2·1. The hypotheses

Let A be an abelian variety of dimension d, defined over a number field k. Let p be an odd prime number and let $F/k$ be a cyclic field extension of degree $p^n$ , for some natural number n. We write $A^t$ for the dual abelian variety.

As in [Reference Bley and Macias Castillo6, Section 2], we will assume the validity of the following list of hypotheses:

1. (a) $p \nmid |A(k)_{\rm tor}| \cdot |A^t(k)_{\rm tor}|$ ;

2. (b) p does not divide the Tamagawa number of A at any place of k at which it has bad reduction;

3. (c) has good reduction at all p-adic places of k;

4. (d) p is unramified in $F/\mathbb Q$ ;

5. (e) no place of bad reduction of A is ramified in $F/k$ ;

6. (f) if a place v of k ramifies in $F/k$ then no point of order p of the reduction of A is defined over the residue field of v;

7. (g) ${\rm{III}}(A_{F})$ is finite;

8. (h) ${\rm{III}}_p(A_{F^H})$ vanishes for all non-trivial subgroups H of G;

9. (i) the group $H^1({\textrm{Gal}}(k(A[p^n])/k),A[p^n])$ vanishes.

The hypothesis (i) recovers hypothesis 10·1 from [Reference Burns and Macias Castillo10] in our setting and will hence allow us to apply Theorem 10·3 of loc. cit. We recall that it is widely satisfied. For instance, it holds whenever multiplication-by-‘ $-1$ ’ belongs to the image of the canonical Galois representation $G_k\to{\rm Aut}_{\mathbb{F}_p}(A[p])$ . In particular, if A is an elliptic curve then this hypothesis excludes only finitely many primes, by a result of Serre. Moreover, if A is an elliptic curve and k does not contain any p-th roots of unity, Lawson and Wuthrich have recently shown that hypothesis (i) is valid in all but certain exceptional cases that, in particular, all have $p\leq 11$ (see [Reference Lawson, Wuthrich, Loeffler and Zerbes21, Theorem 2, Section 6]).

Our main result will concern an ‘equivariant regulator’ ${\rm Reg}_{A,F/k,j}$ in $\mathbb C_p[G]^\times/\mathbb Z_p[G]^\times$ for each isomorphism $j:\mathbb C\cong\mathbb C_p$ , which we now proceed to define. Throughout the construction of this element, we will always use j to implicitly identify $\widehat G$ with ${\textrm{Hom}}_\mathbb Z(G,\mathbb C_p^\times)$ .

2·2. Néron–Tate regulators

For $0\leq r\leq n$ we denote by $J_r$ the subgroup of G of order $p^{n-r}$ and set $F_r\;:\!=\;F^{J_r}$ and $\Gamma_r\;:\!=\;G/J_r$ . Clearly, $[F_r:k] = |\Gamma_r| = p^r$ .

Our hypotheses allow us to apply a result of Yakovlev [Reference Yakovlev31] in order to restrict the Galois structure of the Mordell–Weil groups $A(F)_p$ and $A^t(F)_p$ as follows. For any natural number m we write [m] for the set $\{1,\ldots,m\}$ .

By [Reference Bley and Macias Castillo6, Proposition 2·2 and (2·1)] we may and will fix a set of non-negative integers $\{m_{r}:0\leq r\leq n\}$ and subsets

\begin{eqnarray*}\mathcal{P}_{(r)} &=& \{P_{(r,j)}\;:\;j\in[m_r]\} \subseteq A(F_r), \\\mathcal{P}^t_{(r)}&=&\{P^t_{(r,j)}\;:\;j \in[m_r]\}\subseteq A^t(F_r)\end{eqnarray*}

such that the $\mathbb Z_p[\Gamma_r]$ -modules generated by each of the points $P_{(r,j)}$ and $P^t_{(r,j)}$ are free of rank one and there are direct sum decompositions of $\mathbb Z_p[G]$ -modules

(1) \begin{equation} A(F)_p = \bigoplus_{r=0}^{n}\bigoplus_{j=1}^{m_r}\mathbb Z_p[\Gamma_r]\cdot P_{(r,j)} \,\,\text{ and }\,\, A^t(F)_p = \bigoplus_{r=0}^{n}\bigoplus_{j=1}^{m_r}\mathbb Z_p[\Gamma_r]\cdot P^t_{(r,j)}.\end{equation}

We set

\begin{equation*}\mathcal{P} \;:\!=\; \bigcup_{r=0}^n \mathcal{P}_{(r)}, \quad \mathcal{P}^t \;:\!=\; \bigcup_{r=0}^n \mathcal{P}^t_{(r)}.\end{equation*}

By ordering our fixed choice of points $\mathcal{P}$ and $\mathcal{P}^t$ lexicographically, we obtain a ‘regulator matrix’

\begin{equation*}R_{A,F/k}^{\rm NT}(\mathcal{P}, \mathcal{P}^t)\;:\!=\;\left( \sum_{g \in G}\langle gP^t_{(r,j)},P_{(s,i)}\rangle_{A_F}\cdot g^{-1} \right)_{(r,j),(s,i)}\in M_N(\mathbb R[G])\end{equation*}

with $N \;:\!=\; \sum_{r=0}^n m_r$ , and where $\langle -,-\rangle_{A_F}$ denotes the Néron–Tate height pairing for A over F.

