1 Introduction
1.1 The statement of the main result
Let E be an elliptic curve of conductor N over
$\mathbb {Q}$
and
$p \geq 5$
be a prime of good ordinary reduction for E such that
-
(Im) the mod p Galois representation
$\overline {\rho } : G_{\mathbb {Q}} = \mathrm {Gal}(\overline {\mathbb {Q}}/\mathbb {Q}) \to \mathrm {Aut}_{\mathbb {F}_p}(E[p]) \simeq \mathrm {GL}_2(\mathbb {F}_p)$
is surjective, and -
(Ram)
$\overline {\rho }$
is ramified at every prime dividing N, so p does not divide Tamagawa factors of E.
Let K be an imaginary quadratic field of odd discriminant
$-D_K < -4$
with
$(D_K, Np) = 1$
such that
-
(Spl) p splits in K, and
-
(Na)
$a_p(E) \not \equiv 1\ \pmod {p}$
.
Write
$$ \begin{align*}N = N^+ \cdot N^-,\end{align*} $$
where a prime divisor of
$N^+$
splits in K and a prime divisor of
$N^- $
is inert in K.
-
(Def) Assume that
$N^-$
is a square-free product of an odd number of primes.
Let
$K_\infty $
be the anticyclotomic
$\mathbb {Z}_p$
-extension of K and
$K_n$
be the subextension of K in
$K_\infty $
of degree
$p^n$
for
$n \geq 0$
. Let
$\Lambda _n = \mathbb {Z}_p[\mathrm {Gal}(K_n/K)] \simeq \mathbb {Z}_p[X]/( (1+X)^{p^n} - 1 )$
be the finite layer Iwasawa algebra. Under (Def), denote by
$$ \begin{align*}\theta(E/K_n) = \sum_{\sigma \in \mathrm{Gal}(K_n/K)} a_{\sigma} \cdot \sigma \in \Lambda_n.\end{align*} $$
Bertolini–Darmon’s theta element of E over
$K_n$
which interpolates the square-roots of
$L(E/K, \chi , 1)$
for finite order characters
$\chi $
on
$\mathrm {Gal}(K_n/K)$
. It is reviewed in Section 2. For the natural projection map
$\pi _{n, n-1} : \Lambda _{n} \to \Lambda _{n-1}$
, let
$\nu _{n-1, n} : \Lambda _{n-1} \to \Lambda _n$
be the map defined by
$\sigma \mapsto \sum _{\pi _{n, n-1}(\tau ) = \sigma } \tau $
. For
$0 \leq m \leq n$
, we write
$\nu _{m, n} = \nu _{n-1, n} \circ \nu _{n-2, n-1} \circ \cdots \circ \nu _{m, m+1} $
. Then we have the equality of ideals of
$\Lambda _n$
(Lemma 2.1)
$$ \begin{align*}\left( \theta(E/K_n) , \nu_{n-1, n} \left( \theta(E/K_{n-1}) \right) \right) = \left( \nu_{m, n} \left( \theta(E/K_{m}) \right) : 0 \leq m \leq n \right) ,\end{align*} $$
and it is a principal ideal under (Na) (Lemma 2.2).
The goal of this article is to prove the following anticyclotomic analogue of the “strong” main conjecture of Kurihara [Reference Kurihara14, Conjecture 0.3], which refines the “weak” main conjecture of Mazur and Tate [Reference Mazur and Tate15, Conjecture 3].
Theorem 1.1 Under the assumptions mentioned above, i.e., (Im),(Ram),(Spl),(Na), and (Def), the theta elements over
$K_m$
with
$0 \leq m \leq n$
generate the initial Fitting ideal of dual Selmer groups over
$K_n$
, i.e., we have equality of ideals of
$\Lambda _n$
$$ \begin{align*}\left( \theta(E/K_n) , \nu_{n-1, n} \left( \theta(E/K_{n-1}) \right) \right)^2 = \mathrm{Fitt}_{\Lambda_n} \left( \mathrm{Sel}(K_n, E[p^\infty])^\vee \right) ,\end{align*} $$
which is indeed a principal ideal, where
$\mathrm {Sel}(K_n, E[p^\infty ])$
is the classical Selmer group of
$E[p^\infty ]$
over
$K_n$
and
$(-)^\vee $
means the Pontryagin dual.
Since theta elements interpolate the square-roots of twisted Rankin–Selberg L-values, it is natural that the square of the ideal generated by theta elements appears in the equality.
The strategy of our proof follows that given in [Reference Kim and Kurihara11] and we also add some details on the “p-destabilization” process and on the comparison of various anticyclotomic Selmer groups of elliptic curves.
2 Bertolini–Darmon’s theta elements and anticyclotomic p-adic L-functions
We quickly review the construction of Gross points of conductor
$p^n$
, theta elements, and anticyclotomic p-adic L-functions. See [Reference Chida and Hsieh4, Reference Chida and Hsieh5, Reference Kim10] for details.
2.1 Gross points
Let K be the imaginary quadratic field of odd discriminant
$-D_K < - 4$
. Define
$$ \begin{align*}\vartheta := \dfrac{ D_K - \sqrt{-D_K} }{ 2 } \end{align*} $$
so that
$\mathcal {O}_K = \mathbb {Z} + \mathbb {Z}\vartheta $
. Let
$B_{N^-}$
be the definite quaternion algebra over
$\mathbb {Q}$
of discriminant
$N^-$
. Then there exists an embedding
$\Psi :K \hookrightarrow B_{N^-}$
[Reference Vignéras20]. More explicitly, we choose a K-basis
$(1,J)$
of
$B_{N^-}$
so that
$B_{N^-} = K \oplus K \cdot J$
such that
$\beta := J^2 \in \mathbb {Q}^\times $
with
$\beta <0$
,
$J \cdot t = \overline {t} \cdot J$
for all
$t \in K$
,
$\beta \in \left ( \mathbb {Z}^\times _q \right )^2$
for all
$q \mid pN^+$
, and
$\beta \in \mathbb {Z}^\times _q$
for all
$q \mid D_K$
. Fix a square root
$\sqrt {\beta } \in \overline {\mathbb {Q}}$
of
$\beta $
. For a
$\mathbb {Z}$
-module A, write
$\widehat {A} = A \otimes \widehat {\mathbb {Z}}$
. Fix an isomorphism
$$ \begin{align*}i := \prod i_q : \widehat{B}^{(N^-)}_{N^-} \simeq \mathrm{M}_2(\mathbb{A}^{(N^-\infty)})\end{align*} $$
as follows:
-
• For each finite place
$q \mid N^+p$
, the isomorphism
$i_q : B_{N^-,q} \simeq \mathrm {M}_2(\mathbb {Q}_q)$
is defined by where
$\mathrm {trd}$
and
$\mathrm {nrd}$
are the reduced trace and the reduced norm on B, respectively.
-
• For each finite place
$q \nmid pN^+$
, the isomorphism
$i_q : B_{N^-,q} \simeq \mathrm {M}_2(\mathbb {Q}_q)$
is chosen so that
$i_q \left ( \mathcal {O}_K \otimes \mathbb {Z}_q \right ) \subseteq \mathrm {M}_2(\mathbb {Z}_q) $
.
Under the fixed isomorphism i, for any rational prime q, the local Gross point
${\varsigma _q \in B^\times _{N^-,q}}$
is defined as follows:
-
•
$\varsigma _q := 1$
in
$B^\times _{N^-,q}$
for
$q \nmid pN^+$
. -
•
$\varsigma _q := \frac {1}{\sqrt {D_K}}\cdot \left ( \begin {matrix} \vartheta & \overline {\vartheta } \\ 1 & 1 \end {matrix} \right ) \in \mathrm {GL}_2(K_{\mathfrak {q}}) = \mathrm {GL}_2(\mathbb {Q}_q) $
for
$q \mid N^+$
with
$q = \mathfrak {q} \overline {\mathfrak {q}}$
in
$\mathcal {O}_K$
. -
•
$\varsigma ^{(n)}_p = \left ( \begin {matrix} \vartheta & -1 \\ 1 & 0 \end {matrix} \right ) \cdot \left ( \begin {matrix} p^n & 0 \\ 0 & 1 \end {matrix} \right ) \in \mathrm {GL}_2(K_{\mathfrak {p}}) = \mathrm {GL}_2( \mathbb {Q}_{p} )$
where
$p = \mathfrak {p}\overline {\mathfrak {p}}$
splits in K.
Let
$\widehat {\Psi } : \widehat {K} \hookrightarrow \widehat {B}_{N^-}$
be the adelic version of
$\Psi $
. We define
$x_n : \widehat {K}^\times \to \widehat {B}^\times _{N^-}$
by
$x_n(a) = \widehat {\Psi }(a) \cdot \varsigma ^{(n)} := \widehat {\Psi }(a) \cdot \left ( \varsigma ^{(n)}_p \times \prod _{q \neq p} \varsigma _q \right )$
. The collection
$\left \lbrace x_n(a) : a \in \widehat {K}^\times \right \rbrace $
of points is called the Gross points of conductor
$p^n$
on
$\widehat {B}^\times _{N^-}$
. The fixed embedding
$K \hookrightarrow B_{N^-}$
also induces an optimal embedding of
$\mathcal {O}_n = \mathbb {Z}+p^n \mathcal {O}_K$
into the Eichler order
$B_{N^-} \cap \varsigma ^{(n)}\widehat {R}_{N^+}(\varsigma ^{(n)})^{-1}$
where
$R_{N^+}$
is the Eichler order of level
$N^+$
under the fixed isomorphism i.
2.2 Theta elements
Let
$f(z) = \sum _{n \geq 1} a_n q^n \in S_{2}(\Gamma _0(N))$
be the cuspidal newform of weight two with rational Fourier coefficients corresponding to E via the modularity theorem [Reference Breuil, Conrad, Diamond and Taylor3]. Let
$\phi _f : B^\times _{N^-} \backslash \widehat {B}^{\times }_{N^-} / \widehat {R}^\times _{N^+} \to \mathbb {C}$
be the Jacquet–Langlands transfer of f. Since
$B^\times _{N^-} \backslash \widehat {B}^{\times }_{N^-} / \widehat {R}^\times _{N^+}$
is a finite set and f is a Hecke eigenform, we are able to and do normalize
$$ \begin{align*}\phi_f : B^\times_{N^-} \backslash \widehat{B}^{\times}_{N^-} / \widehat{R}^\times_{N^+} \to \mathbb{Z}_p,\end{align*} $$
such that the image of
$\phi _f $
does not lie in
$p\mathbb {Z}_p$
. This integral normalization is related to the congruence ideals [Reference Kim9, Reference Kim and Ota12, Reference Pollack and Weston16]. Let
$$ \begin{align*}\widetilde{\theta}_n(E/K) = \sum_{[a] \in \mathcal{G}_n} \phi_f (x_n(a) ) \cdot [a] \in \mathbb{Z}_p[\mathcal{G}_n],\end{align*} $$
where
$\mathcal {G}_n = K^\times \backslash \widehat {K}^\times / \widehat {\mathcal {O}}^\times _n $
and
$[a]$
is the image of
$a \in \widehat {K}^\times $
in
$\mathcal {G}_n$
. Then Bertolini–Darmon’s theta element
$\theta (E/K_n)$
of E over
$K_n$
is defined by the image of
$\widetilde {\theta }_n(E/K)$
in
$\Lambda _n$
where the map is naturally induced from the quotient map
$\mathcal {G}_n \to \mathrm {Gal}(K_n/K)$
. It is known that
$\theta (E/K_n)$
interpolates “an half of”
$L(E, \chi , 1)$
where
$ \chi $
runs over characters on
$\mathrm {Gal}(K_n/K)$
. See [Reference Chida and Hsieh5, Remark (iii) after Theorem A] for the precise meaning of “an half of”. Because
$\theta (E/K_n)$
depends on the choice of Gross points,
$\theta (E/K_n)$
is well-defined only up to multiplication by
$\mathrm {Gal}(K_n/K)$
.
2.3 p-adic L-functions
Let
$\alpha , \beta $
be the roots of the Hecke polynomial
$X^2 - a_pX + p$
of f at p. Since f is ordinary at p, one of them, say
$\alpha $
, is a p-adic unit.
The p
-stabilization
$f_\alpha \in S_{2}(\Gamma _0(Np))$
of f is defined by
$$ \begin{align*}f_\alpha (z) = f(z) - \beta \cdot f(pz)\end{align*} $$
whose
$U_p$
-eigenvalue is
$\alpha $
. Then the theta element of
$f_\alpha $
over
$K_n$
is characterized by the following relation:
$$ \begin{align} \theta(f_\alpha/K_n) = \dfrac{1}{\alpha^n} \cdot \left( \theta(E/K_n) - \dfrac{1}{\alpha} \cdot \nu_{n-1, n}( \theta(E/K_{n-1}) ) \right). \end{align} $$
It is known that theta elements of E satisfies the three term relation (e.g., [Reference Darmon and Iovita6, Lemma 2.6])
$$ \begin{align} \pi_{n+1, n} \left( \theta(E/K_{n+1}) \right) = a_p \cdot \theta(E/K_n) - \nu_{n-1, n} \left( \theta(E/K_{n-1}) \right) \end{align} $$
and the theta elements of
$f_\alpha $
satisfy the norm compatibility
$$ \begin{align} \pi_{n+1, n} \left( \theta(f_\alpha/K_{n+1}) \right)= \theta(f_\alpha/K_{n}). \end{align} $$
Let
$\iota $
be the involution on
$\Lambda _n$
defined by inverting group-like elements, so we have
$$ \begin{align*}\iota( \sum_{\sigma \in \mathrm{Gal}(K_n/K)} a_{\sigma} \cdot \sigma ) = \sum_{\sigma \in \mathrm{Gal}(K_n/K)} a_{\sigma} \cdot \sigma^{-1}.\end{align*} $$
We define the anticyclotomic p -adic L -function of E by
$$ \begin{align*}L_p(E/K_\infty) = \varprojlim_n\left( \theta(f_\alpha/K_n) \cdot \iota( \theta(f_\alpha/K_n) ) \right) \in \Lambda = \varprojlim_n \Lambda_n.\end{align*} $$
This element is well-defined. The functional equation for Bertolini–Darmon’s theta elements yields the equality of ideals of
$\Lambda $
(e.g., [Reference Bertolini and Darmon1, Proposition 2.13], [Reference Bertolini and Darmon2, Lemma 1.5])
$$ \begin{align} \left( \theta(f_\alpha/K_n) \right) = \left( \iota( \theta(f_\alpha/K_n) ) \right). \end{align} $$
We prove two useful lemmas.
