rightarrow {\mathrm{GL}}(2n,{\mathbb {C}})\) has the form \((1+x)^i\), where i is equal to the number of non-trivial irreducible summands. Since \({\mathrm{Sp}}(2n,{\mathbb {C}})\le {\mathrm{SL}}(2n,{\mathbb {C}}),\) it follows that for symplectic representations i must be even, and so for \(p=2\) the total Chern class is a polynomial in \(x^2\).

In summary, let \({\widetilde{\rho }}\) be a faithful representation of C in \({\mathrm{Sp}}(2n,F)\). In the case when l is odd, then the total Chern class of \({\widetilde{\rho }}\) is a non-constant polynomial \({\tilde{f(y)=f(x)}}\) in \(y=x^{2l}\) such that f(x) has degree at most 2n, \({\tilde{f}}(y)\) has degree at most n / l, and all roots of \(f,{\tilde{f}}\) lie in \({\mathbb {F}}_p^\times \). In the case when l is even, the total Chern class of \(\rho \) is a non-constant polynomial \({\tilde{f}}(y)=f(x)\) in \(y=x^l\) such that f(x) has degree at most 2n, \({\tilde{f}}(y)\) has degree at most 2n / l, and all roots of both lie in \({\mathbb {F}}_p^\times \). By Proposition 2, it follows that each n(C) is a factor of the number given for \({p^\circ }({\mathrm{Sp}}(2n,{\mathcal {O}}))\), and hence the claim. \(\square \)

Lemma 4

Let \(H\le G\) with \(|G:H|=m\), and let \(\rho \) be a symplectic representation of H on \(V={\mathcal {O}}^{2n}\). The induced representation \(\mathrm{Ind}_H^G(\rho )\) is a symplectic representation of G on \(W:={\mathcal {O}}G\otimes _{{\mathcal {O}}H}V\cong {\mathcal {O}}^{2mn}\).

Proof

Let \(e_1,\ldots ,e_n,f_1,\ldots ,f_n\) be the standard basis for \(V={\mathcal {O}}^{2n}\), so that the bilinear form \(\langle v,w\rangle := v^\mathrm {T}J w\) on V is given by

$$\begin{aligned} \langle e_i,e_j\rangle = 0 = \langle f_i,f_j\rangle ,\,\,\,\, \langle e_i,f_j\rangle = -\langle f_i,e_j\rangle = \delta _{ij}. \end{aligned}$$

The representation \(\rho \) is symplectic if and only if each \(\rho (h)\) preserves this bilinear form.

Let \(t_1,\ldots ,t_m\) be a left transversal to H in G, so that \({\mathcal {O}}G= \oplus _{i=1}^m t_i{\mathcal {O}}H\) as right \({\mathcal {O}}H\)-modules. Define a bilinear form \(\langle \,\, ,\,\,\rangle _W\) on W by

$$\begin{aligned} \left\langle \sum _{i=1}^m t_i\otimes v^i, \sum _{i=1}^m t_i\otimes w^i\right\rangle _W := \sum _{i=1}^m \langle v^i,w^i\rangle . \end{aligned}$$

To see that this bilinear form is preserved by the \({\mathcal {O}}G\)-action on W, fix \(g\in G\) and define a permutation \(\pi \) of \(\{1,\ldots ,m\}\) and elements \(h_1,\ldots , h_m\in H\) by the equations \(gt_i=t_{\pi (i)}h_i\). Now for each i, j with \(1\le i,j\le m,\)

$$\begin{aligned} \left\langle \mathrm{Ind}(\rho (g))t_i\otimes v, \mathrm{Ind}(\rho (g))t_j\otimes w \right\rangle _W&= \left\langle t_{\pi (i)}\otimes \rho (h_i)v, t_{\pi (j)}\otimes \rho (h_j)w \right\rangle _W \\&=\delta _{\pi (i)\pi (j)} \langle \rho (h_i)v,\rho (h_j)w\rangle \\&= \delta _{ij}\langle \rho (h_i)v,\rho (h_i)w\rangle \\&=\delta _{ij} \langle v,w\rangle \\&= \langle t_i\otimes v,t_j\otimes w\rangle _W. \end{aligned}$$

To see that \(\langle \,\, ,\,\, \rangle _W\) is symplectic, define basis elements \({\scriptstyle {E}}_1,\ldots ,{\scriptstyle {E}}_{mn},{\scriptstyle {F}}_1,\ldots ,{\scriptstyle {F}}_{mn}\) for W by the equations

$$\begin{aligned} {\scriptstyle {E}}_{n(i-1)+j}:=t_i\otimes e_j,\,\,\,\hbox {and}\,\,\,{\scriptstyle {F}}_{n(i-1)+j}:= t_i\otimes f_j,\,\,\,\,\, \hbox {for}\,\,1\le i\le m,\,\,1\le j\le n. \end{aligned}$$

It is easily checked that for \(1\le i,j\le mn\)

$$\begin{aligned} \langle {\scriptstyle {E}}_i,{\scriptstyle {E}}_j\rangle _W = 0 = \langle {\scriptstyle {F}}_i,{\scriptstyle {F}}_j\rangle _W,\,\,\,\, \langle {\scriptstyle {E}}_i,{\scriptstyle {F}}_j\rangle _W = -\langle {\scriptstyle {F}}_i,{\scriptstyle {E}}_j\rangle _W= \delta _{ij}, \end{aligned}$$

and so with respect to this basis for W, the bilinear form \(\langle \,\, ,\,\,\rangle _W\) is the standard symplectic form. \(\square \)

Proposition 5

With notation as in Theorem 1, the Yagita invariant \({p^\circ }({\mathrm{Sp}}(2n,{\mathcal {O}}))\) is divisible by the number given in the statement of Theorem 1.