For any matrix $X=(x_{(r,j), (s,i)})$ where the indices $\{ (r,j) : 0\leq r \leq n, j\in[m_r] \}$ are ordered lexicographically and for any $0\leq t\leq n$ , we set

\begin{equation*}X_t\;:\!=\;(x_{(r,j),(s,i)})_{r,s\geq t}\end{equation*}

and, given any matrix Y with entries in $\mathbb C[G]$ , resp. $\mathbb C_p[G]$ , and any $\psi\in\widehat G$ , we write $\psi(Y)$ for the matrix with entries in $\mathbb C$ , resp. $\mathbb C_p$ , obtained after extending $\psi$ to a function on $\mathbb C[G]$ , resp. $\mathbb C_p[G]$ , by linearity and then evaluating $\psi$ at each entry of Y. For any $\psi\in\widehat{G}$ we define an integer $t_\psi$ between 0 and n by the equality $\ker(\psi) = J_{t_\psi}$ and then set

(2) \begin{equation} \varepsilon_\psi(Y) \;:\!=\; \text{det}\left( \psi(Y)_{t_\psi} \right). \end{equation}

We now put

\begin{equation*}m_{\psi}\;:\!=\;\sum_{r=t_\psi}^{n}(n-r)m_r\end{equation*}

and define the $\psi$ -component of the equivariant Néron–Tate regulator by

(3) \begin{equation}{\rm Reg}^{\rm NT}_\psi(\mathcal{P}, \mathcal{P}^t) \;:\!=\;p^{-2m_{\psi}}{\cdot\varepsilon_\psi\bigl(R_{A,F/k}^{\rm NT} (\mathcal{P}, \mathcal{P}^t)\bigr).} \end{equation}

Each regulator term ${\rm Reg}^{\rm NT}_\psi(\mathcal{P}, \mathcal{P}^t)$ coincides with the element $\lambda_\psi(\mathcal{P}, \mathcal{P}^t)$ defined in [Reference Bley and Macias Castillo6, Definition 2·4].

We finally fix a generator $\sigma$ of G and then define a non-zero complex number

(4) \begin{equation}\delta_\psi\;:\!=\;\prod_{r=0}^{t_\psi-1}\left(\psi(\sigma)^{p^r}-1\right)^{m_r}.\end{equation}

2·3. Mazur–Tate regulators

For any $0 \le r \le n$ we recall that, under our given hypotheses, [Reference Burns and Macias Castillo10, Proposition 6·3(ii)] implies that every element of $A^t(F_r)_p$ and $A(F_r)_p$ is ‘locally-normed’. Indeed, from this result one knows that for every finite prime w of F, the G-modules $\mathbb Z_p\otimes_\mathbb Z A^t(F_w)$ and $\mathbb Z_p\otimes_\mathbb Z A(F_w)$ are cohomologically-trivial and from this it follows easily that the p-completions of the subgroups of locally-normed elements in $A^t(F_r)$ and in $A(F_r)$ are equal to the full p-completions $A^t(F_r)_p$ and $A(F_r)_p$ , respectively.

In particular, the construction of Mazur and Tate using the theory of biextensions gives well defined canonical height pairings

(5) \begin{equation}\left\langle\,,\right\rangle_{F/F_r}^{\rm MT}\;:\;A^t(F_r)_p\otimes_{\mathbb Z_p}A(F_r)_p \to J_r\cong I_p(J_r)/I_p(J_r)^2.\end{equation}

Here $I_p(J_r)$ denotes the augmentation ideal in the group ring $\mathbb Z_p[J_r]$ and the isomorphism maps $g\in J_r$ to the class of $g-1$ .

In the sequel we also write $\rho_r$ for the canonical projection $\mathbb Z_p[G]\to\mathbb Z_p[\Gamma_r]$ . For any indices $0\leq r,s\leq n-1$ , any $j\in[m_r]$ and any $i\in[m_s]$ , we set $\ell\;:\!=\;{\rm max}(r,s)$ and then fix any elements $\Psi_{(r,j),(s,i)}$ of $\mathbb Z_p[G]$ which satisfy the equality

(6) \begin{equation} \rho_r(\Psi_{(r,j),(s,i)})\otimes(\sigma^{p^\ell}-1)= \sum_{\gamma\in\Gamma_r}\bigl(\gamma\otimes\langle P^t_{(s,i)},\gamma P_{(r,j)} \rangle_{F/F_\ell}^{\rm MT}\bigr)\end{equation}

in

\begin{equation*}\mathbb Z_p[\Gamma_r]\otimes_{\mathbb Z_p}I_p(J_\ell)/I_p(J_\ell)^2.\end{equation*}

It will be clear from Proposition 3·2 below that such elements always exist.

Remark 2·3. Let $\mathcal I_p(J_r)$ denote the ideal of $\mathbb Z_p[G]$ generated by $I_p(J_r)$ . Then the inclusion $I_p(J_\ell)\subset I_p(J_r)$ induces a canonical inclusion

(7) \begin{equation} {\mathbb{Z}_p}[\Gamma_r] \otimes_{\mathbb{Z}_p} I_p(J_\ell)/I_p(J_\ell)^2 \hookrightarrow {\mathbb{Z}_p}[\Gamma_r] \otimes_{\mathbb{Z}_p} I_p(J_r)/I_p(J_r)^2\cong\mathcal I_p(J_r)/\mathcal I_p(J_r)^2.\end{equation}

Here the isomorphism is canonical (see [Reference Burns, Kurihara and Sano8, Proposition 4·9]). It is then clear that the left-hand side of the equality (6) coincides with the class of $\Psi_{(r,j),(s,i)}\cdot(\sigma^{p^\ell}-1)$ in the quotient $\mathcal I_p(J_r)/\mathcal I_p(J_r)^2$ .

Using these choices we construct a matrix

(8) \begin{equation}\Psi(\mathcal{P}, \mathcal{P}^t)\;:\!=\;\left( \begin{array}{cc|c}\left(\Psi_{(r,j),(s,i)} \right)_{r,s < n} & & \begin{array}{c} 0\\\vdots\\0 \end{array}\\ \hline \begin{array}{ccc} 0 & \ldots & 0 \end{array} && I_{m_{n}}\end{array}\right)\end{equation}

with entries in $\mathbb Z_p[G]$ . Here $I_{m_n}$ denotes the identity $m_n\times m_n$ matrix.

For any $\psi\in\widehat G$ , Lemma 3·4 below will show that the element

(9) \begin{equation}\sum_{\psi\in\widehat G}\varepsilon_\psi(\Psi(\mathcal{P},\mathcal{P}^t))\cdot e_\psi\end{equation}

of $\mathbb C_p[G]$ belongs to $\mathbb C_p[G]^\times$ (so in particular each term $\varepsilon_\psi(\Psi(\mathcal{P},\mathcal{P}^t))$ is non-zero) and is independent, up to multiplication by an element of $\mathbb Z_p[G]^\times$ , of the choices made in (6).

2·4. The equivariant regulator

We may now define our equivariant regulator. We again set $N\;:\!=\;\sum_{r=0}^{n}m_r$ .

Definition 2·4. For a fixed isomorphism $j \colon {\mathbb{C}} \to {\mathbb{C}_p}$ , the class of

(10) \begin{equation}{\rm Reg}_{A, F/k, j} \;:\!=\;(-1)^{N-m_n}\cdot \sum_{\psi \in \widehat G}\frac{j({\rm Reg}^{\rm NT}_\psi(\mathcal{P}, \mathcal{P}^t) \cdot \delta_\psi)} {\varepsilon_{\psi}(\Psi(\mathcal{P},\mathcal{P}^t))}\cdot e_\psi\end{equation}

in $\mathbb{C}_p[G]^\times / \mathbb{Z}_p[G]^\times$ is the (p-primary) equivariant regulator associated to A and $F/k$ .

The claims made in Section 2·3 imply that ${\rm Reg}_{A, F/k, j}$ is independent of the choices of $\mathcal{P}, \mathcal{P}^t, \sigma$ and the matrix $\Psi(\mathcal{P}, \mathcal{P}^t)$ . By abuse of terminology we will often refer to any choice of ${\rm Reg}_{A, F/k, j}$ in $\mathbb{C}_p[G]^\times$ itself as the equivariant regulator.

Remark 2·5. The results of Proposition 3·2 and Lemma 3·4 below combine to show that ${\rm Reg}_{A, F/k, j}$ is precisely the element

\begin{equation*}\sum_{\psi \in \hat G} \lambda_\psi(\mathcal{P}, \mathcal{P}^t) \cdot \varepsilon_\psi(\Phi) \cdot \delta_\psi\cdot e_\psi \quad (\text{ mod } \mathbb{Z}_p[G]^\times)\end{equation*}

which occurs in [Reference Bley and Macias Castillo6, Theorem 2·9]. The main new insight is that the elements $\varepsilon_\psi(\Phi) \in {\mathbb{C}}_p^\times$ which depended upon an essentially unknown matrix $\Phi \in M_N(\mathbb{Z}_p[G])$ can be explicitly determined by (6).

2·5. Statement of the result

The refined Birch and Swinnerton–Dyer conjecture is an equality between analytic and algebraic invariants associated with $A/k$ and $F/k$ . We now briefly describe the analytic part refering the reader to [Reference Bley and Macias Castillo6, Section 2] or [Reference Burns and Macias Castillo10] for further details.

For each $\psi \in \widehat{G}$ we set

\begin{equation*} \mathcal L_\psi^* = \mathcal L_{A, F/k, \psi}^* \;:\!=\; \frac{L_{S_{\rm r}}^*(A, \check{\psi}, 1) \cdot \tau^*(\mathbb{Q}, {\rm ind}_k^{\rm Q}(\psi))^d }{\Omega_A^\psi\cdot w_\psi^d} \in {\mathbb{C}}^\times,\end{equation*}

where;

1. (i) $L^*_{S_r}(A, \psi, 1)$ is the leading term in the Taylor expansion at $z=1$ of the $\psi$ -twisted Hasse–Weil L-function $L_{S_r}(A,\psi,z)$ of A, truncated by removing the Euler factors corresponding to the set $S_{{\rm r}}$ of primes of k which ramify in $F/k$ ;

2. (ii) $\tau^*(\mathbb{Q}, {\rm ind}_k^{\rm Q}(\psi))$ is a suitably modified global Galois–Gauss sum;

3. (iii) $\Omega_A^\psi\cdot w_\psi^d$ is a (suitably normalised) product of periods.

We finally set

\begin{equation*}\mathcal L^* = \mathcal L^*_{A, F/k} \;:\!=\; \sum_{\psi \in \widehat{G}} \mathcal L^*_{A, F/k, \psi} e_\psi \in {\mathbb{C}}[G]^\times\end{equation*}

and note that the element $\mathcal L^*$ defined immediately above [Reference Burns and Macias Castillo10, Theorem 6·5] specialises precisely to our definition.

Without any further mention we will always assume that the functions $L_{S_r}(A,\psi,z)$ have analytic continuation to $z=1$ , so that the above term is well-defined. We also recall that Deligne and Gross have then predicted that these functions should vanish at $z=1$ exactly to order equal to the multiplicity with which the character $\psi$ occurs in the representation $\mathbb C\otimes_\mathbb Z A^t(F)$ of G.

We now formulate the main result of this manuscript. For any $j\;:\;\mathbb C\cong\mathbb C_p$ we write $j_*$ for the associated map $\mathbb C[G]^\times\to \mathbb C_p[G]^\times$ .

Then the p-component of the refined Birch and Swinnerton–Dyer conjecture is valid if and only if, for any $j\;:\;\mathbb C\cong\mathbb C_p$ , the element

(11) \begin{equation}\frac{j_*\left( \mathcal L^*_{A, F/k} \right)}{ {\rm Reg}_{A, F/k, j} }\end{equation}

belongs to $\mathbb{Z}_p[G]^\times$ .

The proof of Theorem 2·1 will occupy the next section.

4·1. The general strategy

We continue to assume the hypotheses of Section 2·1. In particular,as explained in Section 2·3, the result [Reference Burns and Macias Castillo10, Proposition 6·3(ii)] implies that every element of $A^t(k)_p$ and $A(k)_p$ is ‘locally-normed’. Under this condition, Bertolini and Darmon have defined in [Reference Bertolini and Darmon1, Section 3·4·1] and [Reference Bertolini and Darmon2, Section 2·2] a pairing

\begin{equation*}\langle\ , \ \rangle_1 \colon A^t(k)_p \otimes_{\mathbb Z_p} A(k)_p \longrightarrow G \simeq {I}_p(G) / {I}_p(G)^2.\end{equation*}

Although the definition of this pairing is only given in the case that A is an elliptic curve, it extends naturally to our more general setting (see also [Reference Burns and Macias Castillo10,Reference Burns and Macias Castillo10]).

The results of Bertolini and Darmon in [Reference Bertolini and Darmon2, Theorem 2·8 and Remark 2·10] and of Tan in [Reference Tan29, Proposition 3·1] combine to directly show that the pairing $ \langle\ , \ \rangle_1$ coincides with the Mazur–Tate pairing $\langle\ , \ \rangle_{F/k}^{\rm MT}$ . We are therefore left with the task to describe the explicit computation of $\langle\ , \ \rangle_1$ .

Let B be either A or its dual $A^t$ . For a finite set S of non-archimedean places of k we define the generalised Selmer group

\begin{equation*}{\textrm{Sel}}_S^{(p^n)}(B/F) \le H^1(F, B[p^n])\end{equation*}

to be the kernel of the localisation map

\begin{equation*}H^1(F, B[p^n]) \longrightarrow \prod_{w \not\in S(F)} H^1(F_w, B),\end{equation*}

with the product running over all non-archimedean places of F that do not belong to the set S(F) of places that lie above a place in S. We recall that this group is also often referred to as a ‘relaxed Selmer group’.

By Kummer theory we then have

\begin{eqnarray*} {\textrm{Sel}}_S^{(p^n)}(B/F) = \{ \xi \in H^1(F, B[p^n]) \mid {\textrm{res}}_w(\xi) \in \delta_w( B(F_w) / p^nB(F_w)) \text{ for all } w \not\in S(F) \}.\end{eqnarray*}

Here ${\textrm{res}}_w\;:\;H^1(F, B[p^n])\to H^1(F_w,B[p^n])$ denotes the canonical localisation map and $\delta_w\;:\;B(F_w) / p^nB(F_w)\to H^1(F_w,B[p^n])$ is the canonical Kummer map. In particular, when S is taken to be the empty set, one recovers the usual Selmer group ${\textrm{Sel}}^{(p^n)}(B/F)$ associated with multiplication by $p^n$ .

In the following we will employ the notation from [Reference Burns and Macias Castillo10, Section 10·2·1]. We set $Z \;:\!=\; {\mathbb{Z}}/p^n{\mathbb{Z}}$ and $R \;:\!=\; Z [G]$ and define additional R-modules

\begin{equation*}B_S(\mathbb{A}_F) / p^n \;:\!=\; \prod_{w \in S(F)} B(F_w)/p^n B(F_w)\end{equation*}

and

\begin{equation*}H_S^1( \mathbb{A}_F, B[p^n]) \;:\!=\; \prod_{w \in S(F)} H^1(F_w, B[p^n]) .\end{equation*}

In order to define $\langle\ , \ \rangle_1$ , we first recall the construction of a canonical (local duality) perfect pairing

(27) \begin{equation}\langle\ , \ \rangle \colon H_S^1( \mathbb{A}_F, A^t[p^n]) \times H_S^1( \mathbb{A}_F, A[p^n]) \longrightarrow R.\end{equation}

If A is defined over an $\ell$ -adic field L (for some prime $\ell$ ), then we write $\langle \ , \ \rangle_{L, p^n}$ for the local Tate duality pairing obtained by combining the cup product, the Weil pairing and the invariant map as follows:

(28) \begin{eqnarray} H^1(L, A^t[p^n]) \times H^1(L, A[p^n]) &\stackrel{\cup} \longrightarrow& H^2(L, A^t[p^n] \otimes_{\mathbb{Z}_p} A[p^n]) \nonumber \\ &\longrightarrow& H^2(L, \mu_{p^n}) \nonumber\\ &\stackrel{\rm{inv_L}}\longrightarrow& {\mathbb{Q}_p}/{\mathbb{Z}_p}. \end{eqnarray}

Then, for $x=(x_w)_{w \in S(F)} \in H_S^1( \mathbb{A}_F, A^t[p^n])$ and $y=(y_w)_{w\in S(F)} \in H_S^1( \mathbb{A}_F, A[p^n])$ we set

\begin{equation*}\langle x, y \rangle_{S} \;:\!=\; \sum_{w \in S(F)} \langle x_w, y_w \rangle_{F_w, p^n}.\end{equation*}

All values $\langle x, y \rangle_{S}$ in fact belong to $\frac{1}{p^n} {\mathbb{Z}_p}/{\mathbb{Z}_p}$ , which we henceforth identify with $Z = {\mathbb{Z}} / p^n{\mathbb{Z}}$ . We may thus define the pairing (27) by the explicit formula

(29) \begin{equation}\langle x, y \rangle \;:\!=\; \sum_{g \in G} \langle x^g, y \rangle_{S} g^{-1}.\end{equation}

We now recall the explicit definition of $\langle P, Q \rangle_1$ for $P \in A^t(k)$ and $Q \in A(k)$ . Let $\Sigma$ be an admissible set of primes as in [Reference Bertolini and Darmon1, Definition 2·22], [Reference Bertolini and Darmon2, Definition 1·5] or [Reference Burns and Macias Castillo10, Lemma 10·5]. A crucial consequence of the definition of admissibility is that the canonical (diagonal) localisation map

(30) \begin{equation} A(F)/p^nA(F) \longrightarrow A_\Sigma(\mathbb{A}_F) / p^n \end{equation}

is injective.

We write $\delta$ for the canonical global Kummer map and let $\tilde x \in {\textrm{Sel}}_\Sigma^{(p^n)}(A^t/F)^G$ denote the image of P under the canonical composition

\begin{equation*}A^t(k) \longrightarrow (A^t(F)/p^nA^t(F))^G \stackrel{\delta^G}{\longrightarrow} {\textrm{Sel}}_\Sigma^{(p^n)}(A^t/F)^G.\end{equation*}

We also let $\tilde y \in (A_\Sigma(\mathbb{A}_F) / p^n)^G $ be the image of Q under the canonical (diagonal) localisation map

\begin{equation*}A(k) \longrightarrow (A(F)/p^nA(F))^G\longrightarrow (A_\Sigma(\mathbb{A}_F) / p^n)^G.\end{equation*}

By [Reference Bertolini and Darmon1, Section 3·1] the R-modules ${\textrm{Sel}}_\Sigma^{(p^n)}(A^t/F)$ and $A_\Sigma(\mathbb{A}_F) / p^n$ are G-cohomologically trivial. We therefore find elements $x \in {\textrm{Sel}}_\Sigma^{(p^n)}(A^t/F)$ and $y=(y_w)_{w \in \Sigma(F)} \in A_\Sigma(\mathbb{A}_F) / p^n$ such that

\begin{equation*}{\textrm{Tr}}_G(x) = \tilde x, \quad {\textrm{Tr}}_G(y) = \tilde y.\end{equation*}

We next consider the canonical (diagonal) localisation map

\begin{equation*} \lambda_\Sigma \colon {\textrm{Sel}}_\Sigma^{(p^n)}(A^t/F) \subseteq H^1(F, A^t[p^n]) \stackrel{\oplus {\textrm{res}}_w} \longrightarrow H_\Sigma^1( \mathbb{A}_F, A^t[p^n])\end{equation*}

and the product of local Kummer maps

\begin{equation*} \delta_\Sigma \colon A_\Sigma(\mathbb{A}_F) / p^n \stackrel{\oplus \delta_w}\longrightarrow H_\Sigma^1( \mathbb{A}_F, A[p^n]).\end{equation*}

Then, by the definition of $\langle \ , \ \rangle_1$ given in [Reference Bertolini and Darmon2, Section 2·2], we obtain

\begin{equation*}\langle P, Q \rangle_1 \equiv \langle \lambda_\Sigma(x), \delta_\Sigma(y) \rangle \pmod{I_p(G)^2},\end{equation*}

with $\langle\ , \ \rangle$ as in (27). Noting the sign involved in the definition (10) of the equivariant regulator, we are in fact interested in computing the inverse $-\langle P, Q \rangle_1$ . From the definition (29) of $\langle\ , \ \rangle$ , we easily derive the explicit expression

(31) \begin{equation} -\langle P, Q \rangle_1 \equiv \sum_{g \in G} \left( \sum_{w \in \Sigma(F)} \langle {\textrm{res}}_w(x^g), \delta_w(y_w) \rangle_{F_w,p^n} \right) g^{} \pmod{I_p(G)^2},\end{equation}

which is convenient for the explicit evaluations which we will describe in the next subsection.

4·2. Algorithmic evaluation of the pairing

In this subsection we describe how we numerically evaluate the right hand side of (31). From an algorithmic point of view we are mainly interested in the case of elliptic curves and, for this reason, we henceforth assume that $E \;:\!=\; A = A^t$ is an elliptic curve defined over k. Furthermore, we assume that $F/k$ is cyclic of degree p where as before p is an odd prime such that E, $F/k$ and p satisfy the hypotheses of Section 2·1.

The central computational problem in the numerical evaluation of the Mazur–Tate pairing, using the approach of Bertolini and Darmon as described in the previous section, is first of all the computation of the generalised Selmer group ${\textrm{Sel}}^{(p)}_\Sigma(E/F)$ . We will closely follow the method of Schaefer and Stoll [Reference Schaefer and Stoll26] to compute this group.

We fix a finite set V of places of F containing the p-adic places and all places w such that the Tamagawa number $c_w$ of E at w is divisible by p. We also fix an admisible set $\Sigma$ of places of k as in Section 4.1. We then write $H^1(F, E[p]; V\cup \Sigma(F))$ for the group of cohomology classes in $H^1(F, E[p])$ that are unramified outside $V \cup \Sigma(F) $ . Now the result of [Reference Schaefer and Stoll26, Proposition 3·2] shows that ${\textrm{Sel}}^{(p)}(E/F)$ is given by

\begin{equation*} \{ \xi \in H^1(F, E[p]; V \cup \Sigma(F)) \mid {\textrm{res}}_w(\xi) \in \delta_w(E(F_w)/pE(F_w)) \text{ for all } w \in V \cup \Sigma(F) \},\end{equation*}

whereas a slight generalization of the above mentioned result implies that the generalised Selmer group ${\textrm{Sel}}_\Sigma^{(p)}(E/F)$ is equal to

(32) \begin{equation} \{ \xi \in H^1(F, E[p]; V \cup \Sigma(F)) \mid {\textrm{res}}_w(\xi) \in \delta_w(E(F_w)/pE(F_w)) \text{ for all } w \in V \setminus \Sigma(F) \}.\end{equation}

We fix an algebraic closure $F^c$ of F and let $W \subseteq E[p] \setminus \{0\}$ be a $G_F$ -invariant spanning set for E[p]. Thus W is a union of $G_F$ -orbits containing an ${\mathbb{F}_p}$ -basis of E[p]. Note that in the generic case $G_F$ acts transitively on $E[p] \setminus \{0\}$ , so that in such cases we are forced to use $W = E[p] \setminus \{0\}$ .

We write $A^c \;:\!=\; {\textrm{Map}}(W, F^c)$ for the set of maps from W to $F^c$ . Then the Galois group $G_F$ acts on $A^c$ by conjugation, i.e., for $\sigma \in G_F$ , $a \in A^c$ and $P \in W$ one has $(\sigma a)(P) = \sigma( a(\sigma^{-1}(P)) )$ .

We denote by $A = {\textrm{Map}}_{G_F}(W, F^c) \;:\!=\; {\textrm{Map}}(W, F^c)^{G_F}$ the set of $G_F$ -invariant maps in $A^c$ . Then A is a finite dimensional Étale F-algebra. Explicitly, let $P_1, \ldots, P_s \in W$ be a set of $G_F$ -orbit representatives of W and set

\begin{equation*} H_i \;:\!=\; \{ \sigma \in G_F \mid \sigma(P_i) = P_i \} \end{equation*}

and $L_i \;:\!=\; {(F^c)}^{H_i}$ . Then each $L_i/F$ is a finite separable field extension and we have a canonical isomorphism

\begin{equation*}A \longrightarrow \prod_{i=1}^s L_i, \quad a \mapsto \left( a(P_i) \right)_{1 \le i \le s}.\end{equation*}

The Weil pairing $e_p \colon E[p] \times E[p] \longrightarrow \mu_p(F^c)$ defines a map

\begin{equation*}\omega\;\colon E[p] \longrightarrow \mu_p(A^c) \;:\!=\; {\textrm{Map}}(W, \mu_p(F^c)), \quad P \mapsto e_p(P, \ \underline\ \ ).\end{equation*}

This map induces a homomorphism in cohomology

(33) \begin{equation}\bar\omega\;\colon H^1(F, E[p]) \longrightarrow H^1(F, \mu_p(A^c))\end{equation}

which, by [Reference Schaefer and Stoll26, Prop. 4.3], is known to be injective if $p \nmid |{\textrm{Gal}}(F(E[p])/F)|$ or $p \nmid |W|$ .

In addition, by an immediate generalization of Hilbert’s Theorem 90, we have a Kummer isomorphism

(34) \begin{equation} \kappa\;\colon H^1(F, \mu_p(A^c)) \longrightarrow A^\times / A^{\times p}\end{equation}

which combined with $\bar\omega$ defines an embedding (assuming, for example, that $p \nmid |W|$ )

\begin{equation*}H^1(F, E[p]) \hookrightarrow A^\times / A^{\times p}.\end{equation*}

Roughly speaking, the algorithm of Schaefer and Stoll [Reference Schaefer and Stoll26], in a first step, computes $H^1(F, E[p]; V \cup \Sigma(F)) $ as a subset of $A^\times / A^{\times p}$ via the embedding $\kappa \circ \bar\omega$ and then, in a second step, checks the local conditions occurring in (32).

To describe the first step we let $L/F$ be a finite extension. For a finite place t of L we write ${\textrm{ord}}_t \colon L^\times \longrightarrow {\mathbb{Z}}$ for the associated normalised valuation. If T is a finite set of finite places of F, we set

\begin{equation*}L(T,p) \;:\!=\; \{ a \in L^\times/L^{\times p} \mid {\textrm{ord}}_t(a) \in p{\mathbb{Z}} \text{ for all } t \not\in T(L) \},\end{equation*}

and more generally, if $A \simeq \prod_{i=1}^s L_i$ is an étale F-algebra as above, we define

\begin{equation*}A(T, p) \;:\!=\; \prod_{i=1}^s L_i(T, p).\end{equation*}

Then, using the embedding $H^1(F, E[p]) \hookrightarrow A^\times / A^{\times p}$ given by $\kappa\circ\bar\omega$ , the result of [Reference Schaefer and Stoll26, Corollary 5·9] shows the equality

\begin{equation*}H^1(F, E[p]; V \cup \Sigma(F)) = H^1(F, E[p]) \cap A(V \cup \Sigma(F), p).\end{equation*}

The work of Schaefer and Stoll now describes how to compute the group $H^1(F, E[p]) $ inside $A^\times / A^{\times p}$ and then, by intersecting with the finite group $A(V \cup \Sigma(F), p)$ , we obtain $H^1(F, E[p]; V \cup \Sigma(F))$ as a subgroup of $A^\times / A^{\times p}$ . For further details we refer the reader to [Reference Schaefer and Stoll26].

Testing the local conditions in the second step (see again [Reference Schaefer and Stoll26] for details) we finally obtain both ${\textrm{Sel}}^{(p)}(E/F)$ and ${\textrm{Sel}}_\Sigma^{(p)}(E/F)$ as subgroups of $A^\times / A^{\times p}$ , or even better, as subgroups of the finite group $A(V \cup \Sigma(F), p)$ . Note, however, that the computation of $A(V \cup \Sigma(F), p)$ requires the computation of ideal class groups and units in all of the fields $L_i$ , $i = 1,..,s$ .

In this context we recall that in the generic case we have to use $W = E[p] \setminus \{0\}$ . In any such cases we will fix $S_0 \in W$ and set $L \;:\!=\; F(S_0)$ , so that $[L:F] = p^2-1$ and $A \simeq L$ .

For any place w in $\Sigma(F)$ we fix an embedding $\iota_w:F\to F_w$ and set $A_w^c \;:\!=\; F_w^c \otimes_F A$ and $A_w \;:\!=\; F_w \otimes_F A$ . Note that there is a canonical isomorphism

\begin{equation*}A_w \stackrel{\simeq}\longrightarrow {\textrm{Map}}_{G_{F_w}}(W, F_w^c), \quad z \otimes a \longmapsto \left( S \longmapsto z\iota_w(a(S)) \right).\end{equation*}

We then consider the following commutative diagram

(35)

where $\iota_{w,*}$ is induced by composition with $\iota_w$ while the maps $\bar\omega_w$ and $\kappa_w$ are the local analogues of (33) and (34), respectively, and ${\textrm{res}}_w$ is the localisation map.

The local Tate pairing

\begin{equation*}\langle\ , \ \rangle_{F_w,p} \colon H^1(F_w, E[p]) \times H^1(F_w, E[p]) \longrightarrow {\mathbb{Q}_p} / {\mathbb{Z}_p}\end{equation*}

of (28) induces a pairing $\langle\ , \ \rangle_{A_w}$ on the image of $\kappa_w \circ \bar\omega_w$ . In what follows we will describe the computation of $\langle\ , \ \rangle_{A_w}$ .

We write $q_{A_w}$ for the unique quadratic form such that, for all

\begin{equation*}a,b \in (\kappa_w \circ \bar\omega_w)\left( H^1(F_w, E[p]) \right),\end{equation*}

one has

\begin{equation*}\langle a, b \rangle_{A_w} = q_{A_w}(ab) - q_{A_w}(a) - q_{A_w}(b).\end{equation*}

Let

\begin{equation*} \{\ , \ \}_{F_w,p} \colon F_w^\times/F_w^{\times p} \times F_w^\times/F_w^{\times p} \longrightarrow \mu_p\end{equation*}

denote the Hilbert symbol. We henceforth assume that E[p] is contained in $E(F_w)$ (which is always the case for places $w \in \Sigma(F)$ for an admissible set $\Sigma$ of places of k). In this particular case one can shift the problem of computing the local Tate pairing to the computation of Hilbert symbols which is easy for places w which are prime to p by [Reference Neukirch25, Chapter V, Proposition 3·4].

We fix generators $S, T \in E(F_w)$ of E[p]. Then $e_p(T, S)$ generates $\mu_p$ and we define a map

\begin{equation*} \xi_{S,T} \colon \mu_p \longrightarrow {\mathbb{Z}} / p{\mathbb{Z}}\end{equation*}

by $\xi_{S,T}(e_p(T, S)) = 1 + p{\mathbb{Z}}$ . For $\bar a \in A_w^\times / A_w^{\times p}$ with $a \in A_w \simeq {\textrm{Map}}_{G_w}(W, \bar F_w)$ we obtain a well defined map $\bar a \colon W \longrightarrow F_w^\times / F_w^{\times p}$ .

Fisher and Newton in [Reference Fisher and Newton18] have given an explicit description of the local Tate pairing associated with the p-torsion of an elliptic curve in the special case $p=3$ . Their formulas have subsequently been generalised by Visse in [Reference Visse30] to any odd prime p. In particular, from [Reference Visse30, Theorems 3·7 and 3·11] we obtain, for arbitrary odd p, an equality

\begin{equation*}q_{A_w}(\bar a) = \xi_{S,T} \left( \left\{ \bar a(S), \bar a(T) \right\}_{F_w,p} \right)\end{equation*}

for all $\bar a \in (\kappa_w \circ \bar\omega_w)\left( H^1(F_w, E[p]) \right)$ . As a consequence we obtain

(36) \begin{equation} \langle \bar a, \bar b \rangle_{A_w} = \xi_{S,T} \left( \frac{\{\bar a(S), \bar b(T)\}_{F_w,p}}{\{\bar a(T), \bar b(S)\}_{F_w,p}} \right).\end{equation}

We are now in a position to describe the explicit computation of $-\langle P, Q \rangle_1$ using the formula in (31). Let $\tilde y$ denote the image of Q in $E_\Sigma(\mathbb{A}_F)/p$ as in Section 4·1. Assuming that the point Q is not divisible by p ensures that $\tilde y \ne 0$ because of the injectivity of the map (30).

For each $v \in \Sigma$ we fix a place $\hat v$ of F lying over v. For each $w\in\Sigma(F)$ we define

\begin{equation*}y_w \;:\!=\; \begin{cases} \tilde y_{\hat v}, & \text{if } w \mid v \text{ and } w = \hat v, \\ 0, & \text{ otherwise }. \end{cases}\end{equation*}

Since $\tilde y$ is fixed by G it is easy to see that $y \;:\!=\; (y_w)_{w \in \Sigma(F)} \in E_\Sigma(\mathbb{A}_F)/p$ satisfies ${\textrm{Tr}}_G(y) = \tilde y$ . Hence the formula in (31) simplifies to

(37) \begin{equation}-\langle P, Q \rangle_1 \equiv \sum_{g \in G}\left( \sum_{v \in \Sigma} \langle {\textrm{res}}_{\hat v}(x^g), \delta_{\hat v}(y_{\hat v}) \rangle_{F_{\hat v},p} \right) g^{}\pmod{I_p(G)^2}.\end{equation}

Next we explain how to compute $x \in {\textrm{Sel}}^{(p)}_\Sigma(E/F)$ with ${\textrm{Tr}}_G(x) = \tilde x$ . The generalised Selmer group ${\textrm{Sel}}^{(p)}_\Sigma(E/F)$ is computed as a subgroup of $A^\times / A^{\times p}$ which carries an action of G induced by conjugation. Explicitly, for all $\sigma \in G$ , $a \in A$ and $S \in W$ we choose an arbitrary lift $\hat\sigma \in G_k$ of $\sigma$ and set

(38) \begin{equation}(\sigma a)(S) = \hat\sigma(a(\hat\sigma^{-1}(S))).\end{equation}

As sketched in Section 4·3 below this action is explicitly computable. We can therefore compute the ${\mathbb{F}_p}$ -representation of G induced by ${\textrm{Sel}}^{(p)}_\Sigma(E/F)$ and then represent ${\textrm{Tr}}_G$ as a linear map ${\textrm{Sel}}^{(p)}_\Sigma(E/F) \longrightarrow {\textrm{Sel}}^{(p)}_\Sigma(E/F)$ and thus compute a preimage x of $\tilde x$ .

In the same way we can compute the elements $x^g$ occuring in (37). Note that as an element of $A^\times / A^{\times p}$ the element $\tilde x$ is given by $(\kappa\circ\bar\omega\circ\delta)(P)$ . The computation of the map

\begin{equation*} H \;:\!=\; \kappa \circ \bar\omega \circ \delta \end{equation*}

is quite non-trivial and explained in [Reference Schaefer27, 28, Chapter 2] and [Reference Schaefer and Stoll26, Chapter 3], as well as the computation of the local analogue

\begin{equation*} H_w \;:\!=\; \kappa_w \circ \bar\omega_w \circ \delta_w. \end{equation*}

For each $v \in \Sigma$ we now compute the completion $F_{\hat v}$ together with an embedding $\iota_v \colon F \longrightarrow F_{\hat v}$ and define $a_{P, g, \hat v} \;:\!=\; {\textrm{res}}_{\hat v}(x^g) = \iota_v \circ x^g$ .

In order to compute $\delta_{\hat v}(y_{\hat v})$ we use the commutativity of diagram (35) and define

\begin{equation*}a_{Q, \hat v} \;:\!=\; \kappa_{\hat v}( \bar\omega_{\hat v}( \delta_{\hat v}( \iota_{\hat v}(Q) ))) =\iota_{\hat v} \circ ( \kappa( \bar\omega(\delta(Q))) =\iota_{\hat v} \circ H(Q)\end{equation*}

and finally obtain

\begin{equation*}\langle {\textrm{res}}_{\hat v}(x^g), \delta_{\hat v}(y_{\hat v}) \rangle_{F_{\hat v},p} =\xi_{S,T}\left( \frac{\{a_{P,g, \hat v}(S), a_{Q, \hat v}(T)\}_{F_w,p}}{\{a_{P,g,\hat v}(T), a_{Q, \hat v}(S)\}_{F_w,p}}\right).\end{equation*}

4·3. Comments on the implementation

In this subsection we discuss the restrictions on our MAGMA implementation, where we must always assume that $G_F$ acts transitively on $E[p] \setminus \{0\}$ . Hence, for the rest of this subsection, we set $W = E[p] \setminus \{0\}$ and fix a point $S_0 \in W$ . As before, $F/k$ is a cyclic extension of degree p where p is an odd prime.

We choose $\lambda \in \mathbb{Q}$ such that the elements

\begin{equation*}w_S \;:\!=\; y_S + \lambda x_S, \quad S = (x_S, y_S) \in W,\end{equation*}

are pairwise distinct. Since $G_F$ acts transitively on W the polynomial

\begin{equation*}f(x) \;:\!=\; \prod_{S \in W} (x - w_S) \in F[x]\end{equation*}

is irreducible. It is easily seen that $F(S_0) = F(w_{S_0})$ . We therefore may and will use $L \;:\!=\; F(w_{S_0})$ .

This has the following useful consequence for the computation of $a_w \;:\!=\; \iota_w \circ a$ , where a is an element of $A = {\textrm{Map}}_{G_F}(W, F^c)$ and $\iota_w$ an embedding of F into $F_w$ . Namely, if we assume that $a \in A$ corresponds to $\bar h \in F[x] / (f(x))$ under the composite map

\begin{equation*}A \stackrel\alpha\longrightarrow L \stackrel{\beta^{-1}}\longrightarrow F[x]/(f(x)), \quad \alpha(a) \;:\!=\; a(S_0), \quad \beta(\bar h) \;:\!=\; h(w_{S_0}),\end{equation*}

then we have $a(S_0) = h(w_{S_0})$ . Assume further that $\tilde S \in E(F_w)[p]$ and let $S \in W$ be such that $\iota_w(S) = \tilde S$ . Then a straightforward computation shows that $a_w(\tilde S) = (\iota_wh)(w_{\tilde S})$ .

A further useful consequence is a particularly simple description of the action of G on A. Recall that $\sigma \in G$ acts on A by conjugation as in (38). If $\bar h \in F[x]/(f(x))$ corresponds to $a \in A$ , then we claim that $\sigma a$ corresponds to $\overline{\sigma h}$ . Indeed, if $\tau(S_0) = \hat\sigma^{-1}(S_0)$ with $\tau \in G_F$ , then

\begin{equation*}(\sigma a)(S_0) = \hat\sigma(a(\hat\sigma^{-1}(S_0))) = \hat\sigma(\tau(h(w_{S_0}))) = \hat\sigma(h(\tau(w_{S_0}))) = (\sigma h) (w_{S_0}).\end{equation*}

In our implementation we additionally assume $k=\mathbb{Q}$ and $p=3$ . In this case explicit formulae for the computation of the generalised Selmer group based on the work of Schaefer and Stoll in [Reference Schaefer and Stoll26] are given in [Reference Geishauser19, Section 7]. These are used throughout in our implementation.

We summarise the above discussion in the following remark.

4·4. Computations

Our MAGMA implementation can be used to numerically verify the validity of the refined Birch and Swinnerton–Dyer conjecture for pairs $(E,F/k)$ where;

• (i) $k=\mathbb{Q}$ and $E/\mathbb{Q}$ is an elliptic curve with global minimal Weierstrass equation;

• (ii) F is the unique subfield of degree $p=3$ in $\mathbb{Q}(\zeta_\ell)/\mathbb{Q}$ for primes $\ell \equiv 1 \pmod{3}$ ;

• (iii) $\mathrm{rk}(E(F)) = \mathrm{rk}(E(\mathbb{Q})) = r$ for some $r > 0$ ;

• (vi) E and F satisfy the hypotheses (a) - (i) of Section 2·1.

Note that the condition $\mathrm{rk}(E(F)) = \mathrm{rk}(E(\mathbb{Q})) = r$ ensures that $E(F)_p \simeq {\mathbb{Z}}_p^r$ is not projective as a $\mathbb{Z}_p[G]$ -module.

Here is the list of examples for $r=1$ , $\ell < 50$ and conductor less than 100. We specify elliptic curves by their Cremona reference:

• • E is 37a1 and $\ell \in \{13,19\}$ ;

• • E is 43a1 and $\ell \in \{7,13,37\}$ ;

• • E is 53a1 and $\ell \in \{13, 19, 31,43 \}$ ;

• • E is 58a1 and $\ell \in \{7,13,19,31,43 \}$ ;

• • E is 61a1 and $\ell \in \{7, 13, 43 \}$ ;

• • E is 65a2 and $\ell \in \{19, 37, 43 \}$ ;

• • E is 77a1 and $\ell \in \{19, 37 \}$ ;

• • E is 79a1 and $\ell \in \{13, 19, 37 \}$ ;

• • E is 82a1 and $\ell \in \{13, 19, 43 \}$ ;

• • E is 82a2 and $\ell \in \{13, 19, 43 \}$ ;

• • E is 83a1 and $\ell \in \{7, 13, 37 \}$ ;

• • E is 88a1 and $\ell \in \{7,43 \}$ ;

• • E is 89a1 and $\ell \in \{19, 31, 37 \}$ ;

• • E is 91a1 and $\ell \in \{31 \}$ ;

• • E is 389a1 and $\ell \in \{7,13,43 \}$ ;

• • E is 433a1 and $\ell \in \{7, 13, 31, 37 \}$ ;

• • E is 446d1 and $\ell \in \{19 \}$ .

For $r=2$ , $\ell < 50$ and conductor less than 500 we computed the following examples.

• • E is 389a1 and $\ell \in \{7,13,43 \}$ :

• • E is 433a1 and $\ell \in \{7, 13, 31, 37 \}$ ;

• • E is 446d1 and $\ell \in \{19 \}$ ;