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On the Fitting ideals of anticyclotomic Selmer groups of elliptic curves with good ordinary reduction

Chan-Ho Kim Department of Mathematics and Institute of Pure and Applied Mathematics, Jeonbuk National University, 567 Baekje-daero, Deokjin-gu, Jeonju, Jeollabuk-do 54896, Republic of Korea chanho.math@gmail.com
Abstract.

We give a short proof of the anticyclotomic analogue of the “strong” main conjecture of Kurihara on Fitting ideals of Selmer groups for elliptic curves with good ordinary reduction under mild hypotheses. More precisely, we completely determine the initial Fitting ideal of Selmer groups over finite subextensions of an imaginary quadratic field in its anticyclotomic p\mathbb{Z}_{p}-extension in terms of Bertolini–Darmon’s theta elements.

Key words and phrases:
refined Iwasawa theory, anticyclotomic Iwasawa theory
2010 Mathematics Subject Classification:
11F67, 11G40, 11R23
Chan-Ho Kim was partially supported by a KIAS Individual Grant (SP054103) via the Center for Mathematical Challenges at Korea Institute for Advanced Study, by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2018R1C1B6007009, 2019R1A6A1A11051177), by research funds for newly appointed professors of Jeonbuk National University in 2024, and by Global-Learning & Academic research institution for Master’s\cdotPh.D. Students, and Postdocs (LAMP) Program of the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. RS-2024-00443714).

1. Introduction

1.1. The statement of the main result

Let EE be an elliptic curve of conductor NN over \mathbb{Q} and p5p\geq 5 be a prime of good ordinary reduction for EE such that

  • (Im)

    the mod pp Galois representation ρ¯:G=Gal(¯/)Aut𝔽p(E[p])GL2(𝔽p)\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)

    ρ¯\overline{\rho} is ramified at every prime dividing NN, so pp does not divide Tamagawa factors of EE.

Let KK be an imaginary quadratic field of odd discriminant DK<4-D_{K}<-4 with (DK,Np)=1(D_{K},Np)=1 such that

  • (Spl)

    pp splits in KK, and

  • (Na)

    ap(E)1(modp)a_{p}(E)\not\equiv 1\pmod{p}.

Write

N=N+NN=N^{+}\cdot N^{-}

where a prime divisor of N+N^{+} splits in KK and a prime divisor of NN^{-} is inert in KK.

  • (Def)

    Assume that NN^{-} is a square-free product of an odd number of primes.

Let KK_{\infty} be the anticyclotomic p\mathbb{Z}_{p}-extension of KK and KnK_{n} be the subextension of KK in KK_{\infty} of degree pnp^{n} for n0n\geq 0. Let Λn=p[Gal(Kn/K)]p[X]/((1+X)pn1)\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

θn(E/K)=σGal(Kn/K)aσσΛn\theta_{n}(E/K)=\sum_{\sigma\in\mathrm{Gal}(K_{n}/K)}a_{\sigma}\cdot\sigma\in\Lambda_{n}

Bertolini–Darmon’s theta element of EE over KnK_{n} which interpolates the square-roots of L(E/K,χ,1)L(E/K,\chi,1) for finite order characters χ\chi on Gal(Kn/K)\mathrm{Gal}(K_{n}/K). It is reviewed in §\S2. For the natural projection map πn,n1:ΛnΛn1\pi_{n,n-1}:\Lambda_{n}\to\Lambda_{n-1}, let νn1,n:Λn1Λn\nu_{n-1,n}:\Lambda_{n-1}\to\Lambda_{n} be the map defined by σπn,n1(τ)=στ\sigma\mapsto\sum_{\pi_{n,n-1}(\tau)=\sigma}\tau. Then we have the equality of ideals of Λn\Lambda_{n} (Lemma 2.1)

(θn(E/K),νn1,n(θn1(E/K)))=(νm,n(θm(E/K)):0mn),\left(\theta_{n}(E/K),\nu_{n-1,n}\left(\theta_{n-1}(E/K)\right)\right)=\left(\nu_{m,n}\left(\theta_{m}(E/K)\right):0\leq m\leq n\right),

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 [Kur02, Conj. 0.3], which refines the “weak” main conjecture of Mazur and Tate [MT87, Conj. 3].

Theorem 1.1.

Under the assumptions mentioned above, i.e. (Im),(Ram),(Spl),(Na), and (Def), the theta elements over KmK_{m} with 0mn0\leq m\leq n generates the initial Fitting ideal of dual Selmer groups over KnK_{n}, i.e. we have equality of ideals of Λn\Lambda_{n}

(θ(E/Kn),νn1,n(θ(E/Kn1)))2=FittΛn(Sel(Kn,E[p])),\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),

which is indeed a principal ideal, where Sel(Kn,E[p])\mathrm{Sel}(K_{n},E[p^{\infty}]) is the classical Selmer group of E[p]E[p^{\infty}] over KnK_{n} and ()(-)^{\vee} means the Pontryagin dual.

