Comparing anticyclotomic Selmer groups of positive coranks for congruent modular forms -- Part II
aa r X i v : . [ m a t h . N T ] S e p COMPARING ANTICYCLOTOMIC SELMER GROUPS OFPOSITIVE CORANKS FOR CONGRUENT MODULAR FORMS –PART II
JEFFREY HATLEY AND ANTONIO LEI
Abstract.
We study the Selmer group associated to a p -ordinary newform f ∈ S r (Γ ( N )) over the anticyclotomic Z p -extension of an imaginary qua-dratic field K/ Q . Under certain assumptions, we prove that this Selmer grouphas no proper Λ-submodules of finite index. This generalizes work of Bertoliniin the elliptic curve case. We also offer both a correction and an improvementto an earlier result on Iwasawa invariants of congruent modular forms by thepresent authors. Introduction
Let E/ Q be an elliptic curve, K a number field, and p a prime. A fruitful wayto study the arithmetic of E and its associated p -adic Galois representation is todefine a Selmer group Sel( K ∞ , E ) associated to E over a Z p -extension K ∞ /K . ThisSelmer group has the structure of a Λ-module, where Λ ≃ Z p J T K is the Iwasawaalgebra.For many applications, it is useful to know that Sel( K ∞ , E ) has no proper Λ-submodules of finite index. When Sel( K ∞ , E ) is a cotorsion Λ-module, this canbe proved in a wide variety of contexts; see for instance [Gre89, §
7] and [Kid18,Theorem 7.4]. On the other hand, when Sel( K ∞ , E ) is not cotorsion, such aswhen K/ Q is imaginary quadratic satisfying the Heegner hypothesis and K ∞ /K isthe anticycylotomic Z p -extension, one must use different techniques. One of thesetechniques was originally developed by Bertolini [Ber01] in the case when E hasgood ordinary reduction at p , and it has been recently extended to the supersingularsetting by Vigni and the present authors [HLV20].In [HL19], the present authors studied the anticyclotomic Iwasawa theory forSelmer groups associated to p -ordinary modular forms of even weight. Variousauxiliary Selmer groups Sel L ( K ∞ , f ) were introduced, which were shown to haveno proper Λ-submodules of finite index. By studying the relationship betweenSel( K ∞ , f ) and the auxiliary groups Sel L ( K ∞ , f ), results on the variation of Iwa-sawa invariants were obtained in the tradition of Greenberg–Vatsal [GV00], Emerton–Pollack–Weston [EPW06], and Weston [Wes05b].We now summarize the main contributions of this paper: Mathematics Subject Classification.
Key words and phrases.
Anticyclotomic extensions, Selmer groups, modular forms,congruences.The second named author’s research is supported by the NSERC Discovery Grants ProgramRGPIN-2020-04259 and RGPAS-2020-00096. • We generalize Bertolini’s techniques and results from the setting of ellipticcurves to the setting of modular forms. See Theorem 5.1 for the statementof our result. • We correct an inaccuracy in [HL19] which was pointed out to us by MengFai Lim. See § • We correct and greatly simplify the formula for the λ -invariants of congru-ent modular forms given in [HL19, Theorem 5.8]. Our proof is direct anddoes not rely on any auxiliary Selmer groups. Most notably, our improvedresult has no error terms coming from Euler factors or from the cokernel ofthe localization map. See Theorem 4.5 for the statement.As we discuss again in Section 5, once the appropriate objects have been defined,the techniques used in the present paper and in [HLV20] frequently boil down toformal algebraic arguments which are due to Bertolini [Ber01]. That is, Bertolini’sarguments are valid in broad generality once the appropriate algebraic objects havebeen defined. In [HLV20], some of Bertolini’s arguments have been expanded uponin order to highlight their dependence (or lack thereof) on the various algebraicobjects which appear. Acknowledgements.
