-SELMER COMPANION MODULAR FORMS

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ANNALES DE

L’INSTITUT FOURIER

LesAnnales de l’institut Fouriersont membres du Centre Mersenne pour l’édition scientifique ouverte

Somnath Jha, Dipramit Majumdar & Sudhanshu Shekhar p^r-Selmer companion modular forms

Tome 71, n^o1 (2021), p. 53-87.

<http://aif.centre-mersenne.org/item/AIF_2021__71_1_53_0>

Cet article est mis à disposition selon les termes de la licence Creative Commons attribution – pas de modification 3.0 France.

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p

^r

-SELMER COMPANION MODULAR FORMS

by Somnath JHA,

Dipramit MAJUMDAR & Sudhanshu SHEKHAR (*)

Abstract. —The study ofn-Selmer groups of elliptic curves over number fields in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. Given two elliptic curvesE1andE2over a number fieldK, Mazur–

Rubin have defined them to ben-Selmer companionif for every quadratic character χofK, then-Selmer groups ofE^χ₁ andE₂^χoverKare isomorphic. Given a prime p, they have given sufficient conditions for two elliptic curves to be p^r-Selmer companion in terms of mod-p^r congruences between the curves. We discuss an analogue of this for Bloch–Katop^r-Selmer groups of modular forms. We compare the Bloch–Kato Selmer group of a modular form respectively with the Greenberg Selmer group when the modular form is p-ordinary and with the signed Selmer groups of Lei–Loeffler–Zerbes when the modular form is non-ordinary atp. We also indicate the relation between our results and the well-known congruence results for the special values of the correspondingL-functions due to Vatsal.

Résumé. —L’étude de groupes n-Selmer de courbes elliptiques sur des corps de nombres algébriques dans un passé récent a conduit à la découverte de certains résultats profonds en arithmétique des courbes elliptiques. Étant données deux courbes elliptiquesE1etE2sur un corps de nombres algébriquesK, Mazur–Rubin les a définies commen-Selmer compagnon si pour chaque caractère quadratiqueχ deK, les groupesn-Selmer deE₁^χetE₂^χsurKsont isomorphes. Étant donné un nombre premierp, ils ont donné des conditions suffisantes pour que deux courbes elliptiques soient des compagnons p^r-Selmer en termes de congruences mod-p^r entre les courbes. Nous discutons d’un analogue de ce résultat pour les groupesp^r- Bloch–Kato Selmer de formes modulaires. Nous comparons le groupe Bloch–Kato Selmer d’une forme modulaire respectivement avec le groupe Greenberg Selmer lorsque la forme modulaire est p-ordinaire et avec les groupes de Selmer signés de Lei–Loeffler–Zerbes lorsque la forme modulaire est non ordinaire en p. Nous relions aussi nos résultats de congruence bien connus pour les valeurs spéciales des fonctionsLcorrespondantes dues à Vatsal.

Keywords:residual Bloch–Kato Selmer group, congruence of modular forms.

2020Mathematics Subject Classification:11F33, 11R23, 11R34, 11S25, 11G40.

(*) S. Jha acknowledges the support of ECR grant by SERB and INSPIRE faculty grant by DST. D. Majumdar is supported by SPARC project number 445 by MHRD. S.

Shekhar acknowledges the support of DST INSPIRE Faculty grant.

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1. Introduction

The study of n-Selmer groups of elliptic curves over number fields has been of considerable interest in recent past. For example, for certainn, some striking results on the bounds on the average rank ofn-Selmer groups of elliptic curves overQhas been established by Bhargava et al. On the other hand, some deep results related to the rank distribution ofn-Selmer groups for certain n, of a family consisting of all quadratic twists of an elliptic curve, has been studied by Mazur–Rubin and others (cf. [20]). In [20], instead of the rank distribution of then-Selmer group over the family, they formulate the inverse question: given a primepand a number fieldK, what information aboutE is encoded in thep-Selmer group of an elliptic curve EoverK? Motivated by this question, they define the following:

Definition 1.1 ([20, Definition 1.2]). — LetK be a number field and n∈ N be fixed. Two elliptic curves E₁, E₂ are said to be n-Selmer com- panion, if for every quadratic characterχofK, there is an isomorphism of n-Selmer groups ofE₁^χ andE₂^χ overK i.e.Sn(E₁^χ/K)∼=Sn(E₂^χ/K).

That naturally led them to study p^r-Selmer companion elliptic curves for a (fixed) prime pwith r ∈ N. They gave sufficient conditions for two elliptic curves to be p^r-Selmer companion in terms of various conditions related to modp^r-congruences between the curves (see main theorem [20, Theorem 3.1]). They point out in [20, §1] that it would be interesting to investigate this phenomenon more generally forp^r Bloch–Kato Selmer group of motives instead of elliptic curves and that has led us to study this.

