Definition 2·1. The Burnside ring $\mathrm{B}(G)$ is the ring of formal linear combinations of isomorphism classes of finite G-sets, modulo the relations

\begin{equation*}[S] + [T] = [S \sqcup T], \quad [S]\cdot[T] = [S \times T]\end{equation*}

for G-sets S, T ( $S \sqcup T$ being the disjoint union and $S \times T$ the Cartesian product).

Definition 2·2. For $D \leq G$ define the maps $\mathrm{Res}_D^G \,:\, \mathrm{B}(G) \to \mathrm{B}(D)$ , $\mathrm{Ind}_D^G \,:\, \mathrm{B}(D) \to \mathrm{B}(G)$ by

\begin{equation*}\mathrm{Res}_D^G H = \sum_{x \in H \backslash G / D} D \cap H^{x^{-1}},\quad \mathrm{Ind}_D^G H' = H',\end{equation*}

for $H \leq G$ , $H' \leq D$ .

Definition 2·3. Let $K / \mathbb{Q}$ be a finite Galois extension. For $\rho \in \mathrm{Rep}(G)$ with $\mathbb{Q}(\rho) \subset K$ , define

\begin{equation*}N_{K / \mathbb{Q}}(\rho) = \bigoplus_{\sigma \in \mathrm{Gal}(K / \mathbb{Q})} \rho^{\sigma}, \end{equation*}

where $\rho^{\sigma}$ is the representation of G with $\mathrm{Tr} \rho^{\sigma}(g) = \sigma(\mathrm{Tr} \rho(g))$ for all $g \in G$ .

Definition 2·4. Let G be a finite group, and let $K / \mathbb{Q}$ be a finite Galois extension. An element $\Theta \in \mathrm{B}(G)$ is a K-relation for G, if there exists $\rho \in \mathrm{Rep}(G)$ with $\mathbb{Q}(\rho) \subset K$ and

\begin{equation*}\mathbb{C}[G / \Theta] = N_{K / \mathbb{Q}}(\rho).\end{equation*}

Proof. If $\mathbb{C}[G / \Theta] = N_{K /\mathbb{Q}}(\rho)$ with $\mathbb{Q}(\rho) \subset K$ , then $\tau = \bigoplus_{\sigma' \in \mathrm{Gal}(K / L) } \rho^{\sigma'}$ satisfies $\mathbb{Q}(\tau) \subset L$ and $N_{K / \mathbb{Q}}(\rho) = N_{L / \mathbb{Q}}(\tau)$ .

Proof. Let $\mathbb{C}[G / \Theta] = N_{K / \mathbb{Q}}(\rho)$ , $\mathbb{C}[G / \Psi] = N_{K / \mathbb{Q}}(\tau)$ for $\rho$ , $\tau \in \mathrm{Rep}(G)$ . Then $\mathbb{C}[G / \Theta \pm \Psi] = N_{K / \mathbb{Q}}(\rho \pm \tau)$ , $\mathbb{C}[H / \mathrm{Res}_{H}^G \Theta] = N_{K / \mathbb{Q}}(\mathrm{Res}_H^G \rho)$ , and $\mathbb{C}[G' / \mathrm{Ind}_G^{G'}\Theta] = N_{K / \mathbb{Q}}(\mathrm{Ind}_{G}^{G'} \rho)$ , proving (i), (ii), (iii). One has $\mathbb{C}[G / H_i]^{N} = \mathbb{C}[G / N H_i]$ as G-representations (cf. [ Reference Dokchitser and Dokchitser10 , theorem 2·8]). Thus $\mathbb{C}[G / \Theta]^{N} = N_{K / \mathbb{Q}}(\rho^N) = \mathbb{C}[G / N / N\cdot \Theta / N]$ , implying (iv). For (v), it is clear that the relation can be lifted to G.

Remark 2·7 (Brauer relations). When $\rho = 0$ , $\Theta$ is called a Brauer relation (cf. [ Reference Dokchitser and Dokchitser10 , section 2·1]). Non-zero elements of $\Theta \in \mathrm{B}(G)$ with $\mathbb{C}[G / \Theta] = 0$ are instances of non-isomorphic G-sets with isomorphic permutation representations. Indeed, writing $\Theta = X - X'$ as a difference of finite G-sets, one has $X \not\simeq X'$ but $\mathbb{C}[X] = \mathbb{C}[X']$ . The rank of the space of Brauer relations in $\mathrm{B}(G)$ is equal to the number of non-cyclic subgroups of G up to conjugacy [ Reference Dokchitser and Dokchitser10 , remark 2·5]. Brauer relations have been classified by Bartel and Tim Dokchitser in [ Reference Bartel and Dokchitser4 ].

Notation 2·8. There is an inclusion of rings

\begin{equation*} \mathrm{Perm}(G) \subset R_{\mathbb{Q}}(G) \subset \mathrm{Char}_{\mathbb{Q}}(G), \end{equation*}

where $\mathrm{Perm}(G)$ is the ring of virtual permutation representations, $R_{\mathbb{Q}}(G)$ the ring of rational representations, and $\mathrm{Char}_{\mathbb{Q}}(G)$ the ring of representations with $\mathbb{Q}$ -valued characters. These rings have the same rank as $\mathbb{Z}$ -modules. A basis of $\mathrm{Char}_{\mathbb{Q}}(G)$ is given by the distinct $N_{\mathbb{Q}(\chi) / \mathbb{Q}}(\chi)$ as $\chi$ ranges over the complex irreducible representations of G.