We thank Meng Fai Lim for pointing out an inaccuracy in[HL19], as well as for useful comments and suggestions on an earlier version ofthis paper. We also thank Ming-Lun Hsieh, Keenan Kidwell and Stefano Vigni forinteresting discussions on topics related to this paper.2.
Notation and Assumptions
Throughout this paper, p denotes a fixed odd prime. Section 3 is self-containedwith its own set of notation. In the rest of the paper, we will consider modular forms f ∈ S k (Γ ( N )), where N > p ∤ N , and k ≥ K be an imaginary quadratic extension of Q with discriminant coprime to N p in which p = p ¯ p and the prime divisors of N split. We fix embeddings K ֒ → C and Q ֒ → Q p .Let F = Q p ( { a n ( f ) } ) denote the finite extension of Q p (inside of C p ) generatedby the Fourier coefficients of f , and let O denote its valuation ring. We fix auniformizer ̟ of O and denote its residue field by κ . When studying two modularforms f and g , we will enlarge O as necessary so that it contains the coefficients ofboth modular forms.In order to apply results of Bertolini [Ber95], Longo–Vigni [LV19a], and Chida–Hsieh [CH15] we will assume that the triple ( f, K, p ) is admissible in the sensethat: • p does not ramify in F • the p -th Fourier coefficient a p ( f ) is a unit in O• p ∤ N ( k − φ ( N ) h K • if k = 2 then a p ( f ) p .(Here φ denotes the Euler totient function and h K denotes the class number of K .)We will also assume that the residual p -adic representation associated to f ,¯ ρ f : G Q → GL ( κ ) , is absolutely irreducible. NTICYCLOTOMIC SELMER GROUPS OF POSITIVE CORANKS – II 3
Denote by K ∞ the anticyclotomic Z p -extension of K , and for n ≥ K n for the subextension of K ∞ such that K n /K is of degree p n . Let Λ denotethe Iwasawa algebra O [[Γ]], where Γ = Gal( K ∞ /K ) ≃ Z p . For each n ≥
1, writeΓ n = Gal( K ∞ /K n ) and G n = Gal( K n /K ) . Structure of modules over
ΩGiven a finitely generated Λ-module M , we define M tor to be the maximal torsionsubmodule of M . We recall from [Jan89, § § → M tor → M → M ++ → T ( M ) → , where M ++ is the reflexive hull of M , which is free over Λ and T ( M ) is finite.Recall that κ = O / ( ̟ ) denotes the residue field of O . We write Ω for theIwasawa algebra κ [[Γ]] = Λ / ( ̟ ) over κ . Since κ is a field, it follows that Ω is aprincipal ideal domain. In particular, given a finitely generated Ω-module N , thereis an isomorphism of Ω-modules N ∼ = Ω ⊕ r ⊕ t M j =1 Ω / ( F j ) , where F j are polynomials in Ω. We define the characteristic ideal of N bychar Ω ( N ) = t Y j =1 F j Ω . We also define λ ( N ) = t X j =1 deg( F j ) . We note that | N tor | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t M j =1 Ω / ( F j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | κ | λ ( N ) . Lemma 3.1.
Let → A → B → C → be a short exact sequence of finitelygenerated Ω -modules. Suppose that A is torsion over Ω , then (i) rank Ω B = rank Ω C ; (ii) char Ω ( B ) = char Ω ( A )char Ω ( C ) .Proof. This follows from the same proof as [HL19, Proposition 2.1]. (cid:3)
Proposition 3.2.
Let M be a finitely generated Λ -module such that µ ( M ) = 0 .Then, rank Ω ( M/̟ ) = rank Λ M. Furthermore, char Ω ( M/̟ ) = char Ω ( M tor /̟ ) char Ω ( T ( M )[ ̟ ]) . Proof.
