The growth of Tate-Shafarevich groups of $p$-supersingular elliptic curves over anticyclotomic $\mathbb{Z}_p$-extensions at inert primes
Erman Isik Antonio Lei

Checkpoint and Restart: An Energy Consumption Characterization in Clusters
    Marina Morán1 Javier Balladini1 Dolores Rexachs2 Emilio Luque2
    September 9, 2024
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§ ABSTRACT 
    Let E be an elliptic curve defined over , and let K be an imaginary quadratic field. Consider an odd prime p at which E has good supersingular reduction with a_p(E)=0 and which is inert in K. Under the assumption that the signed Selmer groups are cotorsion modules over the corresponding Iwasawa algebra, we prove that the Mordell–Weil ranks of E are bounded over any subextensions of the anticyclotomic _p-extension of K. Additionally, we provide an asymptotic formula for the growth of the p-parts of the Tate–Shafarevich groups of E over these extensions.




§ INTRODUCTION



Let E be an elliptic curve defined over  and p be an odd prime of good reduction.  Let _ denote the cyclotomic _p-extension of , and _(n)  be the unique subextension of _ of degree p^n for n≥ 0. Suppose that the p-primary component of the Tate–Shafarevich group (E/_(n)) is finite for all n, and let p^e_n =| (E/K_n )[p^∞]|. The variation of e_n in n depends on whether the elliptic curve E has ordinary or supersingular reduction at p.




When E has ordinary reduction at p, it follows from <cit.> that the p-primary Selmer group of E over _ is cotorsion over the associated Iwasawa algebra. It then follows from Mazur's control theorem <cit.> that for n≫0, 
 
    e_n-e_n-1  = λ + (p^n-p^n-1) μ - r_∞,

 where r_∞ is the rank of E over _ (which is finite by <cit.>), and λ and μ are the Iwasawa invariants of the Pontryagin dual of the p-primary Selmer group of E over _. 

 


 When E has supersingular reduction at p, the case where a_p(E)=0 has been studied by Kurihara <cit.>, Kobayashi <cit.> and Pollack <cit.>.  For n≫ 0, we have the following formula:
 
    e_n - e_n-1  = _n^∓ + λ_± + μ_±(p^n-p^n-1) - r_∞,

 where _n^∓ is the product of certain cyclotomic polynomials, r_∞ is the rank of E over _, λ_± and μ_± are the Iwasawa invariants of the cotorsion signed Selmer groups of E over _, and the sign ± depends on the parity of n.




In <cit.>, Sprung obtained an analogous formula for the p-supersingular elliptic curves with a_p(E)≠ 0 and abelian varieties of GL(2)-type. See also <cit.> for related work on upper bounds on the growth of the Bloch–Kato–Shafarevich–Tate groups for higher-weight modular forms. Furthermore, the growth of Mordell–Weil ranks and the Tate–Shafarevich groups of an elliptic curve over the cyclotomic _p-extension of certain number fields have been studied in <cit.>.




    :

    e_n - e_n-1  = q_n^∙ + λ_∙ + μ_∙(p^n-p^n-1) - r_∞,  for  n≫ 0,

 where ∙∈{♯, ♭}, q_n^∙ is again a sum of powers of p, λ_∙, μ_∙ are the Iwasawa invariants of certain chromatic Selmer groups defined in <cit.>, where the choice of ∙ is decided by the Modesty Algorithm <cit.>.







The focus of this article is analogous results for the anticyclotomic -extension of an imaginary quadratic field K. Suppose that p splits in K and the two primes above p splits completely in the anticyclotomic -extension of K.  
Under certain hypotheses on the vanishing of the Mordell–Weil rank and the p-part of the Tate–Shafarevich group of E over K, Iovita and Pollack <cit.> derived a formula of the growth of the Tate–Shafarevich groups over anticyclotomic tower that mimics the cyclotomic case. 




 If we assume in addition that p ≥ 5 and that (E,K) satisfies the Heegner hypothesis, Çiperiani proved in <cit.> that the Tate–Shafarevich group of E has trivial corank over the Iwasawa algebra associated with the anticyclotomic _p-extension of K. In <cit.>, a more precise formula of the growth of the Tate–Shafarevich groups inside this tower is given. More recently, Burungale–Kobayashi–Ota <cit.> studied this question when E has complex multiplication by an order in K. 



 §.§ Statement of the main theorem
 
Let K be an imaginary quadratic field such that p is inert in K. Assume that p does not divide the class number h_K of K and a_p(E)=0. Let K_∞ denote the anticyclotomic _p-extension of K. Note that the unique prime of K above p is totally ramified in K_∞. For an integer n ≥ 0, we write K_n for the unique subextension of K_∞ such that [K_n : K] = p^n.





 The Galois group of K_∞ over K is denoted by Γ. In addition, write Γ_n = (K_∞/K_n) and G_n = (K_n/K). We fix once and for all a topological generator γ of Γ. Let Λ denote the Iwasawa algebra _p[[Γ]]=_n[G_n], which we shall identify with the power series ring _p[[X]] by sending γ-1 to X. Let (-)^∨ := (-,_p/_p) denote the Pontryagin duality functor.





 In this article, we prove that the Mordell–Weil rank of E over K_∞ is bounded assuming that the signed Selmer groups ^±(E/K_∞) are Λ-cotorsion. Furthermore, we derive an asymptotic formula for the growth of the p-parts of the Tate–Shafarevich groups (E/K_n ) of E in terms of the Iwasawa invariants of ^±(E/K_∞)^∨. These Selmer groups have recently been studied in <cit.>, and are similar to those defined by Rubin in <cit.> for CM elliptic curves (see Definition <ref> for the precise definition and Remark <ref> for a discussion on how the plus and minus subgroups used to define these groups compare with the counterparts used to define  Kobayashi's plus and minus Selmer groups in <cit.>). The main result of this article is the following theorem.




 



Let E be an elliptic curve defined over  and p an odd prime where E has good supersingular reduction with a_p(E)=0.
Let K be an imaginary quadratic field such that p is inert in K. Assume that p does not divide the class number h_K. Let K_∞ denote the anticyclotomic _p-extension of K.
  Assume that ^±(E/K_∞)^∨ are both Λ-torsion.  Then, we have:
  
        
  * _ E(K_n) is bounded independently of n;
        


        
        
  * Assume that (E/K_n)[p^
        ∞] is finite for all n≥ 0.  Let p^e_n =| (E/K_n )[p^∞]|. Define r_∞=sup_n≥0{_ E(K_n)}. Let λ_± and μ_± be the Iwasawa invariants of ^±(E/K_∞)^∨. Then, for n≫0, 
      
    e_n-e_n-1=
          2s_n-1  + λ_- + ϕ(p^n)μ_-   -r_∞      if  n   is odd,
    
              2s_n-1+λ_+ + ϕ(p^n)μ_+ -r_∞      if  n  is even,

        where ϕ(p^n)=p^n-p^n-1, and s_n=∑_k=1^n(-1)^n-kp^k for n≥ 0.
    


 



     
    q_n=
                 p^n-1- p^n-2+ p^n-3-p^n-4+… +p       if  n   is odd,
    
                  p^n-1-p^n-2+… + p^2-p       if  n  is even.




 Under certain hypotheses, it is known that ^±(E/K_∞)^∨ are Λ-torsion; see <cit.> and Remark <ref> for a more detailed discussion. Our result is complementary to the recent work of <cit.>, where E is assumed to have complex multiplication by an order in K. In their setting, there exists exactly one element ∙∈{+,-} (the choice of which depends on the root number of E) such that ^∙(E/K_∞)^∨ is Λ-torsion. Furthermore, _ E(K_n) is unbounded as n→∞.




 §.§ Organization


In Section <ref>, we review several plus and minus objects. We first review a system of local points introduced in <cit.>, which is used to construct signed (or plus and minus) Coleman maps following <cit.>. These maps are then utilized to define the signed Selmer groups that appear in the statement of Theorem <ref>. After analyzing the images of these Coleman maps, we review results on certain global cohomology groups under the assumption that the signed Selmer groups are cotorsion. We review the definition of Kobayashi ranks for projective systems of ℤ_p-modules in Section <ref>. We derive a formula for the Kobayashi ranks of modules arising from specific 2 × 2 matrices defined over Λ, which is central to our proof of Theorem <ref> and is the principal novelty of this article. In Section <ref>, we establish a connection between Coleman maps and Kobayashi ranks, and show how this relationship enables us to study the growth of certain local modules. We conclude by combining these results to prove Theorem <ref>.