Definition 2·9. Define the group

\begin{equation*}\hat{C}(G) = \mathrm{Char}_{\mathbb{Q}}(G) / \mathrm{Perm}(G).\end{equation*}

Remark 2·10. It follows from Artin’s induction theorem that $\hat{C}(G)$ is finite with exponent dividing $|G|$ . When G is abelian or dihedral, $\hat{C}(G) = 1$ . The order of $N_{\mathbb{Q}(\chi) / \mathbb{Q}}(\chi)$ in $\mathrm{Char}_{\mathbb{Q}}(G) / R_{\mathbb{Q}}(G)$ is the Schur index $m_{\mathbb{Q}}(\chi)$ of $\chi$ . The group $R_{\mathbb{Q}}(G) / \mathrm{Perm}(G)$ is not trivial in general (cf. [ Reference Serre19 , exc. 13·4]). It is trivial when G is a p-group [ Reference Segal18 ]. Given a finite group G, the structure of $\hat{C}(G)$ can be computed using the results of [ Reference Bartel and Dokchitser5 ].

Proof. Suppose that $\mathbb{C}[G / \Theta] = N_{K / \mathbb{Q}}(\rho)$ for some representation $\rho$ of G with $\mathbb{Q}(\rho) \subset K$ . Let $L_{\tau} = K \cap \mathbb{Q}(\chi_{\tau})$ . A basis of representations of G with K-valued character is given by

\begin{equation*}\{ \varphi_{\tau} = \bigoplus_{ \sigma \in \mathrm{Gal}(\mathbb{Q}(\chi_{\tau}) / L_{\tau})} \chi_{\tau}^{\sigma} \,:\, \tau \in \mathrm{Irr}_{\mathbb{Q}}(G) \}.\end{equation*}

Thus $\rho = \bigoplus_{\tau} \varphi_{\tau}^{\oplus n_{\tau}}$ for some $n_{\tau} \in \mathbb{Z}$ and so

\begin{align*}N_{K / \mathbb{Q}}(\rho) & = \bigoplus_{\tau \in \mathrm{Irr}_{\mathbb{Q}}(G)} N_{K / \mathbb{Q}}(\varphi_{\tau})^{\oplus n_{\tau}} = \bigoplus_{\tau \in \mathrm{Irr}_{\mathbb{Q}}(G)} N_{L_{\tau} / \mathbb{Q}}(\varphi_{\tau})^{\oplus n_{\tau} [K \,:\, L_{\tau}]}\nonumber\\& = \bigoplus_{\tau \in \mathrm{Irr}_{\mathbb{Q}}(G)} N_{\mathbb{Q}(\chi_{\tau}) / \mathbb{Q}}(\chi_{\tau})^{\oplus n_{\tau} [K \,:\, L_{\tau}]}.\end{align*}

Example 2·12 (Cyclic Groups). Let $G = C_n$ . For $d \mid n$ , let $\chi_d$ be an order d character of G with $\mathbb{Q}(\chi_d) = \mathbb{Q}(\zeta_d)$ . Then

\begin{equation*}\mathbb{C}[G / \Psi_d] = N_{\mathbb{Q}(\chi_d) / \mathbb{Q}}(\chi_d), \quad \Psi_d = \sum_{d' \mid d} \mu(d/ d') C_{n / d'} \in \mathrm{B}(G),\end{equation*}

where $\mu$ is the Möbius function. Moreover $\Psi_d$ is the unique element of $\mathrm{B}(G)$ with $\mathbb{C}[G / \Psi_d] = N_{\mathbb{Q}(\chi_d) / \mathbb{Q}}(\chi_d)$ . Indeed if $\Psi' \in \mathrm{B}(G)$ also satisfies $\mathbb{C}[G / \Psi'] = N_{\mathbb{Q}(\psi_d) / \mathbb{Q}}(\psi_d)$ , then $\Psi_d - \Psi'$ is a Brauer relation. But cyclic groups have no Brauer relations (cf. [ Reference Dokchitser and Dokchitser10 , exercise 2·2]) so $\Psi_d = \Psi'$ .

If $\Theta$ is a K-relation for G, then by Proposition 2·11 $\Theta = \sum_{d \mid n} a_d \Psi_d$ where $a_d \in \mathbb{Z}_{\geq 1}$ is divisible by $[K \,:\, K \cap \mathbb{Q}(\zeta_d)]$ .

Example 2·13. Let E be an elliptic curve over a number field L, and $F / L$ a finite Galois extension with $G = \mathrm{Gal}(F / L)$ . If $H , H' \leq G$ are conjugate subgroups, then $F^{H} \simeq F^{H'}$ and so $C_{E / F^{H}} = C_{E / F^{H'}}$ ( $C_{E / F^{H}}$ defined in Section 1·5). Thus the function $C \,:\, \mathrm{B}(G) \to \mathbb{Q}^{\times}$ sending $H \leq G$ to $C_{E / F^{H}}$ is well-defined.

Definition 2·14 [ Reference Dokchitser and Dokchitser10 , definition 2·33]. For $D \leq G$ , say a function $f \,:\, \mathrm{B}(G) \to A$ is D-local if there exists a function $f_D$ on $\mathrm{B}(D)$ such that $f(H) = f_D(\mathrm{Res}_D H)$ for all $H \leq G$ . If this is the case, write

\begin{equation*}f = (D, f_D).\end{equation*}

Example 2·15. Let $G = \mathrm{Gal}(F / L)$ be the Galois group of an extension of number fields. Let $\mathfrak{p}$ be a prime of L, and $\mathfrak{q}$ a prime of F above $\mathfrak{p}$ with decomposition group $D_{\mathfrak{q}}$ . For $H \leq G$ , the primes above $\mathfrak{p}$ in $F^H$ are in one-to-one correspondence with elements of $H \backslash G / D_{\mathfrak{q}}$ . For $\lambda \in \mathbb{Q}^{\times}$ , the function $f \,:\, \mathrm{B}(G) \to \mathbb{Q}^{\times }$ ,

\begin{equation*}f(H) = \lambda^{\# \{ \text{primes above $\mathfrak{p}$ in $F^{H}$} \} } = \lambda^{| H \backslash G / D_{\mathfrak{q}}|},\end{equation*}

that counts the primes above $\mathfrak{p}$ is a above $\mathfrak{p}$ is a $D_{\mathfrak{q}}$ -local function with $f = (D_{\mathfrak{q}}, \lambda)$ .