Let us write M ′ for the Λ-torsion-free quotient M/M tor . We begin by con-sidering the following tautological short exact sequence:0 → M tor → M → M ′ → . J. HATLEY AND A. LEI
Since M ′ is Λ-torsion-free, we have M ′ [ ̟ ] = 0. This gives the short exact sequence0 → M tor /̟ → M/̟ → M ′ /̟ → . Our hypothesis that µ ( M ) = 0 implies that M tor is finitely generated over O .Hence, M tor /̟ is a torsion Ω-module. Therefore, Lemma 3.1 implies thatrank Ω M/̟ = rank Ω M ′ /̟ ;(3.2) char Ω ( M/̟ ) = char Ω ( M tor /̟ )char Ω ( M ′ /̟ ) . (3.3)Let r = rank Λ M . Consider the exact sequence (3.1) applied to the Λ-module M ′ . We have 0 → M ′ → Λ ⊕ r → T ( M ) → M ′ tor = 0 and T ( M ) = T ( M ′ ). As Λ[ ̟ ] = 0, we deduce the following exactsequence(3.4) 0 → T ( M )[ ̟ ] → M ′ /̟ θ −→ Ω ⊕ r → T ( M ) /̟ → . Recall that T ( M ) is finite. This tells us that rank Ω M ′ /̟ = rank Ω Im θ = r . Inparticular, if we combine this with (3.2), we see that rank Ω M/̟ = r . Furthermore,Ω ⊕ r is Ω-torsion-free. Thus, Im θ ∼ = Ω ⊕ r as Ω-modules. Therefore, we deduce from(3.4) a short exact sequence0 → T ( M )[ ̟ ] → M ′ /̟ θ −→ Ω ⊕ r → . Lemma 3.1 now tells us thatchar Ω M ′ /̟ = char Ω T ( M )[ ̟ ] . On combining this with (3.3), our result follows. (cid:3)
Proposition 3.2 leads us to introduce the following definition.
Definition 3.3.
Let M be a finitely generated module over Λ . We define c ( M ) tobe the unique integer satisfying the equation | T ( M )[ ̟ ] | = | κ | c ( M ) . Corollary 3.4.
Suppose that A and B are two finitely generated Λ -modules suchthat (i) A/̟ ∼ = B/̟ as Ω -modules; (ii) µ ( A ) = µ ( B ) = 0 ; (iii) Both A and B admit no non-trivial finite Λ -submodules.Then, we have (a) rank Λ A = rank Λ B ; (b) λ ( A ) + c ( A ) = λ ( B ) + c ( B ) .Proof. Our hypothesis (ii) allows us to apply Proposition 3.2 to both A and B .Thus, the hypothesis (i) tells us thatrank Λ A = rank Ω A/̟ = rank Ω B/̟ = rank Λ B and thatchar Ω ( A tor /̟ ) char Ω ( T ( A )[ ̟ ]) = char Ω ( A/̟ )= char Ω ( B/̟ ) = char Ω ( B tor /̟ ) char Ω ( T ( B )[ ̟ ]) . NTICYCLOTOMIC SELMER GROUPS OF POSITIVE CORANKS – II 5
But M tor ∼ = O λ ( M ) as O -modules for both M = A and B thanks to (ii) and (iii).Consequently, M tor /̟ ∼ = κ λ ( M ) as κ -vector spaces. Hence, our result follows. (cid:3) Example . We thank Meng Fai Lim for bringing to our attention the followingexample. Let A = Λ ⊕ O and B the maximal ideal of Λ. Note that neither A nor B admits non-trivial finite Λ-submodules and that µ ( A ) = µ ( B ) = 0. Furthermore,consider the short exact sequence0 → B → Λ → κ → . Since Λ[ ̟ ] = 0, we obtain the following exact sequence:0 → κ → B/̟ θ −→ Ω → κ → . Since both the kernel and cokernel of θ are isomorphic to κ , which is of rank 0over Ω, we see that the image of θ is free of rank one over Ω. This then gives theΩ-isomorphism B/̟ ∼ = Ω ⊕ κ ∼ = A/̟.