 §.§ Outlook

Our proof of Theorem <ref> follows closely the ideas of <cit.>. In loc. cit., the author presented a detailed study of the growth of modules of the form (Λ/⟨ f⟩)_Γ_n, where f is a non-zero element of Λ, as n→∞. In the context of the current article, several Iwasawa modules have rank 2, and we are led to study the growth of modules of the form (Λ^⊕ 2/⟨ u_1,u_2⟩)_Γ_n, where u_1,u_2 are linearly independent elements of Λ^⊕ 2. It should be possible to extend our results to modules arising from Λ^⊕ d, d≥2. This would potentially allow us to remove some of the hypotheses imposed in <cit.> to study elliptic curves defined over more general number fields. In a different vein, it would be interesting to study similar results in the context of the present article without assuming a_p(E)=0. This would require an appropriate extension of the results of <cit.> on local points. We plan to study these questions in the future.





 §.§ Acknowledgements
 We thank Ashay Burungale and Fırtına Küçük for interesting discussions during the preparation of this article. The authors' research is supported by the NSERC Discovery Grants Program RGPIN-2020-04259 and RGPAS-2020-00096. EI is also partially supported by a postdoctoral fellowship from the Fields Institute.






§ PLUS AND MINUS OBJECTS




 §.§ Local points of Burungale–Kobayashi–Ota

For 0 ≤ n ≤∞, let k_n denote the localization of K_n at the unique prime above p (the uniqueness of this prime is a consequence of p∤ h_K). Let _n denote the maximal ideal of the integer ring of k_n. For simplicity, set k:=k_0, and let  denote the integer ring of k. When n <∞, we identify the Galois group (k_n/k) with G_n, and similarly (k_∞/k) with Γ. 



 
 Let  be the formal group of the minimal model of E over .  Let λ:=log_ denote the logarithm of  which is normalized by λ'(0)=1. 

 


 Let Ξ denote the set of finite characters of Γ. For χ∈Ξ, we say that χ is of order p^n if it factors through G_n, but not G_n-1. Define

    Ξ^+   ={χ∈Ξ|the order of χ is a positive even power of  p};
    Ξ^-   ={χ∈Ξ|the order of χ is an odd power of  p}∪{1}.

Here, 1 is the trivial character.






For n≥ 0, let Ξ_n^± denote the set of χ∈Ξ^± factoring through (k_n/k). For χ∈Ξ^±_n, let

    λ_χ(x):=1/p^n∑_σ∈(k_n/k)χ^-1(σ)λ(x)^σ.

Define

    ^±(_n)= {x∈(_n) |λ_χ(x)=0  for all χ∈Ξ_n^∓}.



    For n≥ 0, there exist c_n^+,c_n^-∈(_n) such that:
    
        
  * If (-1)^n+1=±1 and n≥1, then
        
    _n+1/nc_n+1^±= c_n-1^± ,   c_n^±=c_n-1^±,

where _n+1/n: (_n+1)⟶(_n) is the trace map.
 
  * We have c_n^±∈^±(_n).
    
   


    This is <cit.> when p>3. The case where p=3 follows from the same proof after replacing <cit.> by <cit.>.





Define the Iwasawa algebra Λ_:=[[Γ]] and set Λ_,n:= Λ_/(ω_n)=[G_n], where ω_n=(1+X)^p^n-1 for n≥ 0.



    Let n≥ 0 be an integer.
    
        
  * As Λ_,n-modules, we have (_n)=^+(_n)⊕^-(_n).
        
  * ^±(_n) is generated by c_n^± as a Λ_,n-module.
    


    This is <cit.> when p>3. Once again, we replace the input of <cit.> by <cit.> in the case where p=3.






    
    Let q_n=∑_k=0^n(-1)^n-kp^k, for n≥ 0 with q_-1=0. Then 

        _^+(_n)=
            q_n-1    if n  even,

            q_n    if  n  odd;   and   _^-(_n)=
            q_n    if  n  even,

            q_n-1    if  n  odd.
    

    I think this is not correct anymore. But I also don't know where we actually use it.


    Since the -rank of (_n) is p^n, the corollary follows from Theorem <ref> and Lemma <ref> by induction.







Note that c_0^- is an -basis of (_0)=^-(_0), whereas ^+(_0)=^+(_1)={0}. 
       Let n≥ 0 be an integer. The subgroup ^-(_n) can be described in terms of the trace map on the formal group , in the same manner as the cyclotomic counterpart given in <cit.>. However, this does not apply to ^+(_n). In fact, both plus and minus subgroups defined in loc. cit. contain (p), whereas the group ^+(_n) defined here does not contain any non-zero elements of (_0). The plus and minus subgroups studied here correspond to those considered in <cit.>.





  The construction of the local points relies on Rubin's conjecture  <cit.>, which has recently been proven in <cit.> for p≥ 5 and <cit.> for p=3. For p=3, Lemma <ref> and Theorem <ref> can be proven in a similar manner.








 §.§ Local Tate pairing

Let T be the p-adic Tate module of E.  We regard (_n) ⊆ H^1(k_n , T ) via the Kummer
map. Let 
 
    ⟨-,-⟩_n: (_n)×  H^1(k_n,T)  H^2(k_n,(1))

 denote cup-product pairing.
 




    For c∈(_n), we define the [G_n]-morphism
 
    P_n,c: H^1(k_n,T)   ⟶[G_n]
     
          z   ↦∑_σ∈ G_n⟨ c^σ,z⟩_n·σ.


 
 Furthermore, as in <cit.>, the following lemma holds.
 
 
      The following diagram commutes:
     

  
   (Q1) at (0,2) H^1(k_n,T);
     (Q2) at (4,2) Λ_,n;
      (Q3) at (0,0) H^1(k_n-1,T);
    (Q4) at (4,0) Λ_,n-1;

    [->] (Q1)–node [above] P_n,c(Q2);
      [->] (Q1)–node [left] _k_n/k_n-1(Q3);
    [->] (Q3)–node [above] P_n-1,_n/n-1(c)(Q4);
    [->] (Q2)–node [right] (Q4);
           


where the vertical maps are the corestriction and the projection.
 
 

 



 
    We have E(K)[p^∞]=0 and E(K_∞)[p^∞]=0.


This follows from the same proof as <cit.>.




 
     The Kummer map induces an injection ^±(_n)⊗_p/_p ↪ H^1(k_n,E[p^∞]).
 

See <cit.>.







     For ∙∈{+,-}, we define H^1_∙(k_n,T) as the orthogonal complement of ^±(_n)⊗_p/_p⊆ H^1(k_n,E[p^∞]) with respect to the local Tate pairing
       
    H^1(k_n,T)×  H^1(k_n,E[p^∞]) ⟶ k/.







    Define P_n^±:=P_n,c_n^±. Then the kernel of P_n^± is H^1_±(k_n,T).


The proof is the same as that of <cit.>.






For n≥ 0, recall that ω_n=(1+X)^p^n-1=∏_0≤ m≤ nΦ_m, where Φ_m is the p^m-th cyclotomic polynomial in 1+X. We put

    ω^+_n   := X ∏_1≤ m≤ n,
     m:  evenΦ_m ,
    ω_n^-   :=X ∏_1≤ m≤ n, 
     m:  oddΦ_m,

and   _n^±:=ω_n^±/X, ω^+_0 := X, ^-_0:=1. 





    The image of P_n^+ is contained in ω_n^-Λ_,n, whereas the image of P_n^- is contained in _n^+Λ_,n.


 The proof is similar to <cit.>. We illustrate this for P_n^+. Let χ be a character of G_n that belongs to Ξ_n^-. Then, χ sends γ=X+1 to 0 or a p^m-th primitive root of unity for some odd non-negative integer m≤ n. Since 
c_n^+∈^+(_n),
 
    P_n^+(z)(χ)=∑_σ∈ G_n⟨χ(σ)(c_n^+)^σ,z⟩_n=0.

 Therefore, as an element of Λ_,n, P_n^+(z) is divisible by Φ_m for all non-negative odd integers m≤ n. Therefore, 
 
    P_n^+(z)∈ X∏_1≤ m≤ n, 
     m:  oddΦ_mΛ_,n=ω_n^-Λ_,n.