Example 2·16. In Example 2·13, the function C factors as a product of local functions. Recall that $C_{E / F^{H}} = \prod_v C_v(E / F^{H}, \omega)$ (as defined in Section 1·5) for a choice of invariant differential $\omega$ on $E / L$ , where v ranges over all finite places of $F^{H}$ .

For each finite place v of L, let $D_v = \mathrm{Gal}(F_w / L_v) \leq G$ be the decomposition group at v for a choice of place w in F above v. Let $C_{D_v} \,:\, \mathrm{B}(D_v) \to \mathbb{Q}^{\times}$ be the function given by

\begin{equation*} C_{D_v}(H') = C(E / (F_w)^{H'}, \omega),\end{equation*}

for $H' \leq D_v$ . Then $ C = \prod_v (D_v, C_{D_v}),$ with the product ranging over all finite places v of L. Indeed, for $H \leq G$ ,

\begin{equation*} C(H) = \prod_v C_{D_v}(\mathrm{Res}_{D_v} H) = \prod_{v} \prod_{v' \mid v} C(E / (F^{H})_{v'}, \omega) = C_{E / F^{H}}, \end{equation*}

as the double product ranges over all finite places of $F^{H}$ .

Definition 2·17 [ Reference Dokchitser and Dokchitser10 , definition 2·35]. Consider $I \triangleleft D \leq G$ with $D / I$ cyclic and $\psi(e, f)$ a function of e, $f \in \mathbb{N}$ . Define a function on $\mathrm{B}(G)$ by

\begin{equation*}(D, I, \psi) \,:\, H \mapsto \prod_{x \in H \backslash G / D} \psi\left( \tfrac{|I|}{|H \cap I^{x}|} , \tfrac{[D \,:\, I]}{[H \cap D^{x} \,:\, H \cap I^x]} \right).\end{equation*}

This is a D-local function on $\mathrm{B}(G)$ with

\begin{equation*}(D, I, \psi) = \left(D, U \mapsto \psi\left(\tfrac{|I|}{|U \cap I|}, \tfrac{|D|}{|UI|}\right)\right),\end{equation*}

(since $(U \cap D) / (U \cap I) \simeq UI / I$ for $U \leq D$ ).

Definition 2·19. Let $f \,:\, \mathrm{B}(G) \to \mathbb{Q}^{\times}$ be a linear function, K a number field. Say that f is trivial on K-relations for G if $f(\Theta) \equiv 1 \mod N_{K / \mathbb{Q}}(K^{\times})$ whenever $\Theta$ is a K-relation for G.

Example 2·20. Let $G = C_n$ , K a number field and $\Theta$ a K-relation. Consider a function $f \,:\, \mathrm{B}(G) \to \mathbb{Q}^{\times}$ . As in Example 2·12, $\Theta = \sum_{d \mid n} a_d \Psi_d$ , where $a_d$ is divisible by $[K \,:\, \mathbb{Q}(\zeta_d) \cap K ] $ . If $f(\Psi_d) \in N_{\mathbb{Q}(\zeta_d) / \mathbb{Q}} \in N_{\mathbb{Q}(\zeta_d) / \mathbb{Q}}(\mathbb{Q}(\zeta_d)^{\times})$ for each $d | n$ , then by Lemma 2·23 $f(\Theta) \in N_{K / \mathbb{Q}}(K^{\times})$ . Therefore to show that a function f on $\mathrm{B}(G)$ is trivial on K-relations for any field K, it is enough to show that it is trivial on the $\mathbb{Q}(\zeta_d)$ -relation $\Psi_d$ for each $d \mid n$ .

Proof. This follows from Proposition 2·6, using that if $\Theta \in \mathrm{B}(G)$ is a K-relation for G, then $\mathrm{Res}_D \Theta$ is a K-relation for D and $N \cdot \Theta / N$ is a K-relation for $G / N$ .

Proof. Let

\begin{equation*} {}\mathbb{C}[G / \Theta] = N_{K / \mathbb{Q}}(\rho) = \bigoplus_{\tau \in \mathrm{Irr}_{\mathbb{Q}}(G)} N_{\mathbb{Q}(\chi_{\tau}) / \mathbb{Q}}(\chi_{\tau})^{\oplus k_{\tau}} {}\end{equation*}

be a K-relation, where $\chi_{\tau}$ is a complex irreducible constituent of $\tau \in \mathrm{Irr}_{\mathbb{Q}}(G)$ and $[K \,:\, K \cap \mathbb{Q}(\chi_{\tau}) ]$ divides $k_{\tau}$ . Then $\mathbb{C}[G / \Theta] = \bigoplus_{\tau \in \mathrm{Irr}_{\mathbb{Q}}(G)} \mathbb{C}[G / \Theta_{\chi_{\tau}}]^{\oplus k_{\tau}}$ and