In particular, the three hypotheses in Corollary 3.4 hold.It is clear that rank Λ A = rank Λ B = 1, confirming property (a) of the corollary.We have T ( A ) = 0 and T ( B ) = κ . Thus, c ( A ) = 0 and c ( B ) = 1. It is not hardto see that λ ( A ) = 1 and λ ( B ) = 0. Thus, λ ( A ) + c ( A ) = λ ( B ) + c ( B ) = 1 , confirming property (b). From this example, we see that λ ( A ) and λ ( B ) can bedifferent and that the terms c ( A ) and c ( B ) are indispensable in (b). Remark . The preceding results correct some inaccuracies in [HL19, Lemmas2.8 and 2.9] which were brought to our attention by Meng Fai Lim. • In Lemma 2.8, the condition (
C/̟ ) Γ n = 0 is equivalent to saying that C =0. To see this, note that C is finite, and so it is both a discrete and compactmodule. By the p -group fixed point theorem, we have ( C/̟ ) Γ n = 0 if andonly if C/̟ = 0. The latter holds if and only if C = 0 by Nakayama’slemma, which is valid since C is compact. • Lemma 2.9 will not hold if P is finite. Since Γ n will act trivially on P ( i )for n ≫
0, the conclusion can never be attained. The problem in the prooflies in second-to-last line: each summand in the direct sum a priori mapsto P ( i ) rather than being a submodule, but upon taking their sum, thesemaps may introduce a kernel due to the presence of finite modules.Thus, the results from this section of the present paper should be used in lieu ofthose mentioned above.4. Comparing λ -invariants of congruent modular forms Recall from Section 2 that f ∈ S k (Γ ( n )) denotes a modular form of even weight k = 2 r for which the triple ( f, K, p ) is admissible.Denote by ρ f the associated G Q -representation constructed in [Nek92]; it isrealized by a free O -module of rank 2 which we denote by T , and V = T ⊗ F is the r -th Tate twist of the Galois representation constructed by Deligne. Finally, J. HATLEY AND A. LEI let A = V /T . Recall that we assume
T /̟T is irreducible, hence T is unique up toscaling. Our choice of normalization makes both V and T self-dual.For a rational prime ℓ , write G ℓ for the decomposition group at ℓ . As describedin [CH15, § ρ f has the following local properties:(4.1) ρ f | G p ∼ (cid:18) χ − p ǫ r ∗ χ p ǫ − r (cid:19) where χ p is unramified and ǫ is the p -cyclotomic character, and for ℓ | N ,(4.2) ρ f | G ℓ ∼ (cid:18) ± ǫ ∗ ± (cid:19) where ∗ is ramified. This can be deduced via the local Langlands correspondence;see for instance [Wes05a, § N issquarefree.For every integer m ≥
1, let A m = A [ ̟ m ] denote the ̟ m -torsion submoduleof A . There is a canonical isomorphism between A m and T m = T /̟ m T . We willsometimes find it convenient to denote A ∞ = A .In this section, we first recall the various definitions of Selmer groups over an-ticyclotomic extensions following [Gre89], and then we present corrected (and im-proved) versions of our results in [HL19, § Anticyclotomic Selmer Groups.
Since f is p -ordinary and since we areworking with the r -th Tate twist of the associated Galois representation, thereexists a unique G Q p -invariant line F + V ⊂ V on which inertia I Q p acts by ǫ r , where ǫ is the p -cyclotomic character. Let F + A be the image of F + V under the naturalprojection map V → A , and for each m ≥ F + A m = F + A ∩ A m . We define F − T m in a similar way. Then for W ∈ { A m , T m } , we denote by F − W = W/ F + W the unramified quotient.Let E be any finite extension of K and let m ∈ N ∪ {∞} . For any place v of E ,define the ordinary local condition H f ( E v , A m ) = ( ker (cid:0) H ( E v , A m ) → H ( E ur v , A m ) (cid:1) , v ∤ p, ker (cid:0) H ( E v , A m ) → H ( E v , F − A m ) (cid:1) , v | p, where E un v denotes the maximal unramified extension of E v .Recall that p = p ¯ p splits in K . Following [Cas17, Definition 2.2], for v | p and L v ∈ {∅ , Gr , } , set H L v ( E v , A m ) = H ( E v , A m ) if L v = ∅ ,H f ( E v , A m ) if L v = Gr , { } if L v = 0 . If Σ is a finite set of places, we denote by E Σ the maximal extension of E unram-ified outside the places above Σ, and we write H i Σ ( E, ∗ ) for the Galois cohomology H i ( E Σ /E, ∗ ). Definition 4.1.