 Similarly, we have
 
    P_n^-(z)∈∏_1≤ m≤ n, 
     m:  evenΦ_mΛ_,n=_n^+Λ_,n.

 
Note that ω_n^-Λ_,n⊂_n^-Λ_,n. In particular, Proposition <ref> tells us that the image of P_n^+ is contained in _n^-Λ_,n. We follow the strategy of <cit.> to use this weaker congruence to define the corresponding Coleman map in the following section.





 §.§ Signed Coleman maps
  
It is clear from the definitions that ω_n=_n^∓ω_n^±. Thus, the multiplication by ω_n^∓ induces an isomorphism

    Λ_,n^±:=Λ_/(ω_n^±)_n^∓Λ_,n.






   There exist unique morphisms _n^± such that the following diagram commutes:


  
   (Q1) at (0,2) H^1(k_n,T);
     (Q2) at (4,2) Λ_,n^±;
      (Q3) at (0,0) H^1(k_n,T)/H^1_±(k_n,T);
    (Q4) at (4,0) Λ_,n;
        

    [->] (Q1)–node [above] _n^±(Q2);
      [->] (Q1)–(Q3);
    [->] (Q3)–node [above] P_n^±(Q4);
    [right hook->] (Q2)–node [right] ×ω_n^∓(Q4);
               


    
    

 
     

    This follows from Propositions <ref> and <ref>, together with (<ref>).





     

  
   (Q1) at (-7,2) H^1(k_n,T);
     (Q2) at (-3,2) XΛ_,n^-;
      (Q3) at (-7,0) H^1(k_n,T)/H^1_+(k_n,T);
    (Q4) at (-3,0) Λ_,n;
        

    [->] (Q1)–node [above] _n^+(Q2);
      [->] (Q1)–(Q3);
    [->] (Q3)–node [above] P_n^+(Q4);
    [right hook->] (Q2)–node [right] ×ω^+_n(Q4);
    
     (Q5) at (0,2) H^1(k_n,T);
      (Q6) at (4,2) Λ_,n^+;
      (Q7) at (0,0) H^1(k_n,T)/H^1_-(k_n,T);
    (Q8) at (4,0) Λ_,n;
    
    [->] (Q5)–node [above] _n^-(Q6);
    [->] (Q5)–(Q6);
    [->] (Q5)–(Q7);
    [->] (Q7)–node [above] P_n^-(Q8);
    [right hook->] (Q6)–node [right] ×^-_n(Q8);
           

 
   
  Or we can use the following
   

  
   (Q1) at (0,2) H^1(k_n,T);
     (Q2) at (4,2) X^*Λ_,n^∓;
      (Q3) at (0,0) H^1(k_n,T)/H^1_±(k_n,T);
    (Q4) at (4,0) Λ_,n;
        

    [->] (Q1)–node [above] _n^±(Q2);
      [->] (Q1)–(Q3);
    [->] (Q3)–node [above] P_n^±(Q4);
    [right hook->] (Q2)–node [right] ×ω(Q4);
               


where (*,ω)=(1,ω_n^+) for +, and (*,ω)=(0,_n^-) for -. 


    A brief explanation on the definition of Coleman maps.





    The following diagram commutes:
       

  
   (Q1) at (0,2) H^1(k_n+1,T);
     (Q2) at (4,2) Λ_,n+1^±;
      (Q3) at (0,0) H^1(k_n,T);
    (Q4) at (4,0) Λ_,n^±;

    [->] (Q1)–node [above] _n+1^±(Q2);
      [->] (Q1)–(Q3);
    [->] (Q3)–node [above] _n^±(Q4);
    [->] (Q2)–(Q4);
           


where the vertical maps are the corestriction and the projection.


    The proof is the same as <cit.>.




Observe that 

    _n Λ_,n=  _n Λ_/(ω_n^±) = Λ_.

This allows us to give the following definition.


    We define the plus and minus Coleman maps
    
    ^± : H^1_(k_∞,T)⟶Λ_

    as the inverse limits of _n^±: H^1(k_n,T)⟶Λ_,n^±.






 §.§ Image of the signed Coleman maps





    The corestriction map H^1(k_m,T)⟶ H^1(k_n,T) is surjective for all m≥ n.


   The proof is similar to <cit.>.
  




Let I_,n be the augmentation ideal of Λ_,n=[G_n]:

    I_,n=( [G_n]⟶).

We denote I_:=_n I_,n=XΛ_ and I_,n^±=I_,n∩Λ^±_,n=XΛ_,n^±.





     The image of ^- (resp. ^-_n) is equal to Λ_ (resp. Λ_,n^-), whereas the image of ^+ (resp. _n^+) is given by I_ (resp. I_,n^+).


      By Lemma <ref> and Nakayama’s lemma, it is enough to show that ⊷(^-_0)=Λ_,0^-= and ⊷(^+_2)=I_,2^+. Note that (_0) is generated by c_0^- as an -module, so the proof of <cit.> can be extended to show that (^-_0)=. 

We have the following -isomorphism 
    I_,2^+=XΛ_,2^+=XΛ_/(XΦ_2)≃[ζ_p^2],

where ζ_p^2 is a primitive p^2-th root of unity, and Xf(X) is sent to f(ζ_p^2-1) under this isomorphism.
       Since {ζ_p^2^i:0≤ i≤ϕ(p^2)-1} is an -basis of [ζ_p^2], we see that 
            {X(1+X)^i:0≤ i≤ϕ(p^2)-1} is an -basis of I_,2^+.
      
      
      Let σ be the image of our chosen topological generator γ=1+X of Γ in G_2. In particular, it is a generator of the cyclic group G_2, and (k_2/k_1)=⟨σ^p⟩. Since c_2^+∈^+(_2), for all integer j, we have
      
    ∑_i=0^p-1(c_2^+)^σ^j+ip=_2/1(c_2^+)^σ^j=0.

Furthermore, Theorem <ref> tells us that

    (_2)=^+(_2)⊕(_1),

and ^+(_2)=[G_2]· c_2^+.
      Therefore,  {(c_2^+)^σ^i:0≤ i≤ϕ(p^2)-1} is an -basis of ^+(_2). This gives the following [G_2]-isomorphism
      
    ^+(_2)≃ I_,2^+,

      where c_2^+ is sent to X.
          Consequently, the morphism _2^+ can be described as 
      
    H^1(k_2,T)⟶((_2),[G_2]) ⟶(^+(_2),[G_2]) ≃ I_,2^+,

      where the first map is induced by the pairing (<ref>) and the second map is the projection by the decomposition (<ref>), and the last isomorphism is induced from (<ref>). The first arrow is surjective since its dual is the injection (_2)⊗_p/_p ⟶ H^1(k_2,E[p^∞]). It is clear from the definition that the second arrow is surjective. Therefore, the image of _2^+ is I_,2^+, as desired.





 §.§ Signed Selmer groups
  Recall that K_∞ is the anticyclotomic _p-extension of K, and that K_n ⊂ K_∞ denotes the unique subextension such that [K_n : K] = p^n, for an integer n ≥ 0.



For a rational prime ℓ, let

    K_n,ℓ:=K_n⊗__ℓ,   H^1(K_n,ℓ, E[p^∞]):=⊕_λ|ℓ H^1(K_n,λ, E[p^∞]),

where the direct sum runs over all primes of K_n above ℓ. We have the natural restriction map

    _ℓ: H^1(K_n, E[p^∞]) ⟶ H^1(K_n,ℓ, E[p^∞]).


Let H^1_(K_n,ℓ, E[p^∞])⊂ H^1(K_n,ℓ, E[p^∞]) for the Bloch–Kato subgroup. The singular quotient is given by

    H^1_/(K_n,ℓ, X):=H^1(K_n,ℓ, E[p^∞])H^1_(K_n,ℓ, E[p^∞]).






   For ∙∈{+,-}, we define the signed Selmer group of E over K_n by
    
    ^∙(E/K_n):=(H^1(K_n,E[p^∞])⟶∏_ℓ∤ p H^1_/(K_m,ℓ, E[p^∞])×H^1(k_n, E[p^∞])H^1_∙(k_n, E[p^∞])).

    Further, define ^∙(E/K_∞)=_n ^∙(E/K_n).





If _p^∞(E/K_∞) denotes the classical p^∞-Selmer group, then ^∙(E/K_∞)⊂_p^∞(E/K_∞). It follows from <cit.> that _p^∞(E/K_∞) is cofinitely generated over Λ. Hence, so is ^∙(E/K_∞).





    For ∙∈{+,-}, the signed Selmer groups ^∙(E/K_∞) is cotorsion over Λ.



   Let N denote the conductor of E, and p≥ 5. Assume that (D_K,pN)=1, where D_K is the discriminant of K. Write N=N^+N^-, where N^+ (resp. N^-) is divisible only by primes that are split (resp. inert) in K. In <cit.>, it has been proved that Conjecture <ref> is valid under the following hypotheses:
   

       
       
  * N^- is a square-free product of odd number of primes.
       
  * If p=5, the residual representation ρ_E( G_ (μ_p^∞)) contains a conjugate of _2(_p). If p>5, the G_-representation ρ_E is irreducible.
       
  * ρ_E is ramified at the primes ℓ that satisfy one of the following conditions:
       
           
  * ℓ| N^- with ℓ^2 ≡ 1  p,
           
  * ℓ| N^+.
       
   


For the remainder of the article, we assume that Conjecture <ref> holds true.



  For ∙∈{+,-}, we write μ_∙ and λ_∙ for the μ- and λ-invariants of the torsion Λ-module ^∙(E/K_∞)^∨.
  







 §.§ Structure of global cohomologies

In this section, we record several consequences of Conjecture <ref> following <cit.>. 

Let Σ denote a fixed finite set of primes of K containing p, the ramified primes of K/, the archimedean place, and all the bad reduction primes of E. Write K_Σ for the maximal algebraic extension of K which is unramified outside Σ. For any (possibly infinite) extension K ⊆ L ⊆ K_Σ, write G_Σ(L) = (K_Σ/L). 

The signed Selmer group of E over K_∞ can be equivalently defined as follows:

    ^∙(E/K_∞):=(H^1(G_Σ(K_∞),E[p^∞])⟶∏_ℓ∤ p H^1_/(K_∞,ℓ, E[p^∞])×H^1(k_∞, E[p^∞])H^1_∙(k_∞, E[p^∞])).


For i∈{1,2}, we define H^i_,Σ(K_∞,T)=_n H^i(G_Σ(K_n),T), where the transition maps are given by the corestriction maps. Note that H^i_,Σ(K_∞,T) is independent of the choice of Σ (see <cit.>, <cit.> and <cit.>). Since the set Σ is fixed, we will drop the subscript Σ from the notation for simplicity and simply write H^i_(K_∞,T).



    The group H^1_(K_∞,T) is a torsion-free Λ-module. Further, H^1(G_Σ(K),T) is a torsion-free _p-module.

 See <cit.>. 

         
        By considering the low degree terms of the spectral sequence of Jannsen (<cit.>)
    
    _Λ^i(H^j(G_Σ(K_∞),E[p^∞])^∨,Λ) H^i+j_(K_∞,T)

    we obtain the following exact sequence
    
    0⟶_Λ^1((E(K_∞)[p^∞])^∨,Λ) ⟶ H^1_(K_∞,T) ⟶_Λ^0(H^1(G_Σ(K_∞),E[p^∞])^∨,Λ).

    By Lemma <ref>, we have E(K_∞)[p^∞]=0, so the leftmost term vanishes. This in turn implies that H^1_(K_∞,T) injects into an ^0-term. Since the latter is a reflexive Λ-module by <cit.>, H^1_(K_∞,T) must be torsion-free.

    


 For the second assertion, we consider the low degree terms of the spectral sequence
  
    __p^i(H^j(G_Σ(K),E[p^∞])^∨,_p) H^i+j(G_Σ(K),T),

    which yields the following exact sequence:
     
    0⟶__p^1((E(K_∞)[p^∞])^∨,_p)    ⟶ H^1(G_Σ(K),T)
       ⟶__p^0(H^1(G_Σ(K_∞),E[p^∞])^∨,_p).

    By Lemma <ref>, we have E(K)[p^∞]=0, so the leftmost term vanishes. Hence, similar to the previous case, we deduce that H^1(G_Σ(K),T) is a torsion-free _p-module.
  



    One can also show that H^1(G_Σ(K_n), T ) is a torsion-free _p-module for every n.




      For ∙∈{+,-}, the Λ-module ^∙(E/K_∞) is cotorsion if and only if we have H^2(G_Σ(K_∞),E[p^∞])=0 and that the following sequence:
   
    0⟶^∙(E/K_∞)⟶    H^1(G_Σ(K_∞),E[p^∞]) 
       ⟶∏_ℓ∤ p H^1_/(K_∞,ℓ, E[p^∞])×H^1(k_∞, E[p^∞])H^1_∙(k_∞, E[p^∞])⟶ 0,

    is exact.



See <cit.>.

    To simplify the notation, we write J_v(E/K_∞) for each of the local summands. It follows from <cit.>  that we have an exact sequence

    0⟶^∙(E/K_∞)⟶   H^1(G_Σ(K_∞),E[p^∞])⟶∏_v∈ΣJ_v(E/K_∞)
       ⟶^∙(E/K_∞)^∨⟶  H^2(G_Σ(K_∞),E[p^∞])  ⟶ 0,

    where ^∙(E/K_∞) is a Λ-submodule of H^1_(K_∞,T). It follows from the corank calculation <cit.> that we have 
    
    _Λ(H^1(G_Σ(K_∞),E[p^∞])) - _Λ(H^2(G_Σ(K_∞),E[p^∞])) = [K:]=2

and

    _Λ(∏_v∈ΣJ_v(E/K_∞))  = [K:]=2.

    Therefore, we see that ^∙(E/K_∞) is a cotorsion Λ-module if and only if ^∙(E/K_∞) is a torsion Λ-module.  Since ^∙(E/K_∞) is a Λ-submodule of H^1_(K_∞,T), which is torsion-free by Lemma <ref>, the latter statement holds if and only if ^∙(E/K_∞)=0. This is equivalent to having H^2_(K_∞,T)=0 as well as the short exact sequence in the statement of the proposition.




 


We conclude this section with the following statement on the structure of H^i_(K_∞,T).





    If Conjecture <ref> holds,  the following statements are valid:
    
        
  * H^1_(K_∞,T) is a free Λ-module of rank 2.
        
  * H^2_(K_∞,T) is a torsion Λ-module.
    


  See <cit.>.





§ KOBAYASHI RANKS



 §.§ Definition and basic properties

Following <cit.>, we define the Kobayashi ranks as follows.


    Let (M_n)_n≥ 1 be a projective system of finitely generated _p-modules. Given an integer n≥1, if π_n:M_n⟶ M_n-1 has finite kernel and cokernel, we  define

    ∇ M_n:=__p(π_n)-__p( π_n)+__pM_n-1⊗_p.
















Let (M'_n)_n≥1, (M_n)_n≥1 and (M”_n)_n≥1 be projective systems of finitely generated -modules.

    
  * Suppose that for all n≥1, there is an exact sequence
    
    0⟶ (M'_n)⟶ (M_n)⟶ (M”_n)⟶ 0.

    If two of ∇ M_n, ∇ M'_n,∇ M”_n are defined, then the other is also defined, in which case
    
    ∇ M_n=∇ M'_n+∇ M”_n.

    
  * Suppose that M_n are constant. If M_n is finite or the transition map M_n⟶ M_n-1 is given by the multiplication map by p, then ∇ M_n=0.
    


    See <cit.>.






 Let f∈Λ be a non-zero element of Λ. Let (N_n)_n≥1 be a projective system given by N_n= Λ/ (f,ω_n), where the connecting maps are natural projections.


    
  * Suppose that Φ_n∤ f. Then ∇ N_n is defined and is equal to _ϵ_nf(ϵ_n), where ϵ_n=ζ_p^n-1.
   
     
  * Let M be a finitely generated torsion Λ-module with the characteristic polynomial f, and let M_n=M/ω_nM. Consider the natural projective system (M_n)_n≥ 1. Then, for n≫ 0, ∇ M_n is defined and
    
    ∇ M_n= _ϵ_nf(ϵ_n)=λ(M)+ϕ(p^n)μ(M),

where λ(M) and μ(M) are the Iwasawa invariants of M and ϕ is the Euler totient function.






    See <cit.>.



    






    Let M be a finitely generated Λ-module. Then M^δ:=∪_nM^Γ_n is finitely generated as a _p-module and there exists an integer n_0 such that M^δ=M^Γ_n_0.
    
    

    
    In particular, ∇ M^Γ_n=0 for n≫ 0, where the transition maps M^Γ_n+1⟶ M^Γ_n are given by the multiplication by 1+γ_n+…+γ_n^p, where γ_n is a topological generator for Γ_n which is chosen so that γ_n^p=γ_n+1.


    The proof is the same as <cit.>.







 §.§ Kobayashi ranks of modules arising from 2× 2 matrices

We derive a formula for the Kobayashi ranks of modules arising from certain types of 2× 2 matrices defined over Λ. We begin with the following definition.


    Let A=[ a c; b d ]∈ such that (A)0.
   
       
  * We write ⟨ A⟩⊆ for the Λ-module generated by the columns of A. Similarly, given two such matrices A and B, we write ⟨ A,B⟩⊆ for the Λ-module generated by the columns of A and B.
    
  * For n≥0, we write ⟨ A⟩_n for the Λ_n-module generated by the columns of A modulo ω_n inside _n. Furthermore, we write =_n/⟨ A⟩_n=/⟨ω_n I_2,A⟩, where I_2 denotes the 2× 2 identity matrix. 
    
  * We say that A is special relative to an integer n if it satisfies the following property:
    
    Φ_m|(A),  0≤ m≤ n  ⇒ Φ_m | a,b or Φ_m | c,d.


    
    
   


We prove the following generalization of <cit.>.

Let n≥0 be an integer.    If A∈ is special relative to n and Φ_n∤(A), then ∇ is defined and is equal to _ϵ_n( A(ϵ_n)). Here, the connecting map π_n:→ A_(n-1) is the natural projection.



For each integer m such that 0≤ m≤ n-1, we write i_m∈{0,1,2} for the number of columns of A that are divisible by Φ_m. Then, A can be written as

    A=BD,

where B,D∈ such that D is a diagonal matrix whose diagonal entries are square-free products of elements of a subset {Φ_m:0≤ m≤ n-1}, and the number of times Φ_m appears in D is precisely i_m.

By the Chinese remainder theorem, 
    A_(n-1)⊗≃⊕_0≤ m≤ n-1(ζ_p^m)^⊕ 2/⟨ A(ϵ_m)⟩,
 where ⟨ A(ϵ_m)⟩ denotes the image of the (ζ_p^m)-linear endomorphism on (ζ_p^m)^⊕ 2 defined by the matrix A(ϵ_m). The rank of the matrix A(ϵ_m) is equal to 2-i_m. Therefore, the rank-nullity theorem tells us that

    _(ζ_p^m)(ζ_p^m)^⊕ 2/⟨ A(ϵ_m)⟩=i_m.

    Hence,
    
    _A_(n-1)⊗=∑_m=0^n-1ϕ(p^m)i_m=_ϵ_n( D(ϵ_n)).



    As the connecting map π_n is surjective, it remains to show that the length of π_n is given by _ϵ_n(_ϵ_n B(ϵ_n)). The isomorphism theorem tells us that
    
    π_n≃⟨ω_n-1I_2,A⟩/⟨ω_nI_2,A⟩≃⟨ω_n-1I_2⟩/⟨ω_nI_2,A⟩⋂⟨ω_n-1I_2⟩.

   

    Claim: ⟨ω_nI_2,A⟩⋂⟨ω_n-1I_2⟩=⟨ω_n I_2,ω_n-1B⟩.

    It is clear that ⟨ω_n I_2,ω_n-1B⟩⊆⟨ω_n-1 I_2⟩. 
 Since the diagonal entries of the diagonal matrix D are square-free products of elements of a subset of {Φ_m:0≤ m≤ n-1}, we may write ω_n-1I_2=DD' for some diagonal matrix D'.
    
    Thus,
    
    ω_n-1B=BDD'=AD',

    which implies that ⟨ω_n-1B⟩⊆⟨ A⟩. Therefore, we deduce that 
    
    ⟨ω_n I_2,ω_n-1B⟩⊆⟨ω_nI_2,A⟩⋂⟨ω_n-1I_2⟩.


    We now prove the opposite inclusion. Let v∈⟨ω_nI_2,A⟩⋂⟨ω_n-1I_2⟩. We can regard v as a column vector with entries in Λ and write 
    
    v=A[ x; y ]+ω_nu,

      where x,y∈Λ and u∈. Since v∈⟨ω_n-1I_2⟩, we have
    
    A[ x; y ]=BD[ x; y ]≡ 0Φ_m,   0≤ m≤ n-1.

         We consider three cases.
       

          Case 1:  If i_m=0, A(ϵ_m) is an invertible matrix by definition. In this case, (<ref>) implies that x,y≡ 0Φ_m.
            
            Case 2: If i_m=1, there exists a non-zero element [ a; b ]∈ such that
            
    A≡[ a 0; b 0 ]Φ_m  or   A≡[ 0 a; 0 b ]Φ_m

            Let's suppose the first congruence relation holds. Then (<ref>) implies that Φ_m| x. Furthermore, D≡[ d_m   0;   0   0 ]Φ_m for some non-zero d_m∈Λ. Therefore, D[ x; y ]≡0Φ_m. If the second congruence relation in (<ref>) holds, the same argument shows that D[ x; y ]≡0Φ_m.


               Case 3: If i_m=2, then D≡0Φ_m. In particular, D[ x; y ]≡0Φ_m.

            In all three cases, we have D[ x; y ]≡ 0Φ_m. Therefore, we deduce that 
    D[ x; y ]=ω_n-1[ x'; y' ]

    for some x',y'∈Λ. Combined with (<ref>), we have
    
    v=ω_n-1B[ x'; y' ]+ω_nu∈⟨ω_nI_2,ω_n-1B⟩.

    This shows that ⟨ω_nI_2,A⟩⋂⟨ω_n-1I_2⟩⊆⟨ω_n I_2,ω_n-1B⟩, and our claim follows.

We can now rewrite (<ref>) as
    
    π_n≃⟨ω_n-1I_2⟩/⟨ω_n I_2,ω_n-1B⟩≃/⟨Φ_n I_2,B⟩,

    where the last isomorphism is given by
    
    /⟨Φ_n I_2,B⟩   →⟨ω_n-1I_2⟩/⟨ω_n I_2,ω_n-1B⟩
    
            x   ↦ω_n-1x.

    As Φ_n∤ A by assumption, it follows from <cit.> that π_n is finite, with length equal to _ϵ_n( B(ϵ_n)), as desired.







§ PROOF OF THEOREM <REF>


The proof of Theorem <ref> is divided into a number of steps. Following <cit.>, we define for each integer n≥0

    (E/K_n)   :=( H^1(G_Σ(K_n),T) ⟶H^1(k_n,T)/E(k_n)⊗_p).

The key ingredient to studying the growth of _ E(K_n) and (E/K_n )[p^∞] is understanding ∇(E/K_n). Similar to <cit.>, we do so using fine Selmer groups and several auxiliary modules. 


 §.§ Fine Selmer groups

We recall the definition of fine Selmer groups:

    For 0≤ n ≤∞, we define the fine Selmer group  
        
    ^0(E/K_n):=( _p^∞(E/K_n)⟶ H^1(k_n, E[p^∞]) ).

    


Equivalently, we have
 
    ^0(E/K_n):=( H^1(G_Σ(K_n),E/K_n)⟶∏__n∈Σ(K_n) H^1(K_n,_n, E[p^∞]) ),

where Σ(K_n) denotes the set of primes of K_n lying above Σ. The Pontryagin duals of _p^∞(E/K_n), ^±(E/K_n) and  ^0(E/K_n) are denoted by (E/K_n), ^±(E/K_n) and ^0(E/K_n), respectively.





    The natural restriction map
    
    ^0(E/K_n) ⟶^0(E/K_∞)^Γ_n

    has finite kernel and cokernel. Furthermore, their cardinalities stabilize as n→∞.


This is a special case of <cit.> (see also <cit.>).





Assume that Conjecture <ref> holds.   Let n≥ 0 be an integer.
    
        
  * We have the following short exact sequence:
        
    0 ⟶(E/K_n) ⟶(E/K_n) ⟶^0(E/K_n) ⟶ 0.