In this article, we fix a prime pand study Bloch–Katop^r-Selmer groups associated to modular forms and discuss p^r Selmer companion modular forms (see Definition 2.4). To avoid various technical difficulties which naturally arise in our case of Selmer groups of modular forms, throughout the paper we make the restrictive hypothesis that the prime p is odd. Also, we state all our results for Selmer groups defined overQ(see Section 2.3).

However, it can be seen from our proofs that, when the modular forms are p-ordinary, our results can be extended to a general number fieldKat the cost of notation and hypothesis becoming more cumbersome but essentially the same in nature. On the other hand, when the modular forms are non- ordinary atp, for a general number filedK, the theory of signed Selmer groups is not properly developed yet. Hence our main theorem can not be generalized to an arbitrary number field in the non-ordinary reduction case.

For i ∈ {1,2} and for a positive integer N coprime to p, let f_i ∈ Sk(Γ0(N p^tⁱ), i) be a normalized cuspidal Hecke eigenform and let

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K_f₁_,f₂_,₁_,₂ be the number field generated by the Fourier coefficients of f1, f2and values of1, 2. Let πbe a uniformizer of the ring of integers of the completion ofK_f₁_,f₂_,₁_,₂ at a prime abovep. A sample of our results is given in the following theorem which is a special case of our main theorem (Theorem 4.10) for trivial nebentypus i.e. when₁=₂= 1.

Theorem 1.2. — Letp be an odd prime and fori = 1,2, let fi be a normalized cuspidal Hecke eigenform in Sk(Γ0(N p^tⁱ)), where (N, p) = 1, k>2 and ti ∈N∪ {0}. Let r∈ Nand φ :Af₁[π^r] −→Af₂[π^r] be aG_Q linear isomorphism. We assume the following:

(1) N is square-free and ∀ ` ∈ {`prime : ` || N}, cond`(ρf₁) = ` = cond`(ρf₂).

(2) Either(a) or(b)holds.

(a) p>k,f1 andf2 are non-ordinary atp, andt1=t2= 0.

(b) p >2k−3, f₁ and f₂ are ordinary at p, and either(i)or (ii) holds.

(i) t₁=t₂= 0.

(ii) ap(fi)6=±1 (modπ).

Then for every quadratic character χ of G_Q and for every fixed j with 0 6 j 6 k−2, we have an isomorphism of the π^r- Bloch–Kato Selmer groups

S_BK(A_f₁_χ(−j)[π^r]/Q)∼=S_BK(A_f₂_χ(−j)[π^r]/Q).

The basic strategy of the proof is to compare each local factor which arises in the definition of the Selmer groups. Specially, comparing the local factors of Selmer groups at the primeprequires most careful analysis.

Note that in our case of studying Bloch–Kato Selmer groups of modular forms, we do not have the advantage of considering the Kummer map on abelian variety. Also, we do not have an obvious analogue of fppf cohomology on Néron model of elliptic curve, used by Mazur–Rubin for treating the case of local factors of Selmer group at the prime p. Thus we adopt a different strategy. Letf1 andf2 be two weightk normalized cuspforms which are congruent modπ^r. Iff1andf2 arep-ordinary, then we compare π^r-Bloch–Kato Selmer local condition at pwith theπ^r-Greenberg Selmer local condition at p. On the other hand, if f1 andf2 are non-ordinary at p, then we compare π^r-Bloch–Kato Selmer local conditions atpwith the π^r-signed Selmer local conditions atp([12]). The theory of signed Selmer groups for non-ordinary modular forms are developed using works of Lei–

Loeffler–Zerbes ([16, 17]). These signed Selmer groups can be viewed as

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generalizations of the±Selmer groups of supersingular elliptic curves orig- inally developed by S. Kobayashi (cf. [14]). The works of F. Sprung on Iwasawa main conjecture for elliptic curves at supersingular primes are also relevant (cf. [27]).

On the other hand, to compare the local factors of the Selmer groups at primes ` with ` 6= p, we impose some condition on the conductor of the residual mod-`Galois representation and use some Iwasawa theoretic tech- niques. When the modular form has weight 2 and corresponds to an elliptic curve, we have shown in Remark 3.2, that the hypothesis of Mazur–Rubin (see [20, Lemma 2.5]) at a prime`6=pof multiplicative reduction implies our hypothesis on the conductor of the residual mod-` Galois representation. Moreover, we have illustrated in Section 6, this condition at`6=pcan be verified in many cases; for example usinglevel lowering results of Ribet, Serre et al.

We stress that there are some important differences in the notion of being p^r-Selmer companion for elliptic curves and modular forms. We demonstrate this in Remark 2.5, where we give an example of a weight 2 cuspform f such that f and fr are π-Selmer companion for infinitely many distinct weight 2 cuspformsfr. This is in contrast with elliptic curves, where given an elliptic curveE, Mazur–Rubin suggests there would be only finitely many elliptic curvesE⁰, such that the pair (E, E⁰) are p^r-Selmer companion.