The strategy of our proof follows that given in [KK21] and we also add some details on the “pp-destabilization” process and on the comparison of various anticyclotomic Selmer groups of elliptic curves.

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 PhD 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 pp-destabilization process from the discussion with Masato Kurihara at his home, January 3, 2017. The main result of this article partially refines that of [Kim].

2. Bertolini–Darmon’s theta elements and anticyclotomic pp-adic LL-functions

We quickly review the construction of Gross points of conductor pnp^{n}, theta elements, and anticyclotomic pp-adic LL-functions. See [CH15, CH18, Kim19] for details.

2.1. Gross points

Let KK be the imaginary quadratic field of odd discriminant DK<4-D_{K}<-4. Define

ϑ:=DKDK2\vartheta:=\dfrac{D_{K}-\sqrt{-D_{K}}}{2}

so that 𝒪K=+ϑ\mathcal{O}_{K}=\mathbb{Z}+\mathbb{Z}\vartheta. Let BNB_{N^{-}} be the definite quaternion algebra over \mathbb{Q} of discriminant NN^{-}. Then there exists an embedding Ψ:KBN\Psi:K\hookrightarrow B_{N^{-}} [Vig80]. More explicitly, we choose a KK-basis (1,J)(1,J) of BNB_{N^{-}} so that BN=KKJB_{N^{-}}=K\oplus K\cdot J such that β:=J2×\beta:=J^{2}\in\mathbb{Q}^{\times} with β<0\beta<0, Jt=t¯JJ\cdot t=\overline{t}\cdot J for all tKt\in K, β(q×)2\beta\in\left(\mathbb{Z}^{\times}_{q}\right)^{2} for all qpN+q\mid pN^{+}, and βq×\beta\in\mathbb{Z}^{\times}_{q} for all qDKq\mid D_{K}. Fix a square root β¯\sqrt{\beta}\in\overline{\mathbb{Q}} of β\beta. For a \mathbb{Z}-module AA, write A^=A^\widehat{A}=A\otimes\widehat{\mathbb{Z}}. Fix an isomorphism

i:=iq:B^N(N)M2(𝔸(N))i:=\prod i_{q}:\widehat{B}^{(N^{-})}_{N^{-}}\simeq\mathrm{M}_{2}(\mathbb{A}^{(N^{-}\infty)})

as follows:

  • For each finite place qN+pq\mid N^{+}p, the isomorphism iq:BN,qM2(q)i_{q}:B_{N^{-},q}\simeq\mathrm{M}_{2}(\mathbb{Q}_{q}) is defined by

    iq(ϑ)=(trd(ϑ)nrd(ϑ)10),\textstyle{{i_{q}(\vartheta)=\left(\begin{matrix}\mathrm{trd}(\vartheta)&-\mathrm{nrd}(\vartheta)\\ 1&0\end{matrix}\right)},}iq(J)=β(1trd(ϑ)01)\textstyle{{i_{q}(J)=\sqrt{\beta}\cdot\left(\begin{matrix}-1&\mathrm{trd}(\vartheta)\\ 0&1\end{matrix}\right)}}

    where trd\mathrm{trd} and nrd\mathrm{nrd} are the reduced trace and the reduced norm on BB, respectively.

  • For each finite place qpN+q\nmid pN^{+}, the isomorphism iq:BN,qM2(q)i_{q}:B_{N^{-},q}\simeq\mathrm{M}_{2}(\mathbb{Q}_{q}) is chosen so that iq(𝒪Kq)M2(q)i_{q}\left(\mathcal{O}_{K}\otimes\mathbb{Z}_{q}\right)\subseteq\mathrm{M}_{2}(\mathbb{Z}_{q}).

Under the fixed isomorphism ii, for any rational prime qq, the local Gross point ςqBN,q×\varsigma_{q}\in B^{\times}_{N^{-},q} is defined as follows:

  • ςq:=1\varsigma_{q}:=1 in BN,q×B^{\times}_{N^{-},q} for qpN+q\nmid pN^{+}.

  • ςq:=1DK(ϑϑ¯11)GL2(K𝔮)=GL2(q)\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 qN+q\mid N^{+} with q=𝔮𝔮¯q=\mathfrak{q}\overline{\mathfrak{q}} in 𝒪K\mathcal{O}_{K}.

  • ςp(n)=(ϑ110)(pn001)GL2(K𝔭)=GL2(p)\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=𝔭𝔭¯p=\mathfrak{p}\overline{\mathfrak{p}} splits in KK.

Let Ψ^:K^B^N\widehat{\Psi}:\widehat{K}\hookrightarrow\widehat{B}_{N^{-}} be the adelic version of Ψ\Psi. We define xn:K^×B^N×x_{n}:\widehat{K}^{\times}\to\widehat{B}^{\times}_{N^{-}} by xn(a)=Ψ^(a)ς(n):=Ψ^(a)(ςp(n)×qpςq)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 {xn(a):aK^×}\left\{x_{n}(a):a\in\widehat{K}^{\times}\right\} of points is called the Gross points of conductor pnp^{n} on B^N×\widehat{B}^{\times}_{N^{-}}. The fixed embedding KBNK\hookrightarrow B_{N^{-}} also induces an optimal embedding of 𝒪n=+pn𝒪K\mathcal{O}_{n}=\mathbb{Z}+p^{n}\mathcal{O}_{K} into the Eichler order BNς(n)R^N+(ς(n))1B_{N^{-}}\cap\varsigma^{(n)}\widehat{R}_{N^{+}}(\varsigma^{(n)})^{-1} where RN+R_{N^{+}} is the Eichler order of level N+N^{+} under the fixed isomorphism ii.

2.2. Theta elements

Let ϕf:BN×\B^N×/R^N+×\phi_{f}:B^{\times}_{N^{-}}\backslash\widehat{B}^{\times}_{N^{-}}/\widehat{R}^{\times}_{N^{+}}\to\mathbb{C} be the Jacquet–Langlands transfer of ff. Since BN×\B^N×/R^N+×B^{\times}_{N^{-}}\backslash\widehat{B}^{\times}_{N^{-}}/\widehat{R}^{\times}_{N^{+}} is a finite set and ff is a Hecke eigenform, we are able to and do normalize

ϕf:BN×\B^N×/R^N+×p\phi_{f}:B^{\times}_{N^{-}}\backslash\widehat{B}^{\times}_{N^{-}}/\widehat{R}^{\times}_{N^{+}}\to\mathbb{Z}_{p}

such that the image of ϕf\phi_{f} does not lie in ppp\mathbb{Z}_{p}. This integral normalization is related to the congruence ideals [PW11, Kim17, KO23]. Let

θ~n(E/K)=[a]𝒢nϕf(xn(a))[a]p[𝒢n]\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}]

where 𝒢n=K×\K^×/𝒪^n×\mathcal{G}_{n}=K^{\times}\backslash\widehat{K}^{\times}/\widehat{\mathcal{O}}^{\times}_{n} and [a][a] is the image of aK^×a\in\widehat{K}^{\times} in 𝒢n\mathcal{G}_{n}. Then Bertolini–Darmon’s theta element θn(E/K)\theta_{n}(E/K) of EE over KnK_{n} is defined by the image of θ~n(E/K)\widetilde{\theta}_{n}(E/K) in Λn\Lambda_{n}

p[𝒢n]\textstyle{\mathbb{Z}_{p}[\mathcal{G}_{n}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Λn\textstyle{\Lambda_{n}}θ~n(E/K)\textstyle{\widetilde{\theta}_{n}(E/K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}θn(E/K).\textstyle{\theta_{n}(E/K).}

It is known that θn(E/K)\theta_{n}(E/K) interpolates “an half of” L(E,χ,1)L(E,\chi,1) where χ\chi runs over characters on Gal(Kn/K)\mathrm{Gal}(K_{n}/K) [CH18, Kim19]. Because θn(E/K)\theta_{n}(E/K) depends on the choice of Gross points, θn(E/K)\theta_{n}(E/K) is well-defined only up to multiplication by Gal(Kn/K)\mathrm{Gal}(K_{n}/K).

2.3. pp-adic LL-functions

Let f(z)=n1anqnS2(Γ0(N))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 coefficient corresponding to EE via the modularity theorem [BCDT01]. Let α,β\alpha,\beta be the roots of the Hecke polynomial X2apX+pX^{2}-a_{p}X+p at pp. Since ff is ordinary at pp, one of them, say α\alpha, is a pp-adic unit.

The pp-stabilization fαS2(Γ0(Np))f_{\alpha}\in S_{2}(\Gamma_{0}(Np)) of ff is defined by

fα(z)=f(z)βf(pz)f_{\alpha}(z)=f(z)-\beta\cdot f(pz)

whose UpU_{p}-eigenvalue is α\alpha. Then the theta element of fαf_{\alpha} over KnK_{n} is characterized by the following relation:

(2.1) θn(fα/K)=1αn(θn(E/K)1ανn1,n(θn1(E/K))).\theta_{n}(f_{\alpha}/K)=\dfrac{1}{\alpha^{n}}\cdot\left(\theta_{n}(E/K)-\dfrac{1}{\alpha}\cdot\nu_{n-1,n}(\theta_{n-1}(E/K))\right).

It is known that theta elements of EE satisfies the three term relation (e.g. [DI08, Lem. 2.6])

(2.2) πn+1,n(θn+1(E/K))=apθn(E/K)νn1,n(θn1(E/K))\pi_{n+1,n}\left(\theta_{n+1}(E/K)\right)=a_{p}\cdot\theta_{n}(E/K)-\nu_{n-1,n}\left(\theta_{n-1}(E/K)\right)

and the theta elements of fαf_{\alpha} satisfy the norm compatibility

(2.3) πn+1,n(θn+1(fα/K))=θn(fα/K).\pi_{n+1,n}\left(\theta_{n+1}(f_{\alpha}/K)\right)=\theta_{n}(f_{\alpha}/K).