\begin{equation*} {}f(\Theta) = f(\Psi)\prod_{\tau \in \mathrm{Irr}_{\mathbb{Q}}(G)} f(\Theta_{\chi_{\tau}})^{k_{\tau}},\end{equation*}

where $\Psi = \Theta - \sum_{\tau \in \mathrm{Irr}_{\mathbb{Q}}(G)} k_{\tau} \Theta_{\chi_{\tau}}$ is a Brauer relation and $f(\Psi) \in N_{K / \mathbb{Q}}(K^{\times})$ by assumption. For (i), the result follows as $f(\Theta_{\tau}) \in N_{\mathbb{Q}(\chi_{\tau}) / \mathbb{Q}}(\mathbb{Q}(\chi_{\tau})^{\times})$ for all $\tau \in \mathrm{Irr}_{\mathbb{Q}}(G)$ , so that $f(\Theta_{\chi_{\tau}})^{k_{\tau}} \in N_{K / \mathbb{Q}}(K^{\times})$ by Proposition 2·11 and Lemma 2·23 below. For (ii), if $K \not\subset \mathbb{Q}(\chi_{\tau})$ then $k_{\tau}$ is even and so $f(\Theta_{\chi_{\tau}})^{k_{\tau}} \in \mathbb{Q}^{\times 2} \subset N_{K/ \mathbb{Q}}(K^{\times})$ . Thus

\begin{equation*} f(\Theta) \equiv \prod_{\tau\in \mathrm{Irr}_{\mathbb{Q}}(G), K \subset \mathbb{Q}(\chi_{\tau})} f(\Theta_{\chi_{\tau}}) \mod N_{K / \mathbb{Q}}(K^{\times }),\end{equation*}

and it follows that $f(\Theta) \in N_{K / \mathbb{Q}}(K^{\times})$ from the assumption that $f(\Theta_{\chi_{\tau}}) \in N_{K / \mathbb{Q}}(K^{\times})$ .

Proof. One has $x^{[K \,:\, L \cap K]} = N_{K / \mathbb{Q}}(N_{L / L \cap K}(z))$ .

Remark 2·24. The proof of Proposition 2·22 does not hold when $\hat{C}(G) \not= 1$ . For example, the quaternion group $G = Q_8$ has $\hat{C}(G) = \mathbb{Z} / 2 \mathbb{Z}$ , generated by the 2-dimensional irreducible representation $\chi$ . This representation has rational character. One has

\begin{equation*} \chi^{\oplus 2} = \mathbb{C}[G / C_1] \ominus \mathbb{C}[G / C_2],\end{equation*}

and so $\Theta = C_1 - C_2$ is a K-relation for all quadratic fields K. However there does not exist $\Theta' \in \mathrm{B}(G)$ such that $\Theta = \Psi + 2 \Theta'$ for $\Psi$ a Brauer relation.

Proof. Let $\Theta$ be a K-relation for G. Then $\Theta = \sum_{d | n } a_d \Psi_d$ , where $a_d$ is divisible by $[K \,:\, \mathbb{Q}(\zeta_d) \cap K]$ and $\Psi_d \in \mathrm{B}(G)$ is as in Example 2·12. One has

\begin{equation*}g(\Psi_d) = \prod_{d' \mid d} (d')^{\mu(d / d')} = \Phi_d(1) = N_{\mathbb{Q}(\zeta_d) / \mathbb{Q}}(1 - \zeta_d), \end{equation*}

where $\Phi_d$ is the d-th cyclotomic polynomial. Thus $g(\Psi_d) \in N_{\mathbb{Q}(\zeta_d) / \mathbb{Q}}(\mathbb{Q}(\zeta_d)^{\times})$ and so g is trivial on K-relations by Example 2·12.

Proof. For $d \mid n$ , one has $c(\Psi_d) = \alpha^{\sum_{d' | d} \mu(d / d')} = 1$ unless $d = 1$ in which case $c(\Psi_1) = \alpha$ . Thus if $\Theta$ is a K-relation then $c(\Theta) = \alpha^{m [K \,:\, \mathbb{Q}]} = N_{K / \mathbb{Q}}(\alpha^{m})$ where $m \geq 1$ .

Proof. If $k \nmid n$ or if $k = 1$ then this is the constant function which is trivial on K-relations by Lemma 2·26, so we may assume $k \mid n$ and $k \gt 1$ . Let $\chi_k$ be a character of G with $\mathbb{Q}(\chi_k) = \mathbb{Q}(\zeta_k)$ . Then

\begin{equation*}\langle \chi_k, \mathbb{C}[C_n / H] \rangle = \begin{cases} 1, & \text{if } k \mid [C_n \,:\, H], \\ 0, & \text{if } k \nmid [C_n \,:\, H] .\end{cases} \end{equation*}

Thus $c(H) = \alpha^{\langle \chi_k , \mathbb{C}[C_n / H] \rangle }\cdot \beta^{1 - \langle \chi_k, \mathbb{C}[C_n / H] \rangle }$ . Therefore

\begin{equation*}d_{\alpha, \beta}(\Psi_d) = (\alpha / \beta)^{\langle \chi_k, N_{\mathbb{Q}(\chi_d) / \mathbb{Q}} (\chi_d )\rangle } \beta^{\sum_{d' | d} \mu(d / d')} = \begin{cases} \beta, & \text{if } d = 1, \\ \alpha / \beta, & \text{if } d = k ,\\ 1, & \text{if } d \not= k, d \gt 1, \end{cases} \end{equation*}

