On the Fitting ideals of anticyclotomic Selmer groups of elliptic curves with good ordinary reduction
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 -extension in terms of Bertolini–Darmon’s theta elements.
Key words and phrases:
refined Iwasawa theory, anticyclotomic Iwasawa theory2010 Mathematics Subject Classification:
11F67, 11G40, 11R231. Introduction
1.1. The statement of the main result
Let be an elliptic curve of conductor over and be a prime of good ordinary reduction for such that
-
(Im)
the mod Galois representation is surjective, and
-
(Ram)
is ramified at every prime dividing , so does not divide Tamagawa factors of .
Let be an imaginary quadratic field of odd discriminant with such that
-
(Spl)
splits in , and
-
(Na)
.
Write
where a prime divisor of splits in and a prime divisor of is inert in .
-
(Def)
Assume that is a square-free product of an odd number of primes.
Let be the anticyclotomic -extension of and be the subextension of in of degree for . Let be the finite layer Iwasawa algebra. Under (Def), denote by
Bertolini–Darmon’s theta element of over which interpolates the square-roots of for finite order characters on . It is reviewed in 2. For the natural projection map , let be the map defined by . Then we have the equality of ideals of (Lemma 2.1)
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 with generates the initial Fitting ideal of dual Selmer groups over , i.e. we have equality of ideals of
which is indeed a principal ideal, where is the classical Selmer group of over and means the Pontryagin dual.
The strategy of our proof follows that given in [KK21] and we also add some details on the “-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 -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 -adic -functions
We quickly review the construction of Gross points of conductor , theta elements, and anticyclotomic -adic -functions. See [CH15, CH18, Kim19] for details.
2.1. Gross points
Let be the imaginary quadratic field of odd discriminant . Define
so that . Let be the definite quaternion algebra over of discriminant . Then there exists an embedding [Vig80]. More explicitly, we choose a -basis of so that such that with , for all , for all , and for all . Fix a square root of . For a -module , write . Fix an isomorphism
as follows:
-
•
For each finite place , the isomorphism is defined by
where and are the reduced trace and the reduced norm on , respectively.
-
•
For each finite place , the isomorphism is chosen so that .
Under the fixed isomorphism , for any rational prime , the local Gross point is defined as follows:
-
•
in for .
-
•
for with in .
-
•
where splits in .
Let be the adelic version of . We define by . The collection of points is called the Gross points of conductor on . The fixed embedding also induces an optimal embedding of into the Eichler order where is the Eichler order of level under the fixed isomorphism .
2.2. Theta elements
Let be the Jacquet–Langlands transfer of . Since is a finite set and is a Hecke eigenform, we are able to and do normalize
such that the image of does not lie in . This integral normalization is related to the congruence ideals [PW11, Kim17, KO23]. Let
where and is the image of in . Then Bertolini–Darmon’s theta element of over is defined by the image of in
It is known that interpolates “an half of” where runs over characters on [CH18, Kim19]. Because depends on the choice of Gross points, is well-defined only up to multiplication by .
2.3. -adic -functions
Let be the cuspidal newform of weight two with rational Fourier coefficient corresponding to via the modularity theorem [BCDT01]. Let be the roots of the Hecke polynomial at . Since is ordinary at , one of them, say , is a -adic unit.
The -stabilization of is defined by
whose -eigenvalue is . Then the theta element of over is characterized by the following relation:
(2.1) |
It is known that theta elements of satisfies the three term relation (e.g. [DI08, Lem. 2.6])
(2.2) |
and the theta elements of satisfy the norm compatibility
(2.3) |
Using these relations, we prove two useful lemmas.
Lemma 2.1.
We have an equality of ideals of
Proof.
From the three term relation (2.2), we have
Since for some , we have
Then we can deduce
from the three term relation
and for some . By applying this argument recursively, the conclusion follows. ∎
Lemma 2.2.
Under (Na), we have an equality of ideals of
Proof.
By the definition of the -stabilization (2.1), we have one inclusion . Hence, we focus on the opposite inclusion. By the interpolation formula of the anticyclotomic -adic -functions, we have the comparison of -values
Under (Na), we have equality in
so we have . In fact, they are the same ideal. From (2.1) and (2.3), we have
for some . This shows that for some .
We suppose that for some .
for some . This shows that for some . By induction, we have inclusion
so we also have
Since , we have
The conclusion follows. ∎
Let be the involution on defined by inverting group-like elements, so we have
We define the anticyclotomic -adic -function of by
This element is well-defined. The functional equation for Bertolini–Darmon’s theta elements yields the equality of ideals of (e.g. [BD96, Prop. 2.13], [BD05, Lem. 1.5])
(2.4) |
3. Comparison of Selmer groups
3.1. Local properties of Galois representations
Let be the two-dimensional Galois representation associated to .
-
•
Since is good ordinary at , we have
where is the unramified character sending the arithmetic Frobenius at to .
-
•
For dividing exactly, we also have
For a rational prime dividing , we consider the following subspaces:
-
•
For , let be the subspace on which the inertia subgroup acts by .
-
•
For a rational prime dividing , be the on which the inertia subgroup acts by or where is the non-trivial unramified quadratic character of .
Let be an algebraic extension of . For a prime of dividing , we define the ordinary local condition of at by
and the same local conditions for , , , and are defined by propagation.
3.2. -ordinary (residual) Selmer groups
Let be the finite set of places of consisting of the places dividing , and be the maximal extension of unramified outside . We write
-
•
to be the subset of consisting of the places not dividing which splits in , and
-
•
to be the subset of consisting of the places not dividing which are inert in .
For a place of , we write if divides a rational prime contained in , respectively. For every , we define the -ordinary (and -strict) Selmer group of by the kernel of the map
and define . 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 of is defined by the kernel of the map
and the Greenberg Selmer group of is defined by the kernel of the map
where is the inertia subgroup of . Under (Ram), is ramified at every prime dividing , so does not divide any Tamagawa factors. Then by using [PW11, Lem. 3.4], we have an isomorphism
(3.1) |
3.4. The comparison
We recall the final displayed equation in the proof of [PW11, Prop. 3.6]:
where runs over the primes of dividing . The local conditions at primes dividing of minimal Selmer groups and -ordinary Selmer groups coincide since such primes split completely in . Thus, we have inclusion
which is of finite index and is independent of .
Proposition 3.1.
If is -cotorsion with vanishing of -invariant, then is also -cotorsion with vanishing of -invariant.
Proof.
We have since -ordinary Selmer groups of are defined as the injective limit of -ordinary Selmer groups of . By the assumption, is finite as noted in the proof of [KPW17, Cor. 2.3]. Since the inclusion is of finite index, is also finite. By the same reasoning, the conclusion follows. ∎
Proposition 3.2.
Under (Im), if is -cotorsion, then has no proper -submodule of finite index. Thus, we have
Proof.
The following corollary follows from (3.1) and the above two propositions.
Corollary 3.3.
Under (Im) and (Ram), if is -cotorsion with vanishing of -invariant, we have isomorphisms
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 is now completely known for our setting.
Theorem 4.1.
Under (Im),(Ram),(Spl),(Na), and (Def), we have the following statements:
-
(1)
is non-zero.
-
(2)
.
-
(3)
is -cotorsion with vanishing of -invariants.
-
(4)
.
Proof.
∎
Corollary 4.2.
The classical Selmer group has no proper -submodule of finite index; thus,
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 . Under (Im), (Na), and (Def), the restriction map
is injective with the finite cokernel whose size is bounded independently of . If we further assume (Ram), then it is an isomorphism.
4.2. The proof of Theorem 1.1
By the anticyclotomic main conjecture (Theorem 4.1), we have
By Corollary 4.2, the above equality becomes
Under the quotient map , it becomes
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
Theorem 1.1 now follows from Lemma 2.2, and the ideal is principal thanks to the above equality.
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