Using these relations, we prove two useful lemmas.

Lemma 2.1.

We have an equality of ideals of Λn\Lambda_{n}

(θ(E/Kn),νn1,n(θ(E/Kn1)))=(νm,n(θ(E/Km)):0mn).\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).
Proof.

From the three term relation (2.2), we have

νn1,n(πn,n1(θn(E/K)))=apνn1,n(θn1(E/K))νn2,n(θn2(E/K)).\nu_{n-1,n}\left(\pi_{n,n-1}\left(\theta_{n}(E/K)\right)\right)=a_{p}\cdot\nu_{n-1,n}\left(\theta_{n-1}(E/K)\right)-\nu_{n-2,n}\left(\theta_{n-2}(E/K)\right).

Since νn1,n(πn,n1(θn(E/K)))=fnθn(E/K)\nu_{n-1,n}\left(\pi_{n,n-1}\left(\theta_{n}(E/K)\right)\right)=f_{n}\cdot\theta_{n}(E/K) for some fnΛnf_{n}\in\Lambda_{n}, we have

νn2,n(θn2(E/K))(θn(E/K),νn1,n(θn1(E/K))).\nu_{n-2,n}\left(\theta_{n-2}(E/K)\right)\in\left(\theta_{n}(E/K),\nu_{n-1,n}\left(\theta_{n-1}(E/K)\right)\right).

Then we can deduce

νn3,n(θn3(E/K))(θ(E/Kn),νn1,n(θ(E/Kn1)))\nu_{n-3,n}\left(\theta_{n-3}(E/K)\right)\in\left(\theta(E/K_{n}),\nu_{n-1,n}\left(\theta(E/K_{n-1})\right)\right)

from the three term relation

νn2,n(πn1,n2(θn1(E/K)))=apνn2,n(θn2(E/K))νn3,n(θn3(E/K))\nu_{n-2,n}\left(\pi_{n-1,n-2}\left(\theta_{n-1}(E/K)\right)\right)=a_{p}\cdot\nu_{n-2,n}\left(\theta_{n-2}(E/K)\right)-\nu_{n-3,n}\left(\theta_{n-3}(E/K)\right)

and νn2,n(πn1,n2(θn1(E/K)))=fn1θn1(E/K)\nu_{n-2,n}\left(\pi_{n-1,n-2}\left(\theta_{n-1}(E/K)\right)\right)=f_{n-1}\cdot\theta_{n-1}(E/K) for some fn1Λnf_{n-1}\in\Lambda_{n}. By applying this argument recursively, the conclusion follows. ∎

Lemma 2.2.

Under (Na), we have an equality of ideals of Λn\Lambda_{n}

(θ(E/Kn),νn1,n(θ(E/Kn1)))=(θn(fα/K)).\left(\theta(E/K_{n}),\nu_{n-1,n}\left(\theta(E/K_{n-1})\right)\right)=\left(\theta_{n}(f_{\alpha}/K)\right).
Proof.

By the definition of the pp-stabilization (2.1), we have one inclusion \supseteq. Hence, we focus on the opposite inclusion. By the interpolation formula of the anticyclotomic pp-adic LL-functions, we have the comparison of LL-values

θ0(fα/K)=(11α)θ0(E/K).\theta_{0}(f_{\alpha}/K)=\left(1-\dfrac{1}{\alpha}\right)\cdot\theta_{0}(E/K).

Under (Na), we have equality in p\mathbb{Z}_{p}

(11α)1θ0(fα/K)=θ0(E/K),\left(1-\dfrac{1}{\alpha}\right)^{-1}\cdot\theta_{0}(f_{\alpha}/K)=\theta_{0}(E/K),

so we have (θ0(E/K))(θ0(fα/K))(\theta_{0}(E/K))\subseteq(\theta_{0}(f_{\alpha}/K)). In fact, they are the same ideal. From (2.1) and (2.3), we have

θ1(fα/K)\displaystyle\theta_{1}(f_{\alpha}/K) =1α(θ1(E/K)1αν0,1(θ0(E/K)))\displaystyle=\dfrac{1}{\alpha}\cdot\left(\theta_{1}(E/K)-\dfrac{1}{\alpha}\cdot\nu_{0,1}(\theta_{0}(E/K))\right)
=1α(θ1(E/K)1αν0,1((11α)1θ0(fα/K)))\displaystyle=\dfrac{1}{\alpha}\cdot\left(\theta_{1}(E/K)-\dfrac{1}{\alpha}\cdot\nu_{0,1}(\left(1-\dfrac{1}{\alpha}\right)^{-1}\cdot\theta_{0}(f_{\alpha}/K))\right)
=1α(θ1(E/K)1α(11α)1ν0,1(π1,0(θ1(fα/K))))\displaystyle=\dfrac{1}{\alpha}\cdot\left(\theta_{1}(E/K)-\dfrac{1}{\alpha}\cdot\left(1-\dfrac{1}{\alpha}\right)^{-1}\cdot\nu_{0,1}(\pi_{1,0}(\theta_{1}(f_{\alpha}/K)))\right)
=1α(θ1(E/K)1α(11α)1f1θ1(fα/K))\displaystyle=\dfrac{1}{\alpha}\cdot\left(\theta_{1}(E/K)-\dfrac{1}{\alpha}\cdot\left(1-\dfrac{1}{\alpha}\right)^{-1}\cdot f_{1}\cdot\theta_{1}(f_{\alpha}/K)\right)

for some f1Λ1f_{1}\in\Lambda_{1}. This shows that θ1(E/K)=g1θ1(fα/K)\theta_{1}(E/K)=g_{1}\cdot\theta_{1}(f_{\alpha}/K) for some g1Λ1g_{1}\in\Lambda_{1}.

We suppose that θn1(E/K)=gn1θn1(fα/K)\theta_{n-1}(E/K)=g_{n-1}\cdot\theta_{n-1}(f_{\alpha}/K) for some gn1Λn1g_{n-1}\in\Lambda_{n-1}.

θn(fα/K)\displaystyle\theta_{n}(f_{\alpha}/K) =1αn(θn(E/K)1ανn1,n(θn1(E/K)))\displaystyle=\dfrac{1}{\alpha^{n}}\cdot\left(\theta_{n}(E/K)-\dfrac{1}{\alpha}\cdot\nu_{n-1,n}(\theta_{n-1}(E/K))\right)
=1αn(θn(E/K)1ανn1,n(gn1θn1(fα/K)))\displaystyle=\dfrac{1}{\alpha^{n}}\cdot\left(\theta_{n}(E/K)-\dfrac{1}{\alpha}\cdot\nu_{n-1,n}(g_{n-1}\cdot\theta_{n-1}(f_{\alpha}/K))\right)
=1αn(θn(E/K)1αgn1νn1,n(πn,n1(θn(fα/K))))\displaystyle=\dfrac{1}{\alpha^{n}}\cdot\left(\theta_{n}(E/K)-\dfrac{1}{\alpha}\cdot g_{n-1}\cdot\nu_{n-1,n}(\pi_{n,n-1}(\theta_{n}(f_{\alpha}/K)))\right)
=1αn(θ1(E/K)1αgn1fnθn(fα/K))\displaystyle=\dfrac{1}{\alpha^{n}}\cdot\left(\theta_{1}(E/K)-\dfrac{1}{\alpha}\cdot g_{n-1}\cdot f_{n}\cdot\theta_{n}(f_{\alpha}/K)\right)

for some fnΛnf_{n}\in\Lambda_{n}. This shows that θn(E/K)=gnθn(fα/K)\theta_{n}(E/K)=g_{n}\cdot\theta_{n}(f_{\alpha}/K) for some gnΛ1g_{n}\in\Lambda_{1}. By induction, we have inclusion

(θn(E/K))(θn(fα/K)),(\theta_{n}(E/K))\subseteq(\theta_{n}(f_{\alpha}/K)),

so we also have

(νn1,n(θn1(E/K)))(νn1,n(θn1(fα/K))).(\nu_{n-1,n}\left(\theta_{n-1}(E/K)\right))\subseteq(\nu_{n-1,n}\left(\theta_{n-1}(f_{\alpha}/K)\right)).

Since νn1,n(θn1(fα/K))=fnθn(fα/K)\nu_{n-1,n}\left(\theta_{n-1}(f_{\alpha}/K)\right)=f_{n}\cdot\theta_{n}(f_{\alpha}/K), we have

(νn1,n(θn1(fα/K)))(θn(fα/K)).(\nu_{n-1,n}\left(\theta_{n-1}(f_{\alpha}/K)\right))\subseteq(\theta_{n}(f_{\alpha}/K)).

The conclusion follows. ∎

Let ι\iota be the involution on Λn\Lambda_{n} defined by inverting group-like elements, so we have

ι(σGal(Kn/K)aσσ)=σGal(Kn/K)aσσ1.\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}.

We define the anticyclotomic pp-adic LL-function of EE by

Lp(E/K)=limn(θn(fα/K)ι(θn(fα/K)))Λ=limnΛn.L_{p}(E/K_{\infty})=\varprojlim_{n}\left(\theta_{n}(f_{\alpha}/K)\cdot\iota(\theta_{n}(f_{\alpha}/K))\right)\in\Lambda=\varprojlim_{n}\Lambda_{n}.

This element is well-defined. The functional equation for Bertolini–Darmon’s theta elements yields the equality of ideals of Λ\Lambda (e.g. [BD96, Prop. 2.13], [BD05, Lem. 1.5])

(2.4) (θn(fα/K))=(ι(θn(fα/K))).\left(\theta_{n}(f_{\alpha}/K)\right)=\left(\iota(\theta_{n}(f_{\alpha}/K))\right).

3. Comparison of Selmer groups

3.1. Local properties of Galois representations

Let ρ:GAutp(V)=GL2(p)\rho:G_{\mathbb{Q}}\to\mathrm{Aut}_{\mathbb{Q}_{p}}(V)=\mathrm{GL}_{2}(\mathbb{Q}_{p}) be the two-dimensional Galois representation associated to EE.

  • Since EE is good ordinary at pp, we have

    ρ|Gp(χα1χcyc0χα)\rho|_{G_{\mathbb{Q}_{p}}}\sim\begin{pmatrix}\chi^{-1}_{\alpha}\cdot\chi_{\mathrm{cyc}}&*\\ 0&\chi_{\alpha}\end{pmatrix}

    where χα\chi_{\alpha} is the unramified character sending the arithmetic Frobenius at pp to α\alpha.

  • For \ell dividing NN exactly, we also have

    ρ|G(±χcyc0±𝟏).\rho|_{G_{\mathbb{Q}_{\ell}}}\sim\begin{pmatrix}\pm\chi_{\mathrm{cyc}}&*\\ 0&\pm\mathbf{1}\end{pmatrix}.

For a rational prime vv dividing NpN^{-}p, we consider the following subspaces:

  • For v=pv=p, let F+VVF^{+}V\subseteq V be the subspace on which the inertia subgroup IvI_{v} acts by χcyc\chi_{\mathrm{cyc}}.

  • For a rational prime vv dividing NN^{-}, F+VVF^{+}V\subseteq V be the on which the inertia subgroup IvI_{v} acts by χcyc\chi_{\mathrm{cyc}} or χcycτv\chi_{\mathrm{cyc}}\tau_{v} where τv\tau_{v} is the non-trivial unramified quadratic character of GvG_{\mathbb{Q}_{v}}.

Let LL be an algebraic extension of KK. For a prime ww of LL dividing NpN^{-}p, we define the ordinary local condition of VV at ww by

Hord1(Lw,V)=ker(H1(Lw,V)H1(Lw,V/F+V)),\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),

and the same local conditions for TT, T/pkTT/p^{k}T, E[p]E[p^{\infty}], and E[pk]E[p^{k}] are defined by propagation.

3.2. NN^{-}-ordinary (residual) Selmer groups

Let Σ\Sigma be the finite set of places of \mathbb{Q} consisting of the places dividing NpNp\infty, and KΣK_{\Sigma} be the maximal extension of KK unramified outside Σ\Sigma. We write

  • Σ+Σ\Sigma^{+}\subseteq\Sigma to be the subset of Σ\Sigma consisting of the places not dividing pp\infty which splits in K/K/\mathbb{Q}, and

  • ΣΣ\Sigma^{-}\subseteq\Sigma to be the subset of Σ\Sigma consisting of the places not dividing pp\infty which are inert in K/K/\mathbb{Q}.

For a place ww of KK_{\infty}, we write wΣ±w\in\Sigma^{\pm} if ww divides a rational prime \ell contained in Σ±\Sigma^{\pm}, respectively. For every k1k\geq 1, we define the NN^{-}-ordinary (and N+N^{+}-strict) Selmer group of E[pk]E[p^{k}] SelN(K,E[pk])\mathrm{Sel}_{N^{-}}(K_{\infty},E[p^{k}]) by the kernel of the map

H1(KΣ/K,E[pk])wΣ+H1(K,w,E[pk])×wΣ or w|pH1(K,w,E[pk])Hord1(K,w,E[pk])\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|p}\dfrac{\mathrm{H}^{1}(K_{\infty,w},E[p^{k}])}{\mathrm{H}^{1}_{\mathrm{ord}}(K_{\infty,w},E[p^{k}])}

and define SelN(K,E[p])=limkSelN(K,E[pk])\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 [BD05, Def. 2.8].

3.3. Minimal and Greenberg Selmer groups

We follow the convention of [PW11, §3.1]. The minimal Selmer group Selmin(K,E[p])\mathrm{Sel}_{\mathrm{min}}(K_{\infty},E[p^{\infty}]) of E[p]E[p^{\infty}] is defined by the kernel of the map

H1(K,E[p])wpH1(K,w,E[p])×w|pH1(K,w,E[p])Hord1(K,w,E[p]),\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|p}\dfrac{\mathrm{H}^{1}(K_{\infty,w},E[p^{\infty}])}{\mathrm{H}^{1}_{\mathrm{ord}}(K_{\infty,w},E[p^{\infty}])},

and the Greenberg Selmer group SelGr(K,E[p])\mathrm{Sel}_{\mathrm{Gr}}(K_{\infty},E[p^{\infty}]) of E[p]E[p^{\infty}] is defined by the kernel of the map

H1(K,E[p])wpH1(I,w,E[p])×w|pH1(K,w,E[p])Hord1(K,w,E[p])\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|p}\dfrac{\mathrm{H}^{1}(K_{\infty,w},E[p^{\infty}])}{\mathrm{H}^{1}_{\mathrm{ord}}(K_{\infty,w},E[p^{\infty}])}

where I,wI_{\infty,w} is the inertia subgroup of GK,wG_{K_{\infty,w}}. Under (Ram), ρ¯\overline{\rho} is ramified at every prime dividing NN^{-}, so pp does not divide any Tamagawa factors. Then by using [PW11, Lem. 3.4], we have an isomorphism

(3.1) Selmin(K,E[p])SelGr(K,E[p]).\mathrm{Sel}_{\mathrm{min}}(K_{\infty},E[p^{\infty}])\simeq\mathrm{Sel}_{\mathrm{Gr}}(K_{\infty},E[p^{\infty}]).

3.4. The comparison

We recall the final displayed equation in the proof of [PW11, Prop. 3.6]:

0\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}SelN(K,E[pk])\textstyle{\mathrm{Sel}_{N^{-}}(K_{\infty},E[p^{k}])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Selmin(K,E[p])[pk]\textstyle{\mathrm{Sel}_{\mathrm{min}}(K_{\infty},E[p^{\infty}])[p^{k}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}w(E[p])GK,wpk(E[p])GK,w\textstyle{\prod_{w}\dfrac{(E[p^{\infty}])^{G_{K_{\infty,w}}}}{p^{k}(E[p^{\infty}])^{G_{K_{\infty,w}}}}}

where ww runs over the primes of KK_{\infty} dividing N+N^{+}. The local conditions at primes dividing NN^{-} of minimal Selmer groups and NN^{-}-ordinary Selmer groups coincide since such primes split completely in K/KK_{\infty}/K. Thus, we have inclusion

SelN(K,E[pk])Selmin(K,E[p])[pk]\mathrm{Sel}_{N^{-}}(K_{\infty},E[p^{k}])\subseteq\mathrm{Sel}_{\mathrm{min}}(K_{\infty},E[p^{\infty}])[p^{k}]

which is of finite index and is independent of kk.

Proposition 3.1.

If SelN(K,E[p])\mathrm{Sel}_{N^{-}}(K_{\infty},E[p^{\infty}]) is Λ\Lambda-cotorsion with vanishing of μ\mu-invariant, then Selmin(K,E[p])\mathrm{Sel}_{\mathrm{min}}(K_{\infty},E[p^{\infty}]) is also Λ\Lambda-cotorsion with vanishing of μ\mu-invariant.

Proof.

We have SelN(K,E[p])[p]=SelN(K,E[p])\mathrm{Sel}_{N^{-}}(K_{\infty},E[p^{\infty}])[p]=\mathrm{Sel}_{N^{-}}(K_{\infty},E[p]) since NN^{-}-ordinary Selmer groups of E[p]E[p^{\infty}] are defined as the injective limit of NN^{-}-ordinary Selmer groups of E[pk]E[p^{k}]. By the assumption, SelN(K,E[p])\mathrm{Sel}_{N^{-}}(K_{\infty},E[p]) is finite as noted in the proof of [KPW17, Cor. 2.3]. Since the inclusion SelN(K,E[p])Selmin(K,E[p])[p]\mathrm{Sel}_{N^{-}}(K_{\infty},E[p])\subseteq\mathrm{Sel}_{\mathrm{min}}(K_{\infty},E[p^{\infty}])[p] is of finite index, Selmin(K,E[p])[p]\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 Selmin(K,E[p])\mathrm{Sel}_{\mathrm{min}}(K_{\infty},E[p^{\infty}]) is Λ\Lambda-cotorsion, then Selmin(K,E[p])\mathrm{Sel}_{\mathrm{min}}(K_{\infty},E[p^{\infty}]) has no proper Λ\Lambda-submodule of finite index. Thus, we have

charΛSelmin(K,E[p])\displaystyle\mathrm{char}_{\Lambda}\mathrm{Sel}_{\mathrm{min}}(K_{\infty},E[p^{\infty}]) =FittΛSelmin(K,E[p]),\displaystyle=\mathrm{Fitt}_{\Lambda}\mathrm{Sel}_{\mathrm{min}}(K_{\infty},E[p^{\infty}]),
Selmin(K,E[p])\displaystyle\mathrm{Sel}_{\mathrm{min}}(K_{\infty},E[p^{\infty}]) SelN(K,E[p]).\displaystyle\simeq\mathrm{Sel}_{N^{-}}(K_{\infty},E[p^{\infty}]).
Proof.

This follows from [Gre99, Prop. 4.14], which covers the cyclotomic case actually, but the argument generalizes to our setting as mentioned in the proof of [PW11, Prop. 3.6]. ∎

The following corollary follows from (3.1) and the above two propositions.

Corollary 3.3.

Under (Im) and (Ram), if SelN(K,E[p])\mathrm{Sel}_{N^{-}}(K_{\infty},E[p^{\infty}]) is Λ\Lambda-cotorsion with vanishing of μ\mu-invariant, we have isomorphisms

SelN(K,E[p])Selmin(K,E[p])SelGr(K,E[p]).\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}]).

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)(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. (1)

    Lp(E/K)L_{p}(E/K_{\infty}) is non-zero.

  2. (2)

    μ(Lp(E/K))=0\mu(L_{p}(E/K_{\infty}))=0.

  3. (3)

    SelN(K,E[p])\mathrm{Sel}_{N^{-}}(K_{\infty},E[p^{\infty}]) is Λ\Lambda-cotorsion with vanishing of μ\mu-invariants.

  4. (4)

    (Lp(E/K))=charΛ(Sel(K,E[p]))\left(L_{p}(E/K_{\infty})\right)=\mathrm{char}_{\Lambda}\left(\mathrm{Sel}(K_{\infty},E[p^{\infty}])^{\vee}\right).

Proof.
  1. (1)

    It is proved in [Vat02].

  2. (2)

    It is proved in [Vat03].

  3. (3)

    This follows from (1), (2), and the Euler system divisibility

    (Lp(E/K))charΛ(SelN(K,E[p]))\left(L_{p}(E/K_{\infty})\right)\subseteq\mathrm{char}_{\Lambda}\left(\mathrm{Sel}_{N^{-}}(K_{\infty},E[p^{\infty}])^{\vee}\right)

    obtained from the bipartite Euler system argument [BD05, PW11].

  4. (4)

    By using (3) and Corollary 3.3, we can identify SelN(K,E[p])\mathrm{Sel}_{N^{-}}(K_{\infty},E[p^{\infty}]) with the minimal Selmer group. By [Gre99, Prop. 2.1], the minimal Selmer group also coincides with the classical Selmer group. The opposite divisibility

    (Lp(E/K))charΛ(Sel(K,E[p]))\left(L_{p}(E/K_{\infty})\right)\supseteq\mathrm{char}_{\Lambda}\left(\mathrm{Sel}(K_{\infty},E[p^{\infty}])^{\vee}\right)

    follows from [SU14]. Condition (Spl) is needed only for this to invoke [SU14].

Corollary 4.2.

The classical Selmer group Sel(K,E[p])\mathrm{Sel}(K_{\infty},E[p^{\infty}]) has no proper Λ\Lambda-submodule of finite index; thus,

charΛSel(K,E[p])=FittΛSel(K,E[p]).\mathrm{char}_{\Lambda}\mathrm{Sel}(K_{\infty},E[p^{\infty}])=\mathrm{Fitt}_{\Lambda}\mathrm{Sel}(K_{\infty},E[p^{\infty}]).
Proof.

It 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 ωn=ωn(X)=(1+X)pn1pXΛ\omega_{n}=\omega_{n}(X)=(1+X)^{p^{n}}-1\in\mathbb{Z}_{p}\llbracket X\rrbracket\simeq\Lambda. Under (Im), (Na), and (Def), the restriction map

Sel(Kn,E[p])Sel(K,E[p])[ωn]\mathrm{Sel}(K_{n},E[p^{\infty}])\to\mathrm{Sel}(K_{\infty},E[p^{\infty}])[\omega_{n}]

is injective with the finite cokernel whose size is bounded independently of nn. If we further assume (Ram), then it is an isomorphism.

Proof.

See [CH15, Prop. 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

(Lp(E/K))=charΛ(Sel(K,E[p])).\left(L_{p}(E/K_{\infty})\right)=\mathrm{char}_{\Lambda}\left(\mathrm{Sel}(K_{\infty},E[p^{\infty}])^{\vee}\right).

By Corollary 4.2, the above equality becomes

(Lp(E/K))=FittΛ(Sel(K,E[p]))\left(L_{p}(E/K_{\infty})\right)=\mathrm{Fitt}_{\Lambda}\left(\mathrm{Sel}(K_{\infty},E[p^{\infty}])^{\vee}\right)

Under the quotient map ΛΛn=Λ/ωn\Lambda\to\Lambda_{n}=\Lambda/\omega_{n}, it becomes

((θn(fα/K)ι(θn(fα/K))))=FittΛn((Sel(K,E[p])[ωn]))\left(\left(\theta_{n}(f_{\alpha}/K)\cdot\iota(\theta_{n}(f_{\alpha}/K))\right)\right)=\mathrm{Fitt}_{\Lambda_{n}}\left(\left(\mathrm{Sel}(K_{\infty},E[p^{\infty}])[\omega_{n}]\right)^{\vee}\right)

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

(θn(fα/K))2=FittΛn(Sel(Kn,E[p])).\left(\theta_{n}(f_{\alpha}/K)\right)^{2}=\mathrm{Fitt}_{\Lambda_{n}}\left(\mathrm{Sel}(K_{n},E[p^{\infty}])^{\vee}\right).

Theorem 1.1 now follows from Lemma 2.2, and the ideal is principal thanks to the above equality.

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