In Definition 4.14, we take the liberty to extend the definition of Selmer companion for two cuspforms of different weights. Using this definition in Corollary 4.15, we give sufficient conditions for two forms of two different weights (differentp-power level and different nebentypus) to beπ^r-Selmer companion. We give such examples of extendedπ^r-Selmer companion forms within an ordinary Hida family in Example I(4) in Section 6.

We also discussπ^rSelmer companion forms overQcyc, when the congruent modular forms are good, ordinary atp. The proof overQeasily adapts to this case. However, there is a well known congruence result of Vatsal [28]

which shows iff1≡f2(modπ^r) are of weightk>2, levelN and are good, ordinary atp, then for any Dirichlet characterχof conductor prime toN, thep-adic L-functions off1⊗χ and f2⊗χ are congruent mod π^r in the Iwasawa algebra. Thus via the Iwasawa Main Conjecture, our result can be viewed as an algebraic counterpart of Vatsal’s result. However, in case of quadratic characters, our congruence result on Selmer groups works for all possible quadratic characters.

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The structure of the article is as follows. After fixing the notation and basic set-up in Section 2.1, we recall the Galois representation attached to a newform in Sections 2.2 and 2.3 contains the definitions of Green- berg, Bloch–Kato, signedπ^r-Selmer groups of a twisted nomalized cuspidal eigenform. Congruence results on Greenberg Selmer groups are discussed in Section 3. In Section 4, we studyπ^r Bloch–Kato Selmer companion forms and establish our main result (Theorem 4.10). We discussπ^r Selmer companion forms overQcycin Section 5, and point out its relation with the well known congruences of the correspondingp-adicL-functions. We compute several numerical examples verifying our results (at weight 2 as well as at higher weights) in Section 6. In particular, in Section 6 we give explicit numerical examples of π^r-Bloch–Kato Selmer companion modular forms which are (i)p-ordinary and (ii) those which are non-ordinary atpas well.

Acknowledgments

The authors would like to thank David Loeffler and Antonio Lei for useful discussions. We also thank the anonymous referee for valuable comments and suggestions

2. Preliminaries 2.1. Notation and set up:

Throughout we fix an embeddingι_∞ of a fixed algebraic closureQofQ intoCand also an embeddingι_òfQinto a fixed algebraic closureQòf the fieldQòf the`-adic numbers, for every prime`. Fix an odd primepand a positive integerN with (N, p) = 1. Leti∈ {1,2}andf_i∈S_k(Γ₀(N p^tⁱ), _i) where ti ∈ N∪ {0}, be a normalized cuspidal Hecke eigenform which is a newform of conductor N p^tⁱ and nebentypus i. We assume that i is a primitive Dirichlet character of conductor Ci. Then Ci | N p^tⁱ. We can writei=Q

`i,` wherei,`is a primitive Dirichlet character of conductor

=`^n(i,`) withn(i, `)∈N, for every prime divisor`ofC_i. In particular, we can writei=i,p⁰_i, where⁰_i is a Dirichlet character of conductor, say C_i⁰ with (C_i⁰, p) = 1 and C_i⁰ | N. The order of a Dirichlet character τ := the order of the subgroup of the roots of unity inC^∗ generated by the image ofτ. We define a condition (C⁰_i,`) to be used later:

(2.1) (C⁰_i,`) : For every prime`|C_i⁰, the order of²_i,`6=pⁿ for anyn∈N.

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Remark 2.1. — For example, if the order of ⁰_i is co-prime to p, then (C⁰_i,`) is satisfied.

Let K = Kf₁,f₂,₁,₂ be the number field generated by the Fourier coefficients off₁, f₂ and the values of₁, ₂. Let π_K be a uniformizer of the ring of integersOK_p of completion ofKat a primeplying abovepinduced by the embedding ιp. To ease the notation, we often writeO =OK_p and π=πK. Put

S=S_f₁_,f₂ :={`prime :`||N}.

Note that, by definitionp6∈S. For anyOmoduleM,M[π^r] will denote the set ofπ^rtorsion points ofM. For any separable fieldL,GLwill denote the Galois group Gal(L/L). For a groupG acting on a moduleM, we denote by M^G = {m ∈ M|gm = m ∀ g ∈ G}. Also for a number field or a p-adic fieldF, and a discrete Gal(F /F) moduleM,Hⁱ(F, M) will denote the Galois cohomology groupHⁱ(Gal(F /F), M).

2.2. Galois representation of a modular form

Let p be a fixed odd prime and A ∈ N with (A, p) = 1. Let h = Pan(h)qⁿ ∈ Sk(Γ0(Ap^t), ψ) be a normalized eigenform of weight k > 2 and nebentypusψ. Thenhisp-ordinary ifιp(ap(h)) is ap-adic unit. LetKh

be the number field generated by the Fourier coefficients ofhand the values ofψ. LetLbe a number field containingKh andLp denote the completion of this number field at a primep lying abovepinduced by the embedding ιp. LetOL_p denote the ring of integers of Lp and πL be a uniformizer of OL_p. We denote byωp:G_Q−→Z^×p thep-adic cyclotomic character. theo

Theorem 2.2 (Eichler, Shimura, Deligne, Mazur–Wiles, Wiles, etc.). Leth=P

an(h)qⁿ ∈Sk(Γ0(Ap^t), ψ)be a newform of weightk>2where (A, p) = 1. Then there exists a Galois representation,ρh:G_Q−→GL2(Lp) such that

(1) at all primes`-Ap, ρ_h is unramified with the characteristic polynomial of the (arithmetic) Frobenius is given by

trace(ρ_h(Frob_`)) =a_`(h),

det(ρh(Frob`)) =ψ(`)ωp(Frob`)^k−1=ψ(`)`^k−1.

It follows (by the Chebotarev Density Theorem) that det(ρ_h) = ψω_p^k−1.

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(2) LetG_p denote the decomposition subgroup ofG_Qatp. In addition, let us assume h is p-ordinary and denote by αp(h) (respectively β_p(h)) the p-adic unit (respectively non p-adic unit) root of the polynomial X²−ap(h)X +ψ(p)p^k−1. Let λh be the unramified character withλh(Frobp) =αp(h). Then by Mazur–Wiles, Wiles

ρh|G_p∼

λ⁻¹_h ψω_p^k−1 ∗

0 λh

,

LetVh∼=L^⊕2_p denotes the representation space ofρh. By compactness of G_Q, choose anO_L_p latticeT_h∼=O^⊕2_L

p ofV_hwhich is invariant underρ_h. Let ρ_h: G_Q−→GL2

O_L_p πL

be the residual representation ofρ_h.

2.3. Definition of the Selmer groups

We choose and fix a quadratic characterχand setM := cond(χ).Recall fori= 1,2,fi ∈Sk(Γ0(N p^tⁱ), i) are two (fixed) normalized cuspidal eigne- forms with (N, p) = 1 andti∈N∪ {0}. Also recallS=Sf₁,f₂:={`prime :

`||N}. Let Σ be a finite set of primes ofQsuch that Σ ⊃S∪ {p} ∪M. Let QΣ be the maximum algebraic extension of Q unramified outside Σ and set G_Σ(Q) = Gal(QΣ/Q). In this subsection, we use has a notation forf1orf2 i.e.h∈ {f1, f2}. Hence we can takeL=Kf₁,f₂,₁,₂,OL_p =O, πL=π. With theOlatticeTh as above, we have an inducedG_Qaction on the discrete moduleAh:=Vh/Th.We also have the canonical maps

(2.2) 0−→Th−→Vh

ph

−→Ah−→0.

Forj ∈ Z, setV_hχ(−j) =Vh⊗χω_p^−j =Vh⊗Qp(χω_p^−j) with the diagonal action ofG_Q. We further defineT_hχ(−j) =Th⊗χω_p^−j and putA_hχ(−j)=

V_hχ(−j) Thχ(−j).

Let K ⊂ QΣ be a number field. We define Σ_K to be the set primes in Klying over the primes in Σ. In particular for K=Q, Σ_Q= Σ. For every primev∈ΣK, let us choose a subsetH_†¹(Kv, A_hχ(−j))⊂H¹(Kv, A_hχ(−j)).

For this choice, we define†-Selmer groupS†(A_hχ(−j)/K) as (2.3) S†(A_hχ(−j)/K)

:= Ker H¹(QΣ/K, A_hχ(−j))−→ Y

v∈ΣK

H¹(Kv, A_hχ(−j)) H_†¹(Kv, A_hχ(−j))

! .

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For any r∈N, there is a canonical map

(2.4) 0−→A_hχ(−j)[π^r]ⁱ^hχ(−j),r−→ A_hχ(−j) Next we defineπ^r †-Selmer group S_†(A_hχ(−j)[π^r]/K) as (2.5) S_†(A_hχ(−j)[π^r]/K)

:= Ker H¹(QΣ/K, A_hχ(−j)[π^r])−→ Y

v∈ΣK

H¹(K_v, A_hχ(−j)[π^r]) H_†¹(Kv, A_hχ(−j)[π^r])

! ,

where H_†¹(K_v, A_hχ(−j)[π^r]) := i^∗_hχ(−j),r⁻¹ (H_†¹(K_v, A_hχ(−j))) for every v ∈ ΣK. Here

i^∗_hχ(−j),r:H¹(Kv, A_hχ(−j)[π^r])−→H¹(Kv, A_hχ(−j))

is induced from i_hχ(−j),r in (2.4). When there is no confusion, we may denotei^∗_r:=i^∗_hχ(−j),r.

Remark 2.3. — ForK=Q, note that ifA_h[π] is irreducible as aG_Qmod- ule then the natural map from H¹(GΣ(Q), A_hχ(−j)[π^r]) −→ H¹(GΣ(Q), A_hχ(−j))[π^r], induced from the multiplication by π^r map on A_hχ(−j), is an isomorphism. Also it follows from the definition of the Selmer group above that if A_hχ(−j)[π] is irreducible, then the natural map S†(A_hχ(−j)[π^r]/Q)−→S†(A_hχ(−j)/Q)[π^r] is an isomorphism as well.

For a prime v ∈Σ_K, letI_v be the inertia subgroup of Gal(Q/K) at v and p^∗_hχ(−j) : H¹(Kv, V_hχ(−j)) −→ H¹(Kv, A_hχ(−j)) is induced from the mapp_hχ(−j) in (2.2). Now we will make special choice ofH_†¹(Kv, A_hχ(−j)) for every v ∈ Σ_K, to respectively define Bloch–Kato Selmer group and under suitable condition we will also define Greenberg, signed 1 and signed 2 Selmer groups.

First †= BK is defined as follows: For av∈Σ, set H_BK¹ (K_v, A_hχ(−j)) :=

(p^∗_hχ(−j)(H_unr¹ (Kv, Vh)) ifv∈ΣK, v-p p^∗_hχ(−j)(H_f¹(Kv, Vh)) ifv|p.

where,

(2.6) H_unr¹ (Kv, V_hχ(−j))

= Ker

H¹(Kv, V_hχ(−j))−→H¹(Iv, V_hχ(−j))

, v∈ΣK, v-p

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and

(2.7) H_f¹(K_v, V_hχ(−j))

= Ker

H¹(K_v, V_hχ(−j))−→H¹(K_v, V_hχ(−j)⊗_Q_pB_crys) , v|p whereB_crysis the ring of periods, as defined by Fontaine in [7] (see also [2,

§1]). This completes the definition of Bloch–Kato Selmer groups.

For v|p, we also recall the definition of a subgroupH_g¹(Kv, V_hχ(−j)) of H¹(Kv, V_hχ(−j)), to be used later

H_g¹(Kv, V_hχ(−j)) := Ker

H¹(Kv, V_hχ(−j))−→H¹(Kv, V_hχ(−j)⊗_Q_pBdR) , v|p whereBdR is as defined by Fontaine in [7].

Next we take†= Gr and define Greenberg Selmer group. To define this, it is necessary to assume thathis ordinary atp. Then by thep-ordinary property ofρh, by Theorem 2.2 (2),Vh has a filtration as aG_Q_p module (2.8) 0−→V_h⁰−→Vh−→V_h⁰⁰ −→0,

where bothV_h⁰ and V_h⁰⁰ are free Kp vector space of rank 1 and the action ofG_Q onV_h⁰⁰ is unramified atp. We define the image ofV_h⁰ inA to beA⁰_h and setA⁰⁰_h:=A/A⁰_h. Hence we get aG_Q_p module filtration onA_h

(2.9) 0−→A⁰_h−→A_h−→A⁰⁰_h −→0,

where bothA⁰_h^∨andA⁰⁰_h^∨are freeOmodule of rank 1 and the action ofG_Q onA⁰⁰_h is unramified atp. We can also get a similar filtration onA_hχ(−j). We now define

H_Gr¹ (K_v, A_hχ(−j)) :=

(Ker H¹(Kv, A_hχ(−j))−→H¹(Iv, A_hχ(−j))

ifv∈ΣK, v-p Ker H¹(Kv, A_hχ(−j))−→H¹(Iv, A⁰⁰_hχ(−j))

ifv|p.

Now we take † = i and define signed i Selmer group for i ∈ {1,2}

(cf. [12, 16]). To define this, it is necessary to assume thathis “good” at pand also non-ordinary atpi.e.t₁=t₂= 0, (N, p) =1 andv_p(a_p(h))6= 0.

We also assume that the quadratic characterχis trivial i.e. we only define signed Selmer group for f ⊗ω^−j_p . Further, we assume K ⊂ Q(µ_p^∞) :=

S

nQ(µpⁿ). For a primev∈Σ, v-p, we simply define H_i¹(Kv, A_h(−j)) :=H_BK¹ (Kv, A_h(−j)), i= 1,2.

Forv|p, we will now defineH_i¹(Kv, A_h(−j)) fori= 1,2 respectively.

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As v_p(a_p(h))6= 0, a pair of Coleman maps Colh,i:H_Iw¹ (Qp, Th)∼= lim

←−n

H¹(Qp(µpⁿ), Th)−→O[∆][[Γ]]∼=O[∆][[T]]

are defined (see [12, §2] [16] for details). Here ∆ = Gal(Q(µp)/Q). Let PrK_v

be the natural projection map

(2.10) H_Iw¹ (Qp, T_h(−j))^{P r}−→^Kv H¹(K_v, T_h(−j)).

Fori= 1,2, we first define a subsetH_i¹(K_v, T_h(−j))⊂H¹(K_v, T_h(−j)) to be H_i¹(K_v, T_h(−j)) := Pr_K_v(Ker(Col_h,i)⊗ω_p^−j), i= 1,2

Now consider the natural map ι : H¹(K_v, T_h(−j)) −→ H¹(K_v, V_h(−j)) induced by the inclusion of Th in Vh and denote by Vi the subspace of H¹(K_v, V_h(−j)) generated by the image ofH_i¹(K_v, T_h(−j)) underιi.e.V_i= hι(H_i¹(Kv, T_h(−j)))i. Finally, defineH_i¹(Kv, A_h(−j)) := Proj(Vi) where Proj is the natural map Proj :H¹(Kv, V_h(−j))−→ H¹(Kv, A_h(−j)) induced by the projection ofVhtoAh. This completes the definition of signed 1 and 2 Selmer groups ofh⊗ω_p^−j.

Finally, we defineπ^r Selmer companion modular forms.

Definition 2.4. — Let i ∈ {1,2} and f_i ∈ S_k(Γ₀(N_i), _i) be a normalized cuspidal eigenform whereNi ∈ N. Let r ∈ N. We say f1 and f2

are π^r (Bloch–Kato) Selmer companion, if for each critical twist j with 0 6 j 6 k−2 and for every quadratic character χ of G_Q, we have an isomorphism ofπ^r Bloch–Kato Selmer groups off₁⊗χω_p^−j andf₂⊗χω^−j_p overQi.e.

SBK(A_f₁_χ(−j)[π^r]/Q)∼=SBK(A_f₂_χ(−j)[π^r]/Q).

Note that for weightk>2, corresponding tok−1 critical values, there arek−1 many Selmer groups associated to a cuspidal eigenform; and for f1 andf2 to beπ^r-Selmer companion, we need eachj with 06j6k−2 and everyχ, SBK(A_f₁_χ(−j)[π^r]/Q)∼=SBK(A_f₂_χ(−j)[π^r]/Q).

Remark 2.5. — Mazur and Rubin [20, §1] have stated that given an elliptic curve E over a number field K, they expect there are only finitely many elliptic curvesE⁰ overK such thatEandE⁰ are Selmer companion.

Moreover they have shown in [20, Proposition 7.1] that given an elliptic curveE overKthere are only finitely many elliptic curvesE⁰ overKsuch that the pair (E, E⁰) satisfy all the conditions of their main theorem ([20, Theorem 3.1]).

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Our situation is different. Letf ∈S₂(Γ0(N)) be ap-ordinary normalized Hecke eigen newform withap(f)6= ±1 (mod π), N is squarefree and coprime top. We further assume that cond_`(ρ_f) =` for every prime`||N. (In Example I(1), we have given explicit example of such anf.)

Then by Hida theory (see [29]) there exists infinitely many fr ∈ S2(Γ0(N p^r), ψr) such that fr ≡ f (mod π), where π = πf,fr,ψr and ψr

is certain dirichlet character of conductorp^rsatisfyingψris trivial mod π.

Then for eachr, the pair (f, f_r) satisfies conditions (1), (2) and (3) of The- orem 4.10. In particular, f and fr are π Selmer companion for infinitely manyf_r.

Remark 2.6. — In the converse direction, Mazur and Rubin have asked if two elliptic curvesE andE⁰ overKarep^r-Selmer companion, then can we say thatE[p^r]∼=E⁰[p^r] as Gal(K/K)-modules (see [20, Conjecture 7.14 in arxiv version])? It would be interesting to investigate ifπ^r-Selmer companion modular forms of weightk>2 are congruent modπ^r.

3. “Greenberg Selmer companion forms”

In this section, we compare twistedπ^r-Greenberg Selmer groups of twop- ordinaryπ^r congruent cuspforms. We use these results in the next section to study Bloch–Kato Selmer companion forms. Our main result in this section is the following:

Theorem 3.1. — Let p be an odd prime and for i = 1,2, let fi be a p-ordinary normalized eigenform in Sk(Γ0(N p^tⁱ), i), where (N, p) = 1, k>2, ti ∈N∪ {0}. Recall from Section 2.1,C_i⁰ is the tame conductor of i. Letr∈Nandφ:Af₁[π^r]−→Af₂[π^r]be aG_Q linear isomorphism. We assume the following:

(1) N is square-free and∀`∈S,cond`(ρf₁) =`= cond`(ρf₂).

(2) The condition(C⁰_i,`), defined in equation (2.1), is satisfied for i= 1,2.

(3) Assumeω_p^k−1i,p6= 1 (modπ)fori= 1,2.

Then for each fixedjwith06j6k−2, and for every quadratic character χofG_Q there is an isomorphism

SGr(A_f₁_χ(−j)[π^r]/Q)∼=SGr(A_f₂_χ(−j)[π^r]/Q).

Here cond_`(ρ_f₁) and cond_`(ρ_f₂) are defined following Serre and can be found in [18, §1, p. 135]. The proof of the theorem is divided into several

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lemmas and propositions. Before going into the proof, we make the following remark to compare it with the corresponding hypothesis at ` 6= p of Mazur–Rubin.

Remark 3.2. — LetEbe an elliptic curve overQwith squarefree conduc- torN with p-N. LetfE ∈ S2(Γ0(N)) associated withE via modularity and ρ_E = ρ_f_E be the corresponding Galois representation (Section 2.2).

Let`|N be a prime of multiplicative reduction for E. Then note that E has split-multiplicative reduction over an unramified quadratic extension ofQ` and hence E has split multiplicative reduction over Q`(µp). There- foreEp^∞(Q`(µp^∞)) is isomorphic to a subgroup ofQ`(µp^∞)^×/q^Zgenerated by µ_p^∞ and q^1/p^m for some m ∈ N∪ {0} (see [11, Proposition 5.1(ii)]).

Hereq∈Q`(µp)^× is the Tate period of the elliptic curve. The assumption p- ord_`(j(E)) of [20, Lemma 2.5] together with the fact Q`(µ_p^∞) is unramified at ` implies that m= 0 above. Thus Ep^∞(Q`(µp^∞))[p]∼= Z/pZ. In other words,E[p]^I^` ∼=Z/pZ. Recall by the definition of conductor at ` (see [18, §1]),

cond`(ρE) =`^codim^ρÎ`Ê^+sw(ρ^ssÊ⁾ and cond`(ρE) =`^{codim ¯}^ρÎ`Ê^{+sw( ¯}^ρÊ⁾. Also note that using [18, Proposition 1.1], we havesw(ρ^ss_E) =sw(ρE). Now as E has multiplicative reduction at `, using the above facts, we deduce thatE[p]Î^` ∼=Z/pZif and only if cond`(ρE) =`= cond`(ρE). Thus when f is of weight 2 and corresponds to an elliptic curve, we have shown that the condition of [20, Lemma 2.5] implies the hypothesis on conductor given in condition (1) of our Theorem 3.1. Note that [20, Lemma 2.5] is used to show that the condition (iii) of the their main theorem ([20, Theorem 3.1]) holds.

Proposition 3.3. — Let i ∈ {1,2} and fi ∈ Sk(Γ0(N p^tⁱ), i) with (N, p) = 1. Assume N is square-free and ∀ ` ∈ S, cond_`(ρ_f_i) = `. Also assume the hypothesis(C⁰_i,`)defined in equation(2.1). Then for every quadratic characterχ of G_Q, for anyj ∈Z and for every prime `∈Σ\ {p}, cond`(ρf_iχ(−j)) = cond`(ρf_iχ(−j)).

Proof. — Leti∈ {1,2}. Recall,M = conductor ofχandCi= conductor of_i. ThenM =Dor 4DwithDsquare-free. Note asω_pis unramified at`, cond`(ρf_iχ(−j)) = cond`(ρf_iχ) and cond`(ρf_iχ(−j)) = cond`(ρf_iχ).Thus it suffices to show that for every quadratic characterχ ofG_Q, cond`(ρf_iχ) = cond`(ρ_f_i_χ).We prove this by considering various cases.

Case 1: ` ||N and `- M. — Since `-M,χ is unramified at ` and we have cond`(ρf_iχ) = cond`(ρf_i) and cond`(ρf_iχ) = cond`(ρf_i). Now from

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the first assumption of the proposition, we have cond`(ρ_f_i) = cond`(ρ_f_i).

Therefore cond`(ρf_iχ) = cond`(ρf_iχ).

Case 2: `-N and `|M. — Since` -N, ρ_f_i is unramified at `. There- fore cond`(ρf_iχ) = cond`(χ) and cond`(ρf_iχ) = cond`(χ) whereχ denote the residual character modπassociated toχ. Since χis a quadratic character and p 6= 2, cond`(χ) = cond`(χ). This implies that cond`(ρfiχ) = cond`(ρf_iχ).

Case 3: ` || N, ` - C_i⁰ and `|M. — By our assumption, cond_`(ρ_f_i) = cond`(ρf_i) =`. Then from [18, p. 135], we get

cond`(ρf_i) =`^codim^ρ^I`^fi^+sw(ρ

ss fi)

and cond`(ρf_i) =`^{codim ¯}^ρ^I`^fi^{+sw( ¯}^ρ^fi⁾ In particular, codimρ^I_f^`

i 61 and therefore dimρ^I_f^`

i>1. Sinceρ_f_iis ramified at`, dimρ^I_f^`

i = 1. This implies thatρ_f_i has an unramified submodule, say V₁. PutV₂:=V_f_i/V₁whereV_f_i denote the vector space corresponding to ρf_i. Letχ1 andχ2 be the characters associated toV1 andV2respectively.

As`- C_i⁰, we getχ₁χ₂=ω^k−1_p _i. This implies thatχ₂ is also unramified.

Thus we haveρf_i ∼ ^χ^¯_{0 ¯}¹_χ^∗

2

and consequently ρf_iχ =ρf_i⊗χ∼

χ1χ ∗ 0 χ₂χ

.

Now χ being quadratic and ` | M, both χ and χ are ramified at `. In particular, (V_f_i⊗χ)^I^` = 0 i.e. codim(V_f_i⊗χ)^I^` = 2. On the other hand,

ρ_f_i_χ |_G_` ∼

ηiωpχ ∗ 0 ηiχ

,

whereη_iis an unramified character (see [13, Theorem 3.26 (3)]). Again asχ is ramified at`, we deduce (Vf_iχ)^I^` = 0; in other words, codim(Vf_iχ)^I^` = 2 as well. Further by [18, Proposition 1.1],sw(V_f_i⊗χ) =sw((V_f_i_χ)^ss). Hence cond`(ρf_iχ) = cond`(ρf_iχ) holds true in this case.

Case 4: ` || N, ` | C_i⁰ and `|M. — In this case, ρf_i|_I` ∼ ^i,`₀ ⁰₁ and ρf_i|

I` ∼ ^¯^i,`₀ ⁰₁

(see [13, Theorem 3.26 (3)]). Similarly, ρf_iχ|_I` ∼

i,`χ 0

0 χ

and ρf_iχ_|

I` ∼

i,`χ 0

0 χ

.

Note that as`|M, bothχandχare ramified at`as in the previous case.

First we consider the subcase when_iχis ramified at`. This implies that iχis also ramified. Thus codim(Vf_i⊗χ)^I^` = 2 = codim(Vf_iχ)^I^` holds.

Next we assume _iχ is unramified i.e. _i,`χ_` is trivial. Then we have i,`2is trivial. Now by the second assumption of the proposition, the order

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of ²_i,` is not a positive power of p. Hence _i,`² is trivial gives ²_i,` is also trivial. Thusi,` and i,`χ` are both quadratic characters. Hencei,`χ` is trivial implies_i,`χ_` is trivial. Then using ρ_f_i_χ_|

I` ∼ _i,`χ_`⊕χ_` and χ is ramified at `, we deduce codim(V_f_i ⊗χ)^I^` = 1 = codim(V_f_i_χ)^I^`. Again from [18, Proposition 1.1],sw(Vf_i⊗χ) = sw((Vf_iχ)^ss), hence we obtain cond`(ρf_iχ) = cond`(ρf_iχ), as required.

Corollary 3.4. — We keep the hypotheses of Proposition 3.3. Then for every quadratic character χ of G_Q and for the (−j)^th Tate twist of f_i⊗χ, we haveA^I_f^`

iχ(−j)is divisible.

Proof. — This is a direct consequence of Proposition 3.3 and the proof

of [6, Lemma 4.1.2].

Lemma 3.5. — Let us keep the hypotheses of Proposition 3.3. Then for everyq∈Σ\ {p}, everyχ and everyj, we have

H_Gr¹ (Qq, A_f_i_χ(−j)[π^r])

= Ker

H¹(Qq, A_f_i_χ(−j)[π^r])−→H¹(I_q, A_f_i_χ(−j)[π^r]) .

Proof. — Consider the commutative diagram

(3.1)

H¹(Qq, A_f_i_χ(−j)[π^r]) ^φ^r //

i^∗_r

H¹(Iq, A_f_i_χ(−j)[π^r])

s_r

H¹(Qq, A_f_i_χ(−j)) ^φ //H¹(Iq, A_f_i_χ(−j))

Nowb∈H_Gr¹ (Qq, A_f_i_χ(−j)[π^r]) if and only ifb∈Ker(φ◦i^∗_r) = Ker(sr◦φr).

Ass_r is induced by the Kummer map, Ker(s_r)∼= ^(A^fiχ^(−j)⁾

Iq

π^r(A_fiχ(−j))^Iq. It follows from Corollary 3.4 that ker(sr) = 0 which proves the lemma.

Using Lemma 3.5, the following expression of SGr(Af_iχ(−j)[π^r]/Q) is immediate.

Lemma 3.6. — We keep the hypotheses of Proposition 3.3. Then we have an exact sequence

0→S_Gr(A_f_i_χ(−j)[π^r]/Q)−→H¹(G_Σ(Q), A_f_i_χ(−j)[π^r])

−→ Y

q∈Σ,q6=p

H¹(I_q, A_f_i_χ(−j)[π^r])× H¹(Qp, A_f_i_χ(−j)[π^r]) H_Gr¹ (Qp, Af_iχ(−j)[π^r]). Next we study thep-part of the local terms definingSGr(Af_iχ(−j)[π^r]/Q).