Let m ∈ N ∪ {∞} . Then for any finite extension E of K , aset L = {L v } v | p of local conditions at p , and for any finite set Σ of places of E NTICYCLOTOMIC SELMER GROUPS OF POSITIVE CORANKS – II 7 containing those which divide
N p ∞ , we define the Selmer group Sel L ( E, A m ) = ker H ( E, A m ) → Y v ∈ Σ v ∤ p H ( E v , A m ) H f ( E v , A m ) × Y v | p H ( E v , A m ) H L v ( E v , A m ) . We then define
Sel L ( K ∞ , A m ) = lim −→ n Sel L ( K n , A m ) , where the limit is taken with respect to the natural restriction map.Whenever L = { Gr , Gr } , we omit the subscript and just write Sel( K n , A m ) .Remark . It is very often helpful to show that the global-to-local cohomologymap which defines a Selmer group is surjective; for instance, this is the problemwhich is studied by Greenberg in [Gre11], where this surjectivity is used to deducethe non-existence of finite-index Λ-submodules. However, the global-to-local mapdefining our Sel( K ∞ , A ) is not surjective; nevertheless, we deduce that it has nofinite-index submodules in Section 5.Let us set some notation for the rest of the paper. For any 1 ≤ n ≤ ∞ we takeΣ n to be the set of places of K n above the rational primes dividing N p ∞ , and wesimply write Σ instead of Σ ∞ .We note that the set Σ is finite since we have assumed that every prime divid-ing N p splits in K , and since K ∞ /K is anticyclotomic, no rational prime splitscompletely in K ∞ [Bri07, Corollary 1].Recall that A m = A [ ̟ m ] and that A is a divisible O -module. Lemma 4.3.
Let m, n ∈ N ∪ {∞} . We have isomorphisms H ( K, A m ) ≃ H ( K n , A m ) G n , and (4.3) H ( K, A m ) ≃ H ( K, A )[ ̟ m ] . (4.4) Proof.
The isomorphism (4.3) follows from the inflation-restriction exact sequenceand our assumption that ¯ ρ f is absolutely irreducible. To prove (4.4), consider thetautological exact sequence(4.5) 0 → A m → A ̟ m −−→ A → . Taking the long exact sequence in G K -cohomology we obtain the exact sequence H ( K, A ) → H ( K, A m ) → H ( K, A )[ ̟ m ] → H ( K, A ) . Once again, our assumption that ¯ ρ f is irreducible implies that the first term is zero,and the last term is zero since Γ has p -cohomological dimension 1. This concludesthe proof. (cid:3) We now prove the following control theorem. We mention that, in the case L = { Gr , Gr } , this is essentially [CH15, Proposition 1.9(1)], taking S = ∆ = 1 intheir notation. Theorem 4.4.
Let m, m ′ , n, n ′ ∈ N ∪ {∞} with m ≤ m ′ and n ≤ n ′ . There is anisomorphism res m,n : Sel L ( K n , A m ) → Sel L ( K n ′ , A m ′ )[ ̟ m ] Gal( K n ′ /K n ) . J. HATLEY AND A. LEI
Proof.