        
        
  * For n≫ 0, ∇^0(E/K_n) is defined and satisfies the equality
        
    ∇^0(E/K_n)= ∇^0(E/K_∞)_Γ_n.

    


   Assertion (i) follows from the Poitou–Tate exact sequence and the definition of (E/K_n).

    


As ^0(E/K_∞) is a quotient of ^±(E/K_∞), which are assumed to be Λ-torsion, it follows that ^0(E/K_∞) is also Λ-torsion. By Lemma <ref>, we see that the kernel and cokernel of the natural map
    
    ^0(E/K_∞)_Γ_n⟶^0(E/K_n)

    are finite and independent of n. Combining this with Lemma <ref>(ii), the second assertion follows.





 §.§ Calculating certain Kobayashi ranks via special matrices

We introduce auxiliary modules (denoted by M_n) in preparation for the calculation of  ∇(E/K_n).
The composition

    H^1_(K,T)  H^1(k_∞,T) Λ_≃Λ^⊕ 2

 is a Λ-homomorphism between two free Λ-modules of rank 2 (see Proposition <ref>). We write the composition of this map with projection to the two coordinates as
 
    ^±_i:H^1_(K,T)→Λ,   i=1,2.


 

    Let =(u_1,u_2)∈ H^1_(K,T)^⊕ 2 and n≥0 be an integer. We define
    
    F_n()=[ _n^+^-_1(u_1)+_n^-_1^+(u_1) _n^+^-_1(u_2)+_n^-_1^+(u_2); _n^+^-_2(u_1)+_n^-_2^+(u_1) _n^+^-_2(u_2)+_n^-_2^+(u_2) ]∈.

    Furthermore, we define for ∙∈{+,-}
    
    ^∙()=[ _1^∙(u_1) _1^∙(u_2); _2^∙(u_1) _2^∙(u_2) ]∈.




    Let =(u_1,u_2)∈ H^1_(K,T)^⊕ 2 and n≥0 be an integer. For 0≤ m≤ n, there exists a non-zero scalar c_m,n∈(ζ_p^m)^× such that
    F_n()(ϵ_m) is equal to c_m,n^+()(ϵ_m) or c_m,n^-()(ϵ_m), depending on the parity of m.


Recall that _n^+_n^-=∏_1≤ m≤ nΦ_m. Therefore, for each 1≤ m≤ n, exactly one of the two elements of {_n^+(ϵ_m),_n^-(ϵ_m)} vanishes, from which the assertion for m≥1 follows.

If m=0, we have ϵ_m=0. Proposition <ref> tells us that ^+_i(u_j)(0)=0. Therefore,

    F_n()(0)=_n^+(0)^-(),

as desired.



Let ∙∈{+,-}. Suppose that ^∙(E/K_∞) is Λ-torsion.
If =(u_1,u_2)∈ H^1_(K,T)^⊕ 2 such that the quotient of Λ-modules
H^1_(K,T)/⟨ u_1,u_2⟩ is Λ-torsion, then (^∙()) 0.


     By <cit.>, we have the following exact sequence
 
    H^1_(K,T) ⟶H^1(k_∞,T)/^∙⟶^∙(E/K_∞) ⟶^0(E/K_∞) ⟶ 0.

Hence, the result follows from the assumption that ^∙(E/K_∞) is Λ-torsion.

 
    
         Suppose that Conjecture <ref> holds.
Then, there exists =(u_1,u_2)∈ H^1_(K,T)^⊕ 2 such that 
H^1_(K,T)/⟨ u_1,u_2⟩ is Λ-torsion and that F_n() is a special matrix relative to n (in the sense of Definition <ref>) for all n≥0.
    
    
        Let =(z_1,z_2)∈ H^1_(K,T)^⊕ 2 such that {z_1,z_2} be a Λ-basis of  H^1_(K,T).  We write
        
    _n={m: _(ζ_p^m) F_n()(ϵ_m)=1,0≤ m≤ n}.

        It follows from Lemma <ref> that 
          
    _n⊆_n+1

          Furthermore, as ^±() 0 by Lemma <ref>, the number of cyclotomic polynomials dividing either ^+() or ^-() is finite. Therefore, the cardinality of _n is bounded above as n→∞. Let _∞=⋃_n≥0_n.
        
    Let m∈_∞ with 1≤ m≤ n. Let ∙ be the unique element of {+,-} such that
    
    F_n()(ϵ_m)= c_m,n^∙()(ϵ_m)

    as given by Lemma <ref>. By linear algebra, there exists a 2× 2 invertible matrix B_m over (ζ_p^m) such that 
    ^∙()(ϵ_m)B_m=[ 0 *; 0 * ],

    where the second column is non-zero.    
    Identifying (ζ_p^m) with [X]/(Φ_m), the Chinese remainder theorem implies that there exists a 2× 2 matrix B defined over [X] (after multiplying by a power of p, if necessary) such that B(ϵ_m) is a non-zero scalar multiple of B_m for all m∈_∞. Consequently,
    
    F_n()B≡[ 0 *; 0 * ]Φ_m

    for all m∈_n. 
    
    We claim that we can choose B so that Φ_m|(B) for all m≥0. Let M=max_∞. Then, we may choose BΦ_m to be an invertible matrix for all m≤ M by the Chinese remainder theorem. Once such a B is chosen,  any other B' such that B≡ B'ω_M satisfies the same congruence relation modulo Φ_m as B for all m∈_∞. We may choose recursively a sequence of matrices {B_m}_m≥ M such that B_M is our initial choice of B, and B_m+1≡ B_mω_m and Φ_m+1∤ B_m+1. Indeed, suppose that we have chosen 
    
    B_m=[ a b; c d ]
 for some m≥ M. If Φ_m+1∤ B_m, then we can simply choose B_m+1=B_m. If Φ_m+1| B_m, the vectors
    
    [ a(ϵ_m+1); b(ϵ_m+1) ],  [ c(ϵ_m+1); d(ϵ_m+1) ]

are linearly dependent. There exist α,β,γ,δ∈ such that
    
    [ a(ϵ_m+1)+αω_m(ϵ_m+1); b(ϵ_m+1+βω_m(ϵ_m+1)) ],  [ c(ϵ_m+1)+γω_m(ϵ_m+1); d(ϵ_m+1+δω_m(ϵ_m+1)) ]

are linearly independent. Therefore, we can choose

    B_m+1=B_m+ω_m[ α γ; β δ ].

In particular, our claim follows from taking the limit of this sequence.

For this choice of matrix B, we have

    F_n()B≡[ 0 0; 0 0 ]Φ_m

    whenever F_n()Φ_m is the zero matrix, and 
F_n()B(ϵ_m) is invertible whenever Φ_m∤( F_n()). Together with (<ref>), we deduce that F_n()B is special relative to n for all n≥0. The assertion of the proposition now follows from taking

    u_1=az_1+bz_2,   u_2=cz_1+dz_2,

    where B=[ a c; b d ].

 
 
 
 From now on, we fix =(u_1,u_2) satisfying the properties given by Proposition <ref>. For an integer n≥1, we define the following modules
 
    M_n=_n/⟨ F_n()⟩_n,   M_n-1=_n-1/⟨ F_n()⟩_n-1,

 where ⟨ F_n()⟩_m denotes the Λ_m-submodule of Λ_m^⊕ 2 generated by the columns of the matrix F_n().

 
 
 Suppose that Conjecture <ref> holds. For n≫0, we have

    ∇ M_n =
             2  _n^+ + _ϵ_n(^-()(ϵ_n))       if  n   is odd,
    
              2 _n^- +_ϵ_n(^+()(ϵ_n))       if  n  is even.



    This follows from Theorem <ref> and Proposition <ref>.



 
 


 §.§ Final steps of the proof
 
 
 

We study the variation of  (E/K_n) defined at the beginning of the section using the following -modules:

    '(E/K_n)   :=( H^1_(K_∞,T)_Γ_n⟶H^1(k_n,T)/E(k_n)⊗_p),
    ”(E/K_n)   :=( H^1_(K_∞,T)_Γ_n⟶H^1(k_n,T)/H^1_+(k_n,T)⊕H^1(k_n,T)/H^1_-(k_n,T)).

     

        





    For all n≥ 0, we have '(E/K_n)=”(E/K_n).


It follows from Theorem <ref>(i) that there is an isomorphism of Λ-modules

    (_n)⊗/≃((_n)^+⊗/)⊕((_n)^-⊗/).

On taking Pontryagin duals, we deduce:

    H^1(k_n,T)/E(k_n)⊗_p≃H^1(k_n,T)/H^1_+(k_n,T)⊕H^1(k_n,T)/H^1_-(k_n,T),

which implies the assertion.






 Suppose that Conjecture <ref> holds. Let {z_1,z_2} be a Λ-basis of H^1_(K,T) and write =(z_1,z_2). For n≫0, we have

    ∇(E/K_n) =
             2  _n^+ + _ϵ_n(^-()(ϵ_n))       if  n   is odd,
    
              2 _n^- +_ϵ_n(^+()(ϵ_n))       if  n  is even.



    Let Λ_n^-=Λ/(ω_n^-) and I_n^+=XΛ_n^+=XΛ/(ω_n^+).
Consider the following commutative diagram with exact rows


  
        (Q1) at (-3.5,2) 0;
    (Q2) at (-1,2) ( I_n^+⊕Λ_n^-)^⊕ 2;
    (Q3) at (3,2) Λ_n^⊕ 2;    
     (Q4) at (7,2) (Λ_n/ (_n^+,ω_n^- ))^⊕ 2;
      (Q5) at (10,2) 0;

       (Q6) at (-3.5,0) 0;      
      (Q7) at (-1,0) (  I_n-1^+⊕Λ_n-1^-)^⊕ 2;
    (Q8) at (3,0) Λ_n-1^⊕ 2;
(Q9) at (7,0) (Λ_n-1/ (_n^+,ω_n^- ))^⊕ 2;
   (Q10) at (10,0) 0,;


 [->] (Q1)–(Q2);
  [->] (Q2)–node [above] _n^⊕ 2(Q3);
  [->] (Q2)–node [right] pr(Q7);
  [->] (Q3)–node [right] pr(Q8);
  [->] (Q4)–node [right] pr(Q9);
  [->] (Q3)–(Q4);
  [->] (Q4)–(Q5);
   [->] (Q6)–(Q7);
    [->] (Q7)–node [above] _n^⊕ 2(Q8);
   [->] (Q8)–(Q9);
     [->] (Q9)–(Q10);
         
       


where  _n: I_n^+⊕Λ_n^-⟶Λ_n is given by (f,g)↦_n^-f +_n^+gω_n, and _n:   I_n-1^+⊕Λ_n-1^-⟶Λ_n-1 is the map (f,g)↦_n^-f+_n^+gω_n-1. This is well-defined since _n^±ω_n^∓=ω_n.  

It follows from <cit.> that the third vertical map is an isomorphism. Furthermore, Proposition <ref> tells us that the Coleman maps ^±_n give the following identifications

    H^1(k_n,T)/H^1_+(k_n,T)≃ I_,n^+≃ I_n^+,⊕ 2   and   H^1(k_n,T)/H^1_-(k_n,T)≃Λ
    _,n^-≃Λ_n^-,⊕ 2.

Therefore, we may rewrite the commutative diagram above as


  
        (Q1) at (-3.5,2) 0;
    (Q2) at (-1,2) H^1(k_n,T)/H^1_+(k_n,T)⊕H^1(k_n,T)/H^1_-(k_n,T);
    (Q3) at (3,2) Λ_n^⊕ 2;    
     (Q4) at (7,2) (Λ_n/ (_n^+,ω_n^- ))^⊕ 2;
      (Q5) at (10,2) 0;

       (Q6) at (-3.5,0) 0;      
      (Q7) at (-1,0) H^1(k_n-1,T)/H^1_+(k_n-1,T)⊕H^1(k_n-1,T)/H^1_-(k_n-1,T);
    (Q8) at (3,0) Λ_n-1^⊕ 2;
(Q9) at (7,0) (Λ_n-1/ (_n^+,ω_n^- ))^⊕ 2;
   (Q10) at (10,0) 0,;


 [->] (Q1)–(Q2);
  [->] (Q2)–node [above] (Q3);
  [->] (Q2)–node [right] pr(Q7);
  [->] (Q3)–node [right] pr(Q8);
  [->] (Q4)–node [right] ≃(Q9);
  [->] (Q3)–(Q4);
  [->] (Q4)–(Q5);
   [->] (Q6)–(Q7);
    [->] (Q7)–node [above] (Q8);
   [->] (Q8)–(Q9);
     [->] (Q9)–(Q10);
         
       


where the first two horizontal maps are given by the compositions of _n^⊕2 with (^+_n,^-_n), and _n^⊕2 with (^+_n-1,^-_n-1), respectively.
Consequently, if we write
  
    ”'(E/K_n)=(⟨ u_1,u_2⟩⟶H^1(k_n,T)/H^1_+(k_n,T)⊕H^1(k_n,T)/H^1_-(k_n,T)),

we have the following commutative diagram with exact rows:

  

  
   (Qa) at (-3.5,2) 0;
    (Qb) at (-3.5,0) 0;
    (Qc) at (10,0) 0.;
    (Qd) at (10,2) 0;

    
     (Q2) at (-1,2) ”'(E/K_n);
     (Q3) at (6,2)  (Λ_n/ (_n^+,ω_n^- ))^⊕ 2;
      
      (Q5) at (-1,0) ”'(E/K_n-1);
    (Q6) at (6,0) (Λ_n-1/ (_n^+,ω_n^- ))^⊕ 2;
(Q7) at (2.5,0) M_n-1;
    (Q8) at (2.5,2) M_n;

   
    [->] (Q2)–(Q8);
    [->] (Q5)–(Q7);
      [->] (Q7)–(Q6);
    [->] (Q8)–(Q3);
     [->] (Q8)– (Q7);
    [->] (Q3)–node [right] ≃ (Q6);
     [->] (Qa)–(Q2);
     [->] (Q3)–(Qd);
      [->] (Qb)–(Q5);
     [->] (Q6)–(Qc);
      [-] (Q2)–(Q5);
       


Therefore, ∇”'(E/K_n)=∇ M_n.


By definition, there is a short exact sequence

    0⟶H^1_(K,T)_Γ_n/⟨ u_1,u_2⟩_Γ_n⟶”'(E/K_n)⟶”(E/K_n)⟶ 0.

Furthermore, the first term of this exact sequence is isomorphic to (H^1_(K,T)/⟨ u_1,u_2⟩)_Γ_n. Let A∈ be the change of basis matrix so that 
    [ u_1 u_2 ]=[ z_1 z_2 ]A.

Then (A) generates the Λ-characteristic ideal of H^1_(K,T)/⟨ u_1,u_2⟩. Therefore, for n≫0, 

    ∇H^1_(K,T)_Γ_n/⟨ u_1,u_2⟩_Γ_n=_ϵ_n((A)(ϵ_n))

by Lemma <ref>(ii). Thus, for n≫0, ∇”(E/K_n) is defined and equal to ∇ M_n-_ϵ_n((A)(ϵ_n)).
Furthermore, ^±()=^±()(A) by definition. Hence, after combining Lemmas <ref> and <ref>, we deduce that

    ∇'(E/K_n) =
             2  _n^+ + _ϵ_n(^-()(ϵ_n))       if  n   is odd,
    
              2 _n^- +_ϵ_n(^+()(ϵ_n))       if  n  is even.

It follows from an adaptation of <cit.> and <cit.> to the current setting that for n≫ 0, 
    
    ∇(E/K_n)=∇'(E/K_n),

    which concludes the proof.




 
 
 Suppose that Conjecture <ref> holds.     For n≫ 0, we have
    
    ∇(E/K_n)=
    
      2 s_n-1   + λ_- + (p^n-p^n-1)μ_-       if  n   is odd,
    
            2s_n-1+λ_+ + (p^n-p^n-1)μ_+       if  n  is even,

    where s_n=∑_k=1^n(-1)^n-kp^k for n≥ 0.
  
 
 
Let ∙∈{+,-}. Let f_∙∈Λ (resp. f_0) denote a characteristic element of the torsion Λ-module ^∙(E/K_∞) (resp. ^0(E/K_∞)).  We have the following Poitou–Tate exact sequence of Λ-torsion modules:
 
    0⟶⊷(^∙)/⟨^∙()⟩⟶^∙(E/K_∞)⟶^0(E/K_∞)⟶0.

 As ⊷(^-)= and ⊷(^+)=(XΛ)^⊕2, this gives the equality of Λ-ideals
 
    (X^n^∙^∙()f_0)=(f^∙),

 where n^-=0 and n^+=-2.
 Lemma <ref> tells us that
 
    ∇(E/K_n)=∇(E/K_n)+∇^0(E/K_n).

 Hence, the corollary follows from Lemma <ref> and Proposition <ref>.
 



 Let {z_1,z_2} be a Λ-basis of H^1_(K,T). Let us write ^±∘_p (z_i)=(z_i1^±,z_i2^±) for i=1,2. Set f_±:=(z_ij^±)∈Λ.










We consider the following commutative diagram with exact rows



  
        (Q1) at (-3.5,2) 0;
    (Q2) at (-1,2) ( Λ_n^-⊕  I_n^+)^⊕ 2;
    (Q3) at (3,2) Λ_n^⊕ 2;    
     (Q4) at (7,2) (Λ_n/ (_n^+,ω_n^- ))^⊕ 2;
      (Q5) at (10,2) 0;

       (Q6) at (-3.5,0) 0;      
      (Q7) at (-1,0) ( Λ_n-1^-⊕ I_n-1^+)^⊕ 2;
    (Q8) at (3,0) Λ_n-1^⊕ 2;
(Q9) at (7,0) (Λ_n-1/ (_n^+,ω_n^- ))^⊕ 2;
   (Q10) at (10,0) 0;


 [->] (Q1)–(Q2);
  [->] (Q2)–node [above] F_n^⊕ 2(Q3);
  [->] (Q2)–node [right] pr(Q7);
  [->] (Q3)–node [right] pr(Q8);
  [->] (Q4)–node [right] pr(Q9);
  [->] (Q3)–(Q4);
  [->] (Q4)–(Q5);
   [->] (Q6)–(Q7);
    [->] (Q7)–node [above] F̂_n^⊕ 2(Q8);
   [->] (Q8)–(Q9);
     [->] (Q9)–(Q10);
         
       


where  F_n:  Λ_n^-⊕  I_n^+⟶Λ_n is given by (f,g)↦_n^+f+ω_n^-gω_n, and F̂_n:  Λ_n-1^-⊕  I_n-1^+⟶Λ_n-1 is the map (f,g)↦_n^+f+ω_n^-gω_n-1. This is well-defined since Λ_n^-≃Λ/ω_n^-,  I_n^+≃Λ/ω_n^+ and ω_n^+ω_n^-=ω_n^-ω_n^+=ω_n.






The projection Λ_n/ (_n^+,ω_n^- )⟶Λ_n-1/ (_n^+,ω_n^- ) is bijective.


  This is <cit.>.





Define

    M_n:=Λ_n^⊕ 2/⟨ F_n^⊕ 2(z_1),F_n^⊕ 2(z_2) ⟩   and    M_n-1:=Λ_n-1^⊕ 2/⟨F̂_n^⊕ 2(z_1),F̂_n^⊕ 2(z_2) ⟩,

where F_n^⊕ 2(z_i), i=1,2 denotes the image of z_i under the composition

    H^1_(K,T)  H^1(k_∞,T)⟶ H^1(k_n,T)Λ_,n^-⊕ I_,n^+≃(Λ_n^-⊕ I_n^+)^⊕ 2Λ_n^⊕ 2,

and F̂_n^⊕ 2(z_i) are defined similarly. Our goal is to study ∇ M_n, via special matrices introduced in Definition <ref>. 








    For n≫0,  
    ∇ M_n =
             2  (_n^+(X)) + _ϵ_n(f_-(ϵ_n))        if  n   is odd,
    
              2 (ω_n^-(X)) +_ϵ_n(f_+(ϵ_n))       if  n  is even,

where ϵ_n=ζ_p^n-1.


Since the projection map π_n: M_n ⟶ M_n-1 is surjective and M_n-1 is finite, we have

    ∇ M_n=__p(π_n)
        .

  One can see that π_n≃(Λ/⟨Φ_n⟩)^2/⟨ F_n^⊕ 2(c_1^-,+),F_n^⊕ 2(c_2^-,+) ⟩. Therefore, it follows from <cit.> that 
  
    __p(π_n)=
             2  (_n^+(X)) + _ϵ_n(f_-(ϵ_n))        if  n   is odd,
    
              2 (ω_n^-(X)) +_ϵ_n(f_+(ϵ_n))       if  n  is even.







    For n≫ 0, we have
    
    ∇(E/K_n)=
    
      2( q_n-1-1)   + λ_- + (p^n-p^n-1)μ_-       if  n   is odd,
    
            2(  q_n-1+1)+λ_+ + (p^n-p^n-1)μ_++1       if  n  is even.




 It follows from Lemma <ref> that we have

    ∇ M_n =
             2  (_n^+(X)) + _ϵ_n(f_-(ϵ_n))        if  n   is odd,
    
              2 (ω_n^-(X)) +_ϵ_n(f_+(ϵ_n))       if  n  is even,

where ϵ_n=ζ_p^n-1. By Lemma <ref> and the commutative diagram above, we have
  
    

  
   (Qa) at (-3.5,2) 0;
    (Qb) at (-3.5,0) 0;
    (Qc) at (10,0) 0.;
    (Qd) at (10,2) 0;

    
     (Q2) at (-1,2) ”(E/K_n);
     (Q3) at (6,2)  (Λ_n/ (_n^+,ω_n^- ))^⊕ 2;
      
      (Q5) at (-1,0) ”(E/K_n-1);
    (Q6) at (6,0) (Λ_n-1/ (_n^+,ω_n^- ))^⊕ 2;
(Q7) at (2.5,0) M_n-1;
    (Q8) at (2.5,2) M_n;

   
    [->] (Q2)–(Q8);
    [->] (Q5)–(Q7);
      [->] (Q7)–(Q6);
    [->] (Q8)–(Q3);
     [->] (Q8)– (Q7);
    [->] (Q3)–node [right] = (Q6);
     [->] (Qa)–(Q2);
     [->] (Q3)–(Qd);
      [->] (Qb)–(Q5);
     [->] (Q6)–(Qc);
      [-] (Q2)–(Q5);
       


It then follows from Lemma <ref> that ∇'(E/K_n)=∇ M_n, for n≫ 0. By Lemma <ref> (i), Proposition <ref> and Lemma <ref> we have 

    ∇(E/K_n)=∇ M_n +∇^0(E/K_n).

On the other hand, we have ( ^+(E/K_∞))=( f_0(X)f_+(X)/X ) and ( ^-(E/K_∞))=( f_0(X)f_-(X)). Hence, the result follows.






We are now ready to conclude the proof of Theorem <ref>.


  Suppose that Conjecture <ref> holds. Under our running hypotheses, we have:
    
        
  * _ E(K_n) is bounded independently of n;
        


        
        
  * Assume that (E/K_n )[p^∞] is finite for all n≥ 0. Define r_∞=sup_n≥0{_ E(K_n)}. Then, for n≫ 0, we have
        
    ∇(E/K_n )[p^∞]=
          2s_n-1   + λ_- + ϕ(p^n)μ_-   -r_∞      if  n   is odd,
    
              2s_n-1 +λ_+ + ϕ(p^n)μ_+ -r_∞      if  n  is even,

        where ϕ(p^n)=p^n-p^n-1, and s_n=∑_k=1^n(-1)^n-kp^k, for n≥ 0.
    


      Since ∇(E/K_n) is defined when n is sufficiently large by Corollary <ref>, the kernel and cokernel of the natural map (E/K_n+1)→(E/K_n) are finite for such n. In particular, ∇(E/K_n) is bounded independently of n. Hence, the first assertion follows from the well-known exact sequence
    
    0 ⟶ E(K_n)⊗_p/_p ⟶_p^∞(E/K_n) ⟶(E/K_n )[p^∞] ⟶ 0.

        
    It follows from the exact sequence (<ref>) and Lemma <ref> that
    
    ∇(E/K_n )[p^∞] = ∇(E/K_n)-∇ E(K_n)⊗_p.

    Part (i) tells us that ∇ E(K_n)⊗_p=r_∞ for n≫0.  Thus, the assertion (ii) follows from Corollary <ref>.




   Similarly to <cit.>, it would be possible to give an alternative proof of part (i) of Theorem <ref> via analogues of the control theorems in 9 of op. cit.



alpha 