Lemma 2.1 We have an equality of ideals of
$\Lambda _n$
$$ \begin{align*}\left( \theta(E/K_n) , \nu_{n-1, n} \left( \theta(E/K_{n-1}) \right) \right) = \left( \nu_{m, n} \left( \theta(E/K_{m}) \right) : 0 \leq m \leq n \right) .\end{align*} $$
Proof From the three term relation (2.2), we have
$$ \begin{align*}\nu_{n-1, n} \left( \pi_{n, n-1} \left( \theta(E/K_{n}) \right) \right) = a_p \cdot \nu_{n-1, n} \left( \theta(E/K_{n-1}) \right) - \nu_{n-2, n} \left( \theta(E/K_{n-2}) \right)\end{align*} $$
for
$n \geq 2$
. Since
$\nu _{n-1, n} \left ( \pi _{n, n-1} \left ( \theta (E/K_{n}) \right ) \right ) = f_n \cdot \theta (E/K_{n})$
for some
$f_n \in \Lambda _n$
, we have
$$ \begin{align*}\nu_{n-2, n} \left( \theta(E/K_{n-2}) \right) \in \left( \theta(E/K_n) , \nu_{n-1, n} \left( \theta(E/K_{n-1}) \right) \right) \subseteq \Lambda_n.\end{align*} $$
In the same manner, we can obtain
$$ \begin{align*}\nu_{n-3, n-1} \left( \theta(E/K_{n-3}) \right) \in \left( \theta(E/K_{n-1}) , \nu_{n-2, n-1} \left( \theta(E/K_{n-2}) \right) \right) \subseteq \Lambda_{n-1}.\end{align*} $$
By taking
$\nu _{n-1, n}$
, we have
$$ \begin{align*} \nu_{n-3, n} \left( \theta(E/K_{n-3}) \right) & \in \left( \nu_{n-1, n} \left( \theta(E/K_{n-1}) \right) , \nu_{n-2, n} \left( \theta(E/K_{n-2}) \right) \right) \\ & \subseteq \left( \theta(E/K_{n}), \nu_{n-1, n} \left( \theta(E/K_{n-1}) \right) \right) \\ & \subseteq \Lambda_{n}. \end{align*} $$
By applying this argument recursively, the conclusion follows.
Lemma 2.2 Under (Spl) and (Na), we have an equality of ideals of
$\Lambda _n$
$$ \begin{align*}\left( \theta(E/K_n) , \nu_{n-1, n} \left( \theta(E/K_{n-1}) \right) \right) = \left( \theta(f_\alpha/K_{n}) \right) .\end{align*} $$
Proof By the definition of the p-stabilization (2.1), we have one inclusion
$\supseteq $
. Hence, we focus on the opposite inclusion. By the interpolation formula of the anticyclotomic p-adic L-functions [Reference Chida and Hsieh5, Theorem A] under (Spl), we have the comparison of (the square-roots of) L-values
$$ \begin{align*}\theta(f_\alpha/K) = \left( 1 - \dfrac{1}{\alpha} \right) \cdot \theta(E/K).\end{align*} $$
Under (Na), we have equality in
$\mathbb {Z}_p$
$$ \begin{align*}\left( 1 - \dfrac{1}{\alpha} \right)^{-1} \cdot \theta(f_\alpha/K) = \theta(E/K),\end{align*} $$
so we have
$(\theta (E/K)) \subseteq (\theta (f_\alpha /K))$
. In fact, they are the same ideal. From (2.1) and (2.3), we have
$$ \begin{align*} \theta(f_\alpha/K_1) & = \dfrac{1}{\alpha} \cdot \left( \theta(E/K_1) - \dfrac{1}{\alpha} \cdot \nu_{0, 1}( \theta(E/K) ) \right) \\ & = \dfrac{1}{\alpha} \cdot \left( \theta(E/K_1) - \dfrac{1}{\alpha} \cdot \nu_{0, 1}( \left( 1 - \dfrac{1}{\alpha} \right)^{-1} \cdot \theta(f_\alpha/K) ) \right) \\ & = \dfrac{1}{\alpha} \cdot \left( \theta(E/K_1) - \dfrac{1}{\alpha} \cdot \left( 1 - \dfrac{1}{\alpha} \right)^{-1} \cdot \nu_{0, 1}( \pi_{1,0} ( \theta(f_\alpha/K_1)) ) \right) \\ & = \dfrac{1}{\alpha} \cdot \left( \theta(E/K_1) - \dfrac{1}{\alpha} \cdot \left( 1 - \dfrac{1}{\alpha} \right)^{-1} \cdot f_1 \cdot \theta(f_\alpha/K_1) \right) \end{align*} $$
for some
$f_1 \in \Lambda _1$
. This shows that
$\theta (E/K_{1}) = g_{1} \cdot \theta (f_\alpha /K_{1})$
for some
$g_{1} \in \Lambda _{1}$
.
We suppose that
$\theta (E/K_{n-1}) = g_{n-1} \cdot \theta (f_\alpha /K_{n-1})$
for some
$g_{n-1} \in \Lambda _{n-1}$
.
$$ \begin{align*} \theta(f_\alpha/K_n) & = \dfrac{1}{\alpha^n} \cdot \left( \theta(E/K_n) - \dfrac{1}{\alpha} \cdot \nu_{n-1, n}( \theta(E/K_{n-1}) ) \right) \\ & = \dfrac{1}{\alpha^n} \cdot \left( \theta(E/K_{n}) - \dfrac{1}{\alpha} \cdot \nu_{n-1, n}( g_{n-1} \cdot \theta(f_\alpha/K_{n-1}) ) \right) \\ & = \dfrac{1}{\alpha^n} \cdot \left( \theta(E/K_n) - \dfrac{1}{\alpha} \cdot g_{n-1} \cdot \nu_{n-1, n}( \pi_{n,n-1} ( \theta(f_\alpha/K_n)) ) \right) \\ & = \dfrac{1}{\alpha^n} \cdot \left( \theta(E/K_1) - \dfrac{1}{\alpha} \cdot g_{n-1} \cdot f_{n} \cdot \theta(f_\alpha/K_n) \right) \end{align*} $$
for some
$f_n \in \Lambda _n$
. This shows that
$\theta (E/K_{n}) = g_{n} \cdot \theta (f_\alpha /K_{n})$
for some
$g_{n} \in \Lambda _{n}$
. By induction, we have inclusion
$$ \begin{align*}(\theta(E/K_n)) \subseteq (\theta(f_\alpha/K_n)),\end{align*} $$
so we also have
$$ \begin{align*}( \nu_{n-1, n} \left( \theta(E/K_{n-1}) \right) ) \subseteq ( \nu_{n-1, n} \left( \theta(f_\alpha/K_{n-1}) \right)).\end{align*} $$
Since
$\nu _{n-1, n} \left ( \theta (f_\alpha /K_{n-1}) \right ) = f_{n} \cdot \theta (f_\alpha /K_n)$
, we have
$$ \begin{align*}( \nu_{n-1, n} \left( \theta(f_\alpha/K_{n-1}) \right)) \subseteq ( \theta(f_\alpha/K_{n}) ).\end{align*} $$
The conclusion follows.
3 Comparison of Selmer groups
3.1 Local properties of Galois representations
Let
$\rho : G_{\mathbb {Q}} \to \mathrm {Aut}_{\mathbb {Q}_p}(V) = \mathrm {GL}_2(\mathbb {Q}_p)$
be the two-dimensional Galois representation associated with E.
-
• Since E is good ordinary at p, we have
where
$$ \begin{align*}\rho \vert_{ G_{\mathbb{Q}_p} } \sim \begin{pmatrix} \chi^{-1}_{\alpha} \cdot \chi_{\mathrm{cyc}} & * \\ 0 & \chi_{\alpha} \end{pmatrix}\end{align*} $$
$\chi _{\alpha }$
is the unramified character sending the arithmetic Frobenius at p to
$\alpha $
.
-
• For
$\ell $
dividing N exactly, we also have
$$ \begin{align*}\rho \vert_{ G_{\mathbb{Q}_\ell} } \sim \begin{pmatrix} \pm \chi_{\mathrm{cyc}} & * \\ 0 & \pm \mathbf{1} \end{pmatrix} .\end{align*} $$
For a rational prime v dividing
$N^-p$
, we consider the following subspaces:
-
• For
$v= p$
, let
$F^+V \subseteq V$
be the subspace on which the inertia subgroup
$I_v$
acts by
$\chi _{\mathrm {cyc}}$
. -
• For a rational prime v dividing
$N^-$
,
$F^+V \subseteq V$
be the subspace on which the inertia subgroup
$I_v$
acts by
$\chi _{\mathrm {cyc}}$
or
$\chi _{\mathrm {cyc}} \tau _v$
where
$\tau _v$
is the non-trivial unramified quadratic character of
$G_{\mathbb {Q}_v}$
.
Let L be an algebraic extension of K. For a prime w of L dividing
$N^-p$
, we define the ordinary local condition of V at w by
$$ \begin{align*}\mathrm{H}^1_{\mathrm{ord}}(L_w, V) = \mathrm{ker} \left( \mathrm{H}^1(L_w, V) \to \mathrm{H}^1(L_w, V/F^+V) \right).\end{align*} $$
Denote by
$T = \varprojlim _k E[p^k]$
the p-adic Tate module of E, so we have
$T \otimes _{\mathbb {Z}_p} \mathbb {Q}_p = V$
, and by
$E[p^\infty ] = \varinjlim _k E[p^k]$
the p-power torsion points of E. Then the same local conditions for T,
$T/p^kT$
,
$E[p^\infty ]$
, and
$E[p^k]$
are defined by propagation.
3.2
$N^-$
-ordinary (residual) Selmer groups
Let
$\Sigma $
be the finite set of places of
$\mathbb {Q}$
consisting of the places dividing
$Np\infty $
, and
$K_\Sigma $
be the maximal extension of K unramified outside
$\Sigma $
. We write
-
•
$\Sigma ^+ \subseteq \Sigma $
to be the subset of
$\Sigma $
consisting of the places not dividing
$p\infty $
which split in
$K/\mathbb {Q}$
, and -
•
$\Sigma ^- \subseteq \Sigma $
to be the subset of
$\Sigma $
consisting of the places not dividing
$p\infty $
which are inert in
$K/\mathbb {Q}$
.
For a place w of
$K_\infty $
, we write
$w \in \Sigma ^{\pm }$
if w divides a rational prime
$\ell $
contained in
$\Sigma ^{\pm }$
, respectively. For every
$k \geq 1$
, we define the
$N^-$
-ordinary (and
$N^+$
-strict) Selmer group of
$E[p^k]$
$\mathrm {Sel}_{N^-}(K_\infty , E[p^k])$
by the kernel of the map
$$ \begin{align*}\mathrm{H}^1(K_\Sigma/ K_\infty, E[p^k]) \to \prod_{w \nmid \Sigma^+} \mathrm{H}^1(K_{\infty, w}, E[p^k]) \times \prod_{w \in \Sigma^- \textrm{ or } w \vert p} \dfrac{\mathrm{H}^1(K_{\infty,w}, E[p^k])}{\mathrm{H}^1_{\mathrm{ord}}(K_{\infty,w}, E[p^k])},\end{align*} $$
and define
$\mathrm {Sel}_{N^-}(K_\infty , E[p^\infty ]) = \varinjlim _k \mathrm {Sel}_{N^-}(K_\infty , E[p^k])$
. This is the Selmer group used in the bipartite Euler system argument [Reference Bertolini and Darmon2, Definition 2.8].
3.3 Minimal and Greenberg Selmer groups
We follow the convention of [Reference Pollack and Weston16, Section 3.1]. The minimal Selmer group
$\mathrm {Sel}_{\mathrm {min}}(K_\infty , E[p^\infty ])$
of
$E[p^\infty ]$
is defined by the kernel of the map
$$ \begin{align*}\mathrm{H}^1(K_\infty, E[p^\infty]) \to \prod_{w \nmid p} \mathrm{H}^1(K_{\infty, w}, E[p^\infty]) \times \prod_{w \vert p} \dfrac{\mathrm{H}^1(K_{\infty,w}, E[p^\infty])}{\mathrm{H}^1_{\mathrm{ord}}(K_{\infty,w}, E[p^\infty])},\end{align*} $$
and the Greenberg Selmer group
$\mathrm {Sel}_{\mathrm {Gr}}(K_\infty , E[p^\infty ])$
of
$E[p^\infty ]$
is defined by the kernel of the map
$$ \begin{align*}\mathrm{H}^1(K_\infty, E[p^\infty]) \to \prod_{w \nmid p} \mathrm{H}^1(I_{\infty, w}, E[p^\infty]) \times \prod_{w \vert p} \dfrac{\mathrm{H}^1(K_{\infty,w}, E[p^\infty])}{\mathrm{H}^1_{\mathrm{ord}}(K_{\infty,w}, E[p^\infty])},\end{align*} $$
where
$I_{\infty , w}$
is the inertia subgroup of
$G_{K_{\infty ,w}}$
. Under (Ram),
$\overline {\rho }$
is ramified at every prime dividing N, so p does not divide any Tamagawa factors. Then by using [Reference Pollack and Weston16, Lemma 3.4], we have an isomorphism
$$ \begin{align} \mathrm{Sel}_{\mathrm{min}}(K_\infty, E[p^\infty]) \simeq \mathrm{Sel}_{\mathrm{Gr}}(K_\infty, E[p^\infty]). \end{align} $$
3.4 The comparison
We recall the final displayed equation in the proof of [Reference Pollack and Weston16, Proposition 3.6]:
where w runs over the primes of
$K_\infty $
dividing
$N^+$
. The local conditions at primes dividing
$N^-$
of minimal Selmer groups and
$N^-$
-ordinary Selmer groups coincide since such primes split completely in
$K_\infty /K$
. Thus, we have inclusion
$$ \begin{align*}\mathrm{Sel}_{N^-}(K_\infty, E[p^k]) \subseteq \mathrm{Sel}_{\mathrm{min}}(K_\infty, E[p^\infty])[p^k]\end{align*} $$
which is of finite index and is independent of k.
Proposition 3.1 If
$\mathrm {Sel}_{N^-}(K_\infty , E[p^\infty ])$
is
$\Lambda $
-cotorsion with vanishing of
$\mu $
-invariant, then
$\mathrm {Sel}_{\mathrm {min}}(K_\infty , E[p^\infty ])$
is also
$\Lambda $
-cotorsion with vanishing of
$\mu $
-invariant.
Proof We have
$\mathrm {Sel}_{N^-}(K_\infty , E[p^\infty ])[p] = \mathrm {Sel}_{N^-}(K_\infty , E[p])$
since
$N^-$
-ordinary Selmer groups of
$E[p^\infty ]$
are defined as the injective limit of
$N^-$
-ordinary Selmer groups of
$E[p^k]$
. By the assumption,
$\mathrm {Sel}_{N^-}(K_\infty , E[p])$
is finite as noted in the proof of [Reference Kim, Pollack and Weston13, Corollary 2.3]. Since the inclusion
$\mathrm {Sel}_{N^-}(K_\infty , E[p]) \subseteq \mathrm {Sel}_{\mathrm {min}}(K_\infty , E[p^\infty ])[p]$
is of finite index,
$\mathrm {Sel}_{\mathrm {min}}(K_\infty , E[p^\infty ])[p]$
is also finite. By the same reasoning, the conclusion follows.
Proposition 3.2 Under (Im), if
$\mathrm {Sel}_{\mathrm {min}}(K_\infty , E[p^\infty ])$
is
$\Lambda $
-cotorsion, then
$\mathrm {Sel}_{\mathrm {min}}(K_\infty , E[p^\infty ])$
has no proper
$\Lambda $
-submodule of finite index. Thus, we have
$$ \begin{align*} \mathrm{char}_{\Lambda} \mathrm{Sel}_{\mathrm{min}}(K_\infty, E[p^\infty]) & = \mathrm{Fitt}_{\Lambda} \mathrm{Sel}_{\mathrm{min}}(K_\infty, E[p^\infty]) ,\\ \mathrm{Sel}_{\mathrm{min}}(K_\infty, E[p^\infty]) & \simeq \mathrm{Sel}_{N^-}(K_\infty, E[p^\infty]). \end{align*} $$
Proof This follows from [Reference Greenberg and Viola7, Proposition 4.14], which covers the cyclotomic case actually, but the argument generalizes to our setting as mentioned in the proof of [Reference Pollack and Weston16, Proposition 3.6].
The following corollary follows from (3.1) and the above two propositions.
Corollary 3.3 Under (Im) and (Ram), if
$\mathrm {Sel}_{N^-}(K_\infty , E[p^\infty ])$
is
$\Lambda $
-cotorsion with vanishing of
$\mu $
-invariant, we have isomorphisms
$$ \begin{align*}\mathrm{Sel}_{N^-}(K_\infty, E[p^\infty]) \simeq \mathrm{Sel}_{\mathrm{min}}(K_\infty, E[p^\infty]) \simeq \mathrm{Sel}_{\mathrm{Gr}}(K_\infty, E[p^\infty]).\end{align*} $$
4 The proof of the main theorem via Iwasawa theory
We first gather some tools from Iwasawa theory and give a proof of Theorem 1.1.
4.1 Iwasawa theory
The anticyclotomic main conjecture for
$(E,p, K)$
is now completely known for our setting.
Theorem 4.1 Under (Im),(Ram),(Spl),(Na), and (Def), we have the following statements.
-
(1)
$L_p(E/K_\infty )$
is non-zero. -
(2)
$\mu (L_p(E/K_\infty ))= 0 $
. -
(3)
$\mathrm {Sel}_{N^-}(K_\infty , E[p^\infty ])$
is
$\Lambda $
-cotorsion with vanishing of
$\mu $
-invariants. -
(4)
$\left ( L_p(E/K_\infty ) \right ) = \mathrm {char}_\Lambda \left ( \mathrm {Sel}(K_\infty , E[p^\infty ])^\vee \right )$
.
Proof
-
(1) It is proved in [Reference Vatsal18].
-
(2) It is proved in [Reference Vatsal19].
-
(3) This follows from (1), (2), and the Euler system divisibility
obtained from the bipartite Euler system argument [Reference Bertolini and Darmon2, Reference Pollack and Weston16]. Condition (Na) is implicitly used in the Euler system argument. See [Reference Kim, Pollack and Weston13, Assumption 1.1 and Remark 1.4] for this issue.
$$ \begin{align*}\left( L_p(E/K_\infty) \right) \subseteq \mathrm{char}_\Lambda \left( \mathrm{Sel}_{N^-}(K_\infty, E[p^\infty])^\vee \right)\end{align*} $$
-
(4) By using (3) and Corollary 3.3, we can identify
$\mathrm {Sel}_{N^-}(K_\infty , E[p^\infty ])$
with the minimal Selmer group. By [Reference Greenberg and Viola7, Proposition 2.1], the minimal Selmer group also coincides with the classical Selmer group. The opposite divisibility follows from [Reference Skinner and Urban17]. Condition (Spl) is needed only for this last statement to invoke [Reference Skinner and Urban17].
$$ \begin{align*}\left( L_p(E/K_\infty) \right) \supseteq \mathrm{char}_\Lambda \left( \mathrm{Sel}(K_\infty, E[p^\infty])^\vee \right)\end{align*} $$
Corollary 4.2 Under (Im),(Ram),(Na), and (Def), the classical Selmer group
$\mathrm {Sel}(K_\infty , E[p^\infty ])$
has no proper
$\Lambda $
-submodule of finite index; thus,
$$ \begin{align*}\mathrm{char}_{\Lambda} \mathrm{Sel}(K_\infty, E[p^\infty]) = \mathrm{Fitt}_{\Lambda} \mathrm{Sel}(K_\infty, E[p^\infty]) .\end{align*} $$
Proof By Theorem 4.1.(3),
$\mathrm {Sel}_{N^-}(K_\infty , E[p^\infty ])$
is
$\Lambda $
-cotorsion. By Proposition 3.1,
$\mathrm {Sel}_{\mathrm {min}}(K_\infty , E[p^\infty ])$
is also
$\Lambda $
-cotorsion. The conclusion follows from Proposition 3.2 and the identification of the minimal Selmer group and the classical Selmer group.
We recall the control theorem.
Proposition 4.3 (Control theorem)
Let 
. Under (Im), (Na), and (Def), the restriction map
$$ \begin{align*}\mathrm{Sel}(K_n, E[p^\infty]) \to \mathrm{Sel}(K_\infty, E[p^\infty])[\omega_n]\end{align*} $$
is injective with the finite cokernel whose size is bounded independently of n. If we further assume (Ram), then it is an isomorphism.
Proof See [Reference Chida and Hsieh4, Proposition 1.9] with the identifications of Selmer groups in Corollaries 3.3 and 4.2.
4.2 The proof of Theorem 1.1
By the anticyclotomic main conjecture (Theorem 4.1), we have
$$ \begin{align*}\left( L_p(E/K_\infty) \right) = \mathrm{char}_{\Lambda} \left( \mathrm{Sel}(K_\infty, E[p^\infty])^\vee \right) .\end{align*} $$
By Corollary 4.2, the above equality becomes
$$ \begin{align*}\left( L_p(E/K_\infty) \right) = \mathrm{Fitt}_{\Lambda} \left( \mathrm{Sel}(K_\infty, E[p^\infty])^\vee \right)\end{align*} $$
Under the quotient map
$\Lambda \to \Lambda _n = \Lambda / \omega _n$
, it becomes
$$ \begin{align*}\left( \left( \theta(f_\alpha/K_n) \cdot \iota( \theta(f_\alpha/K_n) ) \right) \right) = \mathrm{Fitt}_{\Lambda_n} \left( \left( \mathrm{Sel}(K_\infty, E[p^\infty]) [\omega_n]\right)^\vee \right)\end{align*} $$
since Fitting ideals are compatible with base change. By using the functional equation of theta elements (2.4) and the control theorem (Proposition 4.3), we have
$$ \begin{align*}\left( \theta(f_\alpha/K_n) \right)^2 = \mathrm{Fitt}_{\Lambda_n} \left( \mathrm{Sel}(K_n, E[p^\infty])^\vee \right) .\end{align*} $$
Theorem 1.1 now follows from Lemma 2.2, and the ideal is principal thanks to the above equality.
Acknowledgement
I would like to thank Rob Pollack and Masato Kurihara very much. Rob Pollack first suggested me to think about this problem when I was a Ph.D. student and I was able to solve it after having the discussions with Masato Kurihara. In particular, I learned the idea of the proof of Lemma 2.2 on the p-destabilization process from the discussion with Masato Kurihara at his home, January 3, 2017. The main result of this article partially refines that of [Reference Kim8]. We are grateful to thank the referee for pointing out inaccuracies and unrefined writings. The explanation is greatly improved in the revised version following the referee’s suggestions.
Funding
C.-H.K. was partially supported by research funds for newly appointed professors of Jeonbuk National University in 2024, and by Global-Learning & Academic research institution for Master’s Ph.D. Students, and Postdocs (LAMP) Program of the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. RS-2024-00443714).
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