Proof

To give lower bounds for \({p^\circ }({\mathrm{Sp}}(2n,{\mathcal {O}})),\) we use finite subgroups. Firstly, consider the semidirect product \(H=C_p {\rtimes }C_{p-1}\), where \(C_{p-1}\) acts faithfully on \(C_p\); equivalently, this is the group of affine transformations of the line over \({\mathbb {F}}_p\). It is well known that the image of \(H^*(G;{\mathbb {Z}})\) inside \(H^*(C_p;{\mathbb {Z}})\cong {\mathbb {Z}}[x]/(px)\) is the subring generated by \(x^{p-1}\). It follows that \(2(p-1)\) divides \({p^\circ }(G)\) for any G containing H as a subgroup. The group H has a faithful permutation action on p points, and hence a faithful representation in \({\mathrm{GL}}(p-1,{\mathbb {Z}})\), where \({\mathbb {Z}}^{p-1}\) is identified with the kernel of the H-equivariant map \({\mathbb {Z}}\{1,\ldots , p\}\rightarrow {\mathbb {Z}}\). Since \({\mathrm{GL}}(p-1,{\mathbb {Z}})\) embeds in \({\mathrm{Sp}}(2(p-1),{\mathbb {Z}}),\) we deduce that H embeds in \({\mathrm{Sp}}(2n,{\mathcal {O}})\) for each \({\mathcal {O}}\) and for each \(n\ge p-1\).

To give a lower bound for the p-part of \({p^\circ }({\mathrm{Sp}}(2n,{\mathcal {O}})),\) we use the extraspecial p-groups. For p odd, let E(p, 1) be the non-abelian p-group of order \(p^3\) and exponent p, and let E(2, 1) be the dihedral group of order 8. (Equivalently in each case E(p, 1) is the Sylow p-subgroup of \({\mathrm{GL}}(3,{\mathbb {F}}_p)\).) For \(m\ge 2\), let E(p, m) denote the central product of m copies of E(p, 1), so that E(p, m) is one of the two extraspecial groups of order \(p^{2m+1}\). Yagita showed that \({p^\circ }(E(p,m))=2p^m\) for each m and p [6]. The centre and commutator subgroup of E(p, m) are equal and have order p, and the abelianization of E(p, m) is isomorphic to \(C_p^{2m}\). The irreducible complex representations of E(p, m) are well understood: there are \(p^{2m}\) distinct one-dimensional irreducibles, each of which restricts to the centre as the trivial representation, and there are \(p-1\) faithful representations of dimension \(p^m\), each of which restricts to the centre as the sum of \(p^m\) copies of a single (non-trivial) irreducible representation of \(C_p\). The group \(G=E(p,m)\) contains a subgroup H isomorphic to \(C_p^{m+1}\), and each of its faithful \(p^m\)-dimensional representations can be obtained by inducing up a one-dimensional representation \(H\rightarrow C_p\rightarrow {\mathrm{GL}}(1,{\mathbb {C}})\).

According to Bürgisser, \(C_p\) embeds in \({\mathrm{Sp}}(2l,{\mathcal {O}})\) (resp. in \({\mathrm{Sp}}(l,{\mathcal {O}})\) when l is even) provided that \({\mathcal {O}}\) is integrally closed in \({\mathbb {C}}\) [1]. Here as usual, \(l:=|F[\zeta _p],F|\) and F is the field of fractions of \({\mathcal {O}}\). If instead \(\zeta _p\in {\mathcal {O}}\), then \(l=1\) and clearly \(C_p\) embeds in \({\mathrm{GL}}(1,{\mathcal {O}})\) and hence also in \({\mathrm{Sp}}(2,{\mathcal {O}})={\mathrm{Sp}}(2l,{\mathcal {O}})\). Taking this embedding of \(C_p\) and composing it with any homomorphism \(H\rightarrow C_p,\) we get a symplectic representation \(\rho \) of H on \({\mathcal {O}}^{2l}\) for any l (resp. on \({\mathcal {O}}^l\) for l even). For a suitable homomorphism, we know that \(\mathrm{Ind}_H^G(\rho )\) is a faithful representation of G on \({\mathcal {O}}^{2lp^m}\) (resp. on \({\mathcal {O}}^{lp^m}\) for l even) and by Lemma 4 we see that \(\mathrm{Ind}_H^G(\rho )\) is symplectic. Hence, we see that E(m, p) embeds as a subgroup of \({\mathrm{Sp}}(2lp^m,{\mathcal {O}})\) for any l and as a subgroup of \({\mathrm{Sp}}(lp^m,{\mathcal {O}})\) in the case when l is even. Since \({p^\circ }(E(m,p))=2p^m\), this shows that \(2p^m\) divides \({p^\circ }({\mathrm{Sp}}(2lp^m,{\mathcal {O}}))\) always and that \(2p^m\) divides \({p^\circ }({\mathrm{Sp}}(lp^m,{\mathcal {O}}))\) in the case when l is even. \(\square \)

Corollary 3 and Proposition 5 together complete the proof of Theorem 1.

We finish by pointing out that we have not computed \({p^\circ }({\mathrm{Sp}}(2n,{\mathcal {O}}))\) for general \({\mathcal {O}}\) when \(n<p-1\); to do this one would have to know which metacyclic groups \(C_p{\rtimes }C_k\) with k coprime to p admit low-dimensional symplectic representations.