Let us first show that restriction gives us an isomorphismSel L ( K, A m ) → Sel L ( K n , A m ) G n . In light of (4.3) from the previous lemma, this will follow from showing the injec-tivity of the maps H ( K ur v , A m ) → H ( K ur n,v , A m ) , v ∤ p,H ( K v , F − A m ) → H ( K n,v , F − A m ) , v | p,H ( K v , A m ) → H ( K n,v , A m ) , v | p. (Here and at other times we abuse notation and write v for a compatible pair ofplaces above the same rational prime in both K and K n .)For the case when v ∤ p , note that since K ∞ /K is an anticyclotomic Z p -extension, v divides a rational prime ℓ which is not ramified at K ∞ , so K ur v = K ur n,v . For v | p ,[CH15, Lemma 1.8] says that H ( K n,v , F − A ) = 0, and it is clear from the localdescription (4.1) that H ( K n,v , A ) = 0, so injectivity of the second and third mapsfollows from the inflation-restriction exact sequence; note that in the case k = 2,this uses the admissibility assumption a p ( f ) p .Now let us prove that we have an isomorphismSel( K, A m ) → Sel(
K, A )[ ̟ m ] . The theorem will then follow from the composition of these various isomorphisms.In light of (4.4), it suffices to show the injectivity of the maps H ( K ur v , A m ) → H ( K ur v , A ) , v ∤ p,H ( K v , F − A m ) → H ( K v , F − A ) , v | pH ( K v , A m ) → H ( K v , A ) , v | p. Taking the long-exact sequence corresponding to the tautological exact sequence(4.5), we see that the kernels of the maps above are given respectively by A I v / (cid:0) ̟ m A I v (cid:1) v ∤ p,H ( K v , F − A ) v | p, or H ( K v , A ) v | p. where we have written I v for the inertia group at v . Thus, the case when v = p follows by the same argument that we gave above.Now consider v ∤ p . If A is unramified at v , then A I v = A is divisible, we seethat A I v /̟ m A I v = 0 as desired. Otherwise, suppose v | N and A is ramified at v .In this case, by (4.2), the local representation restricted to inertia has the form ρ f (cid:12)(cid:12) I v ∼ (cid:18) ± ∗ ± (cid:19) with ∗ nontrivial. Thus A I v is either 0 or isomorphic to F p / O , in which case it isonce again divisible, and this finishes the proof. (cid:3) Variation of Iwasawa invariants.
In this section we give an application ofthe results in Section 5. Throughout this section, assume that we have two modularforms f and g , of levels N f and N g respectively (possibly of different weights). Let NTICYCLOTOMIC SELMER GROUPS OF POSITIVE CORANKS – II 9 K/ Q be an imaginary quadratic field and p a prime such that the triples ( f, K, p )and ( g, K, p ) are both admissible in the sense of §
2, and assume further that¯ ρ f ≃ ¯ ρ g . So in particular, modifying our previous notation to handle two modular forms bywriting A f and A g , we have an isomorphism of G Q -modules(4.6) A f [ ̟ ] ≃ A g [ ̟ ] . We expand our sets of places Σ n to include the places above the rational primesdividing N g .We now wish to begin studying the Iwasawa invariants of our Selmer groups.Denote by X ( f ) the Pontryagin dual of the Selmer group Sel( K, A f ) and similarlyfor g . Write µ ( f ) = µ ( X ( f )) and λ ( f ) = λ ( X ( f ))as defined in Section 3, and similarly for g . Recall the notation from Definition 3.3,and set c ( f ) = c (Sel( K ∞ , A f )) . and similarly for g .As explained in Example 3.5, there is an inaccuracy in the statements of [HL19,Lemmas 2.8 and 2.9] due to the omission of the error terms c ( f ) and c ( g ). Thisin turn leads to an inaccuracy in the statement of [HL19, Theorem 5.8]. In whatfollows, we will both correct and significantly improve this theorem: we will nowcorrect it by including the terms c ( f ) and c ( g ), and we will improve it by usingTheorem 4.4 to completely bypass the use of non-primitive or auxiliary Selmergroups, thus eliminating all of the other “error terms” in the statement of [HL19,Theorem 5.8]. Theorem 4.5.