where $\chi_d$ is a character of G of order d. If $\Theta$ is a K-relation for G, then $d_{\alpha, \beta}(\Theta) = \beta^{x[K \,:\, \mathbb{Q}]}\cdot(\alpha / \beta)^{y [K \,:\, K \cap \mathbb{Q}(\zeta_k)]}$ for some $x, y \in \mathbb{Z}$ . One has $\beta^{x[K \,:\, \mathbb{Q}]} = N_{K / \mathbb{Q}}(\beta^x)$ , and $(\alpha / \beta)^{y [K \,:\, K \cap \mathbb{Q}(\zeta_k)]} \in N_{K / \mathbb{Q}}(K^{\times})$ by Lemma 2·23 and the assumption that $\alpha / \beta \in N_{\mathbb{Q}(\zeta_k) / \mathbb{Q}}(\mathbb{Q}(\zeta_k)^{\times})$ . Thus $d_{\alpha, \beta}(\Theta)$ is the norm of an element of K.

Proof. If $2 \nmid n$ , then this is always 1, so assume $2 \mid n$ . Let $\chi_2$ be a character of G of order 2. Then g can be expressed as $g(H) = [C_n \,:\, H]^{\langle \chi_2 ,\ \mathbb{C}[C_n / H] \rangle}$ . For $d \mid n$ ,

\begin{equation*} {}g(\Psi_d) = \prod_{d' \mid d} (d')^{\mu(d / d')\cdot \langle \chi_2, \mathbb{C}[C_n / C_{n / d'}] \rangle} = \begin{cases} {} {}2, & \text{if } d = 2, \\ p, & \text{if } d = 2 p,\ p \text{ prime,}\\ 1, & \text{otherwise}. {}\end{cases} {}\end{equation*}

In each case it is clear that $g(\Psi_d) \in N_{\mathbb{Q}(\zeta_d) / \mathbb{Q}}(\mathbb{Q}(\zeta_d)^{\times})$ and so the result follows by Example 2·12.

Notation 2·30. Let V be a vector space over $\mathcal{K}$ , with $\langle, \rangle$ a non-degenerate $\mathcal{K}$ -bilinear $\mathcal{L}$ -valued pairing on V, where $\mathcal{L}$ is some finite extension of $\mathcal{K}$ . Let $\det(\langle, \rangle | V) \in \mathcal{L}^{\times} / \mathcal{K}^{\times 2}$ denote $\det(\langle e_i, e_j \rangle_{i, j})$ in any $\mathcal{K}$ -basis $\{e_1, \ldots e_n \}$ of V.

Definition 2·31 (Regulator constant). Let G be a finite group, $\tau$ a self-dual $\mathcal{K} G$ -representation. Let $\langle, \rangle$ be a G-invariant, non-degenerate, $\mathcal{K}$ -bilinear pairing on $\tau$ taking values in an extension $\mathcal{L}$ of $\mathcal{K}$ . Let $\Theta = \sum_i n_i H_i \in \mathrm{B}(G)$ be a K-relation for G. Define the regulator constant

\begin{equation*} \mathcal{C}^{\langle, \rangle}_{\Theta}(\tau) = \prod_i \det\left(\frac{1}{|H_i|} \langle, \rangle \mid \tau^{H_i} \right)^{n_i} \in \mathcal{L}^{\times} / \mathcal{K}^{\times 2}.\end{equation*}

Remark 2·32. If $\tau$ and $\langle, \rangle$ are as in the definition, then for each $H \leq G$ the restriction of $\langle, \rangle$ to $\tau^{H}$ is non-degenerate [ Reference Dokchitser and Dokchitser10 , lemma 2·15]. Thus $\det(\langle, \rangle \mid \tau^{H}) \not=0$ so $\mathcal{C}_{\Theta}^{\langle, \rangle}(\tau)$ is non-zero.

Proof. Follow proof of [ Reference Dokchitser and Dokchitser10 , theorem 2·17] where this is proved for $\mathbb{C}[G / \Theta] = 0$ .

Proof. It is enough to prove this for a particular choice of basis of each $\tau^{H_i}$ . Suppose firstly that $\tau = \alpha \oplus \beta$ with $\alpha, \beta$ $\mathbb{Q} G$ -representations and $\mathrm{Hom}_G(\alpha, \beta^{\vee}) = 0$ , where $\beta^{\vee}$ denotes the dual representation of $\beta$ . Then $\alpha$ and $\beta$ are orthogonal with respect to $\langle, \rangle_1$ , $\langle, \rangle_2$ . Since $\tau^{H} = \alpha^H \oplus \beta^H$ for all $H \leq G$ , choosing bases of $\tau^{H_i}$ that respects this decomposition reduces the problem to $\alpha$ and $\beta$ separately. Thus we may assume that $\tau = \rho^{\oplus n}$ where $\rho$ is an irreducible $\mathbb{Q} G$ -representation. As $\rho$ is irreducible it is of the form $\rho = N_{\mathbb{Q}(\chi) / \mathbb{Q}}(\chi)^{\oplus m_{\chi}}$ where $\chi$ is a complex irreducible representation of G and $m_{\chi}$ is the Schur index of $\chi$ .

Let $\langle, \rangle$ be a non-degenerate G-invariant $\mathbb{Q}$ -valued pairing on $\rho$ . Then any other such pairing is of the form $\langle, \rangle_{\alpha}$ for $\alpha \in \mathrm{End}_G(\rho)$ with $\langle x, y \rangle_{\alpha} = \langle \alpha(x), y \rangle$ for x, $y \in \rho$ . Similarly the non-degenerate G-invariant $\mathbb{Q}$ -valued pairings on $\tau$ are in one-to-one correspondence with $\mathrm{End}_G(\tau)$ . Note that

\begin{equation*}\mathrm{End}_G(\tau) = \mathrm{End}_G(\rho^{\oplus n }) \simeq M_n(\mathrm{End}_G(\rho)) \simeq M_n(D),\end{equation*}

where $D = \mathrm{End}_G(\rho)$ is a finite-dimensional division algebra over $\mathbb{Q}$ with centre $\mathbb{Q}(\chi)$ (cf. [ Reference Serre19 , section 12·2]).

Fix a subgroup $H \leq G$ . Viewing $\rho^H$ as a D-vector space with basis $\{v_1, \ldots v_k \}$ , $k = \dim_D \rho^H$ , an endomorphism $\alpha \in D$ acts by the matrix $\alpha I_k$ . The algebra D has dimension $m_{\chi}^2$ over $\mathbb{Q}(\chi)$ , and there exists a splitting field $\mathbb{Q}(\chi) \subset L \subset D$ such that $[D \,:\, L] = \,:\, L] = m_{\chi}$ and $D \otimes_{\mathbb{Q}(\chi)} L \simeq M_{m_{\chi}}(L)$ . Let $\{f_1, \ldots f_{m_{\chi}} \}$ be a basis for D over L. With respect to the basis $\{ f_i v_j \}$ of $\rho^H$ , $\alpha$ acts by a block diagonal matrix consisting of k blocks corresponding to multiplication by $\alpha$ on D as an L-vector space. If we further consider a basis $\{ e_1, \ldots e_t\}$ of L as a $\mathbb{Q}$ -vector space, then $ \mathcal{B} = \{ e_i f_j v_k \}$ is a $\mathbb{Q}$ -basis of $\rho^{H}$ . Induce this to a basis of $(\rho^{H})^{\oplus n}$ , and let C be the matrix of $\langle, \rangle$ on $\rho^H$ with respect to $\mathcal{B}$ . The matrix of $\langle, \rangle_1$ on $\tau^H$ is of the form

\begin{equation*}\left(\begin{smallmatrix}C &\, 0 &\, \cdots &\, 0 \\0 &\, \ddots &\, \ddots &\, \vdots \\\vdots &\, \ddots &\, \ddots &\, 0 \\0 &\, \cdots &\, 0 &\, C\end{smallmatrix}\right)\left(\begin{smallmatrix}A_{11} &\, A_{12} &\, \cdots &\, A_{1n} \\A_{21} &\, \ddots &\, \ddots &\, \vdots \\\vdots &\, \ddots &\, \ddots &\, \vdots \\A_{n1} &\, \cdots &\, \cdots &\, A_{nn}\end{smallmatrix}\right),\quad A_{ij} =\left(\begin{smallmatrix}\alpha_{ij} &\, 0 &\, \cdots &\, 0 \\0 &\, \alpha_{ij} &\, 0 &\, \vdots \\\vdots &\, \ddots &\, \ddots &\, 0 \\0 &\, \cdots &\, 0 &\, \alpha_{ij}\end{smallmatrix}\right)\end{equation*}

with each $A_{i j} \in M_{k \cdot\dim_{\mathbb{Q}}D}(\mathbb{Q})$ a block diagonal matrix consisting of k-blocks $\alpha_{i j}$ , where $\alpha_{i j}$ is the matrix of multiplication by an element of D with respect to the $\mathbb{Q}$ -basis $\{ e_i f_j \}$ of D. In particular, the matrix of $({1}/{|H|}) \langle, \rangle_1$ has determinant

\begin{equation*}\det\left(\tfrac{1}{|H|} \langle, \rangle_1 \mid \tau^{H} \right) = \det\left(\tfrac{1}{|H|} C \right)^n \cdot \det(\Lambda_{\alpha})^{\dim_D \rho^H},\end{equation*}

where $\Lambda_{\alpha}$ is the block matrix $\Lambda_{\alpha} = (\alpha_{i j })_{i, j}$ which does not depend on H.

The determinant of $\Lambda_{\alpha}$ is the norm of an element of $\mathbb{Q}(\chi)^{\times}$ . To see this, view $\Lambda_{\alpha}$ as a block matrix $(m_{x_{i j }})_{i, j}$ , where $m_{x_{i, j}}$ corresponds to the multiplication by $x_{i,j} \in L$ map on L as a $\mathbb{Q}$ -vector space with basis $\{e_i\}$ . Let $R = \{ m_y \,:\, y \in L\}$ . This is a commutative subring of $M_{[L \,:\, \mathbb{Q}]}(\mathbb{Q})$ . We can view $\Lambda_{\alpha}$ as a matrix in $M_n(R)$ , and consider the determinant $\det_R(\Lambda_{\alpha}) \in R$ as R is commutative. By [ Reference Silvester21 , theorem 1],

\begin{equation*}\det(\Lambda_{\alpha}) = \det(\det\nolimits_R(\Lambda_{\alpha})) = \det(m_z) = N_{L / \mathbb{Q}}(z), \quad z = \det(x_{i,j})_{i,j} \in L,\end{equation*}

hence $\det(\Lambda_{\alpha}) \in N_{\mathbb{Q}(\chi) / \mathbb{Q}}(\mathbb{Q}(\chi)^{\times})$ .

The matrix of $\langle , \rangle_2$ with respect to our chosen basis is of the same form and

\begin{equation*}\det\left(\tfrac{1}{|H|} \langle , \rangle_2 \mid \tau^{H}\right) = \det\left(\tfrac{1}{|H|} C\right)^n \cdot \det(\Lambda_{\beta})^{\dim_D \rho^H}\end{equation*}

with $\Lambda_{\beta}$ not depending on H and $\det(\Lambda_{\beta}) \in N_{\mathbb{Q}(\chi) / \mathbb{Q}}(\mathbb{Q}(\chi)^{\times})$ . Note that

\begin{align*} \sum_i n_i \dim_D \rho^{H_i} & = \tfrac{1}{[D \,:\, \mathbb{Q}]}\sum_i n_i \dim_{\mathbb{Q}} \rho^{H_i} = \tfrac{1}{[D \,:\, \mathbb{Q}]} \sum_i n_i \langle \mathrm{Ind}_{H_i}^G \unicode{x1D7D9}, \rho \rangle\\& = \tfrac{1}{[D \,:\, \mathbb{Q}]} \langle \mathbb{C}[G / \Theta] , \rho \rangle = \langle \mathbb{C}[G / \Theta], \chi \rangle.\end{align*}

Taking the product of the ratio of the determinants over all $H_i$ we have

\begin{equation*}\prod_i \big{(}\det(\tfrac{1}{|H_i|} \langle, \rangle_1 \mid \tau^{H_i})^{n_i} / \det(\tfrac{1}{|H_i|} \langle, \rangle_2 \mid \tau^{H_i})^{n_i}\big{)} = N_{\mathbb{Q}(\chi) / \mathbb{Q}}(s)^{\langle \mathbb{C}[G / \Theta], \chi \rangle}\end{equation*}

for some $s \in \mathbb{Q}(\chi)$ . The result follows as by Proposition 2·11, $\langle \mathbb{C}[G / \Theta], \chi \rangle$ is divisible by $[K \,:\, K \cap \mathbb{Q}(\chi)]$ , and by Lemma 2·23 $N_{\mathbb{Q}(\chi) / \mathbb{Q}}(s)^{[K \,:\, K \cap \mathbb{Q}(\chi)]} \in N_{K / \mathbb{Q}}(K^{\times})$ .

Example 2·37 (Permutation representations, [ Reference Dokchitser and Dokchitser10 , example 2·19]). Consider $\tau = \mathbb{Q}[G / D]$ for some subgroup $D \leq G$ . By considering the standard ( $\mathbb{Q}$ -valued) pairing on $\tau$ , the elements of $G /D$ become an orthonormal basis. The space of invariants $\rho^H$ for $H \leq G$ has a basis a basis consisting of H-orbit sums of these basis vectors. Then $\det(\langle, \rangle \mid \tau^H)$ is a product of the lengths of these orbits, and

\begin{equation*}\det\left(\tfrac{1}{|H|} \langle, \rangle \mid \tau^H \right) = \prod_{w \in H \backslash G / D} \tfrac{1}{|H|} \cdot \tfrac{|H|}{|\mathrm{Stab}_H(wD)|} = \prod_{w \in H \backslash G / D} \frac{1}{|H \cap D^{w}|}.\end{equation*}

By Corollary 2·36(2), this computation can be used to compute $\mathcal{C}_{\Theta}(\rho)$ for any rational representation $\rho$ that lies in the virtual permutation ring of G and any K-relation $\Theta$ for G.

Proof. This is due to $\tau \otimes \mathbb{C}$ admitting a non-degenerate alternating G-invariant pairing, so that in an appropriate basis the relevant determinant is a square (see [ Reference Dokchitser and Dokchitser10 , theorem 2·24, Corollary 2·25]).

Definition 2·39. Let $\tau$ be a $\mathbb{Q} G$ -representation with a non-degenerate $\mathbb{Q}$ -valued G-invariant bilinear pairing $\langle, \rangle$ . Define

\begin{equation*} \mathcal{D}_{\tau} \,:\, H \mapsto \det\left(\tfrac{1}{|H|} \langle, \rangle \mid \tau^{H} \right) \in \mathbb{Q}^{\times} / \mathbb{Q}^{\times 2},\end{equation*}

where the determinant is taken with respect to any rational basis of $\tau^{H}$ .

Example 2·40. Using the bases and pairings defined in Example 2·37, $\mathcal{D}_{\unicode{x1D7D9}}$ is the function $H \mapsto {1}/{|H|}$ and $\mathcal{D}_{\mathbb{Q}[G]}$ is the constant function $H \mapsto 1$ .

Proof. The proof of the analogous statement for Brauer relations [ Reference Dokchitser and Dokchitser10 , lemma 2·43] works for K-relations also.

Proof. (i) follows from Proposition 2·41, (ii) follows from the definition and noting that for $H \leq D$ , $(\mathrm{Res}_D \tau)^{H} \simeq \tau^{H}$ as vector spaces.

Proof. By Corollary 2·43, $\mathcal{D}_{\mathbb{Q}[G / D]} \sim_K (D, H \mapsto [D \,:\, H])\stackrel{\text{Lem } 2.25}{\sim_K} 1$ , which proves (i). For (ii), since G is cyclic, every rational representation of G can be written as a combination of permutation representations. Thus $\tau = \oplus_i \mathbb{Q}[G / H_i]^{\oplus n_i}$ for $H_i \leq G$ and so $\mathcal{C}_{\Theta}(\tau) \equiv \prod_i \mathcal{C}_{\Theta}(\mathbb{Q}[G / H_i])^{n_i} \equiv 1 \mod N_{K / \mathbb{Q}}(K^{\times})$ by (i) as each $H_i$ is cyclic.

Proof. By multiplicativity of regulator constants, it is enough to show this for irreducible rational representations. If $\tau \in \mathrm{Irr}_{\mathbb{Q}}(G)$ is non-trivial, then $\tau \otimes_{\mathbb{Q}} \mathbb{C} = \chi \oplus \overline \chi$ where $\chi$ is not self-dual. Thus $\mathcal{C}_{\Theta}(\tau) \in N_{K / \mathbb{Q}}(K^{\times})$ by Proposition 2·38. Therefore it remains to show that $\mathcal{D}_{\unicode{x1D7D9}} \sim_K 1$ and one has

\begin{equation*} \mathcal{D}_{\unicode{x1D7D9}} \sim_K \mathcal{D}_{\mathbb{Q}[G]} \stackrel{\text{Cor 2.43}}{\sim_K} 1. \end{equation*}

Notation 2·36.

1. (1) $E / \mathcal{K}$ has semistable reduction.

   1. (i) $(\text{1G})$ , E has good reduction, $\mathcal{L} = \mathcal{K}$ .

   2. (ii) $(\text{1S})$ , E has split multiplicative reduction, $\mathcal{L} = \mathcal{K}$ .

   3. (iii) $(\text{1NS})$ , E has non-split multiplicative reduction, $\mathcal{L}$ is quadratic unramified.

2. (2) $E / \mathcal{K}$ has additive reduction. In this case assume that $\mathcal{K}$ has residue characteristic $l \gt 3$ . Let $\Delta_{E}$ and $c_6$ be the standard invariants for some model of $E / \mathcal{K}$ and let $\mathfrak{e} = {12}/{\mathrm{gcd}(12, \mathrm{ord} \Delta_{E})}$ .

   1. (i) $(\text{2C})$ , E has potentially good reduction, $\mu_{\mathfrak{e}} \subset \mathcal{K}$ , $\mathcal{L} = \mathcal{K}(\sqrt[\mathfrak{e}]{\Delta_{E}}) {} {} {} {}$ is a cyclic extension of $\mathcal{K}$ .

   2. (ii) $(\text{2D})$ , E has potentially good reduction, $\mu_{\mathfrak{e}} \not\subset \mathcal{K}$ , $\mathcal{L} = \mathcal{K}(\mu_{\mathfrak{e}}, {} {} {} {} {} {}\sqrt[\mathfrak{e}]{\Delta_{E}})$ is a dihedral extension of $\mathcal{K}$ .

   3. (iii) $(\text{2M})$ , E has potentially multiplicative reduction, $\mathcal{L} = \mathcal{K}(\sqrt{-c_6})$ .

These are the minimal Galois extensions over which E attains split semistable reduction. As in [ Reference Dokchitser and Dokchitser10 , hypothesis 3·6], we claim that we may assume that $\mathcal{L}$ is a subfield of $\mathcal{F}$ when proving 3·4.

Proof. Let $G = \mathrm{Gal}(\mathcal{M} / \mathcal{K})$ , $N = \mathrm{Gal}(\mathcal{M} / \mathcal{F})$ . If $\tau$ is a self-dual representation of $G / N$ then we can inflate it to a representation of G and one has $({-}1)^{\langle \tau, \mathcal{W} \rangle_{G / N}} = ({-}1)^{\langle \tau, \mathcal{V} \rangle_{G}} = w(E / \mathcal{K}, \tau) / w(\tau)^2$ by assumption. Let $K / \mathbb{Q}$ be quadratic and let $\Theta$ be a K-relation for $G / N$ . By Proposition 2·6(5), the lift $\tilde{\Theta}$ of $\Theta$ to $\mathrm{B}(G)$ is a K-relation for G, and moreover $C_{G / N}(\Theta) = C_G(\tilde{\Theta})$ . The virtual representation $\mathcal{V} \ominus \mathcal{W}$ has $(\mathcal{V} \ominus \mathcal{W})^{N} = 0$ , so that $\mathcal{C}_{\tilde{\Theta}}(\mathcal{V} \ominus \mathcal{W}) = 1$ . Since $\mathcal{C}_{\tilde{\Theta}}(\mathcal{V} \ominus \mathcal{W}) = \mathcal{C}_{\tilde{\Theta}}(\mathcal{V}) / \mathcal{C}_{\tilde{\Theta}}(\mathcal{W})$ by Corollary 2·36(ii), we have $\mathcal{C}_{\tilde{\Theta}}(\mathcal{V}) = \mathcal{C}_{\tilde{\Theta}}(\mathcal{W}) = \mathcal{C}_{\Theta}(\mathcal{W})$ . The result follows since $C_G(\tilde{\Theta}) \equiv \mathcal{C}_{\tilde{\Theta}}(\mathcal{V}) \mod N_{K / \mathbb{Q}}(K^{\times})$ .

Notation 3·10. We break up the functions

\begin{equation*}C_D = c \cdot \omega, \quad \mathcal{D}_{V} = a \cdot d,\end{equation*}

with

\begin{equation*}\begin{array}{l@{\quad}l} {}c(H) = c(E / \mathcal{F}^{H}), & a(H) = \det(\langle, \rangle \mid V^{H}), \\ \\ {}\omega(H) = \left|\frac{\omega_{E / \mathcal{K}}^0}{\omega_{E / \mathcal{F}^{H}}^0} \right|_{\mathcal{F}^{H}}, & d(H) = |H|^{-\dim V^{H}}.\end{array}\end{equation*}

The rational pairing on V is obtained by writing V as a combination of permutation representations and using the pairings defined in Example 2·37. These functions are well-defined on conjugacy classes and define functions on $\mathrm{B}(D)$ taking values in $\mathbb{Q}^{\times} / \mathbb{Q}^{\times 2}$ .

Proof. Follow the proof of [ Reference Dokchitser and Dokchitser10 , lemma 3·13].