Let f and g be modular forms such that ( f, K, p ) and ( g, K, p ) areadmissible in the sense of § ¯ ρ f ≃ ¯ ρ g between their (irreducible) residual Galois representations. Then µ ( f ) = 0 ⇐⇒ µ ( g ) = 0 . Suppose that µ ( f ) = µ ( g ) = 0 and Theorem 5.1 holds for f and g . Then λ ( f ) + c ( f ) = λ ( g ) + c ( g ) . Proof.
Our assumption that A f [ ̟ ] ≃ A g [ ̟ ] implies, thanks to Theorem 4.4, thatwe have isomorphisms Sel( K, A f )[ ̟ ] ≃ Sel(
K, A g )[ ̟ ] , and taking duals, this gives an isomorphism X ( f ) /̟ ≃ X ( g ) /̟. As noted in [HL19, Lemma 3.6(i)], both X ( f ) and X ( g ) have Λ-rank 1, so we mayapply [HL19, Corollary 2.4] to conclude that µ ( f ) = 0 ⇐⇒ µ ( g ) = 0 . Assume that the µ -invariants vanish. Then by Theorem 5.1 we may apply Corollary3.4 to conclude the proof. (cid:3) Remark . In [HL19, Proposition 3.12] it is shown that neither Sel Gr , ∅ ( K, A )nor Sel ∅ , ( K, A ) have any nontrivial finite-index Λ-submodules, and [HL19, Corol-lary 3.8] describes their Λ-ranks, so the statement and proof of Theorem 4.5 alsoapply to these alternative Selmer groups.5.
Non-existence of submodules of finite index
The goal of this section is to prove the following theorem.
Theorem 5.1.
Assume ( f, K, p ) is admissible in the sense of § and that hypotheses (non. triv.) and (Sha) hold. Then Sel( K ∞ , A ) contains no proper Λ -submodulesof finite index. Hypothesis (non. triv.) is defined in § .
1, while (Sha) is defined in § f ∈ S (Γ ( N )) is the modular form associatedto a p -ordinary elliptic curve E/ Q . In this case, Theorem 5.1 is due to Bertolini[Ber01, Theorem 7.1]. Thus, we aim to extend this result to the more general classof modular forms which we have been studying in this paper.Bertolini’s result was recently extended to the case of elliptic curves with super-singular reduction at p in [HLV20] by Vigni and the present authors. The carefulreader will notice that, once certain objects have been properly defined and theirproperties established, the proof of [Ber01, Theorem 7.1] goes through quite for-mally. Our strategy in the present paper will thus be to put all of the necessarypieces into place and then describe the main steps of the proof, referring the readerto the proofs in [Ber01, Section 7] and [HLV20, Section 5] for more details.5.1. Heegner cycles.
Recall that by work of Bertolini-Darmon-Prasana [BDP13](see also [CH18, How04, LV19a]), one may use generalized Heegner cycles in orderto construct a system of compatible cohomology classes which aid in the study ofour Selmer groups.Recall that T m = T /̟ m T and A m = A [ ̟ m ], and in fact these are isomor-phic, with the notation suggesting compatibility with projective and direct limits,respectively. We define, for n ∈ N ,Sel( K n , T ) = lim ←− m Sel( K n , A m ) , and then we define ˆ S ( K ∞ , T ) = lim ←− n Sel( K n , T ) , where the limits are taken with respect to the multiplication-by- ̟ and corestrictionmaps.In what follows, we shall write M ∧ for the Pontryagin dual of a O -module M . Lemma 5.2.
There is a canonical isomorphism of Λ -modules ˆ S ( K ∞ , T ) ≃ Hom Λ (Sel( K ∞ , A ) ∧ , Λ) . Proof.
The argument of [PR87, Lemme 5] holds in our setting upon replacing thecontrol theorem used in [PR87] with Theorem 4.4. (cid:3)
Definition 5.3.
For each n ≥ , we denote by z n the generalized Heegner class z n ∈ Sel( K n , T ) constructed in [CH18, § . NTICYCLOTOMIC SELMER GROUPS OF POSITIVE CORANKS – II 11
These Heegner classes satisfy the following “three-term relation”:(5.1) cores K n +1 /K n ( z n +1 ) = a p ( f ) z n − p k − res K n +1 /K n ( z n − ) . Here, cores K n +1 /K n and res K n +1 /K n denote the natural corestriction and restrictionmaps at the level of Galois cohomology.Let R m,n = ( O /̟ m O )][ G n ], where G n = Gal( K n /K ). Definition 5.4.
Denote by A m,n the R m,n -submodule of Sel( K n , A m ) generated bythe Heegner class z n . Taking limits, we obtain the Λ -module A ∞ = lim −→ m A m,m ⊂ Sel( K ∞ , A ) and its Pontryagin dual H ∞ = A ∧∞ = lim ←− m ( A m,m ) ∧ ⊂ ˆ S ( K ∞ , T ) . We also define A m,n = A Γ n ∞ [ ̟ m ] . When ( m, n ) = (1 , n ) we shall omit m from the notation and simply write R n , A n , and A n . The following lemma will allow us to henceforth consider A m,n a R m,n -submoduleof Sel( K n , T m ). Lemma 5.5.
For each m, n there exist injections of R m,n -modules A m,n ֒ → Sel( K n , T m ) and A m,n ֒ → A m,n +1 . The image of the latter map is precisely A Gal( K n +1 /K n ) m,n +1 .Proof. Remembering that A m ≃ T m , this is immediate from Definitions 5.3 and 5.4and Theorem 4.4. (cid:3) The non-triviality of Heegner cycles mod p is an open question. We will hence-forth impose the following assumption. (non. triv.) For n ≫ A n = 0.In particular, this assumption guarantees A ∞ = 0. The proof of the next proposi-tion is similar to that of [LV19b, Proposition 4.7]. Proposition 5.6.
The R n -module A n is cyclic.Proof. Let n ≫ A n,n = 0. Then since it is a cyclic R n -module, we havea natural surjection R n,n → A n,n so by duality we have an injection A ∧ n,n ֒ → R ∧ n,n ≃ ( O /̟ n O )[ G n ] . Taking limits, we obtain an injection H ∞ ֒ → Λ. Since Λ is Noetherian, thisimplies H ∞ is a finitely-generated and torsion-free Λ-module of rank at most 1.Since Λ is a domain, torsion-free implies free, and our assumption that A ∞ = 0then implies that H ∞ has rank 1. Thus, A n = A Γ n ∞ [ ̟ ]= ( H ∧∞ ) Γ n [ ̟ ] ≃ ( H ∞ /̟ ) Γ n ≃ ( O /̟ O ) [Γ] Γ n ≃ R n which proves the claim. (cid:3) Pairings and universal norms.
For each m, n ∈ N write I m,n for the aug-mentation ideal of R m,n . The following result generalizes [Ber01, Proposition 6.3]to our current setting. Proposition 5.7.
Let m, n ∈ N . There is a perfect pairing of Tate cohomologygroups h· , ·i m,n : ˆ H (cid:0) G n , Sel( K n , A m ) (cid:1) × ˆ H − (cid:0) G n , Sel( K n , A m ) (cid:1) −→ I m,n /I m,n . Proof.
This proof is almost entirely formal, relying on arguments regarding ab-stract Galois cohomology, as in the proofs of [Ber01, Proposition 6.3] and [HLV20,Proposition 5.1]. One generalizes from the elliptic curve setting to the modularform setting by using the appropriate version of local Tate duality as described in[CH15, § (cid:3) Now we will reinterpret this pairing by calculating the Tate cohomology groups.
Lemma 5.8.
We have an isomorphism lim −→ n lim −→ m ˆ H − (cid:0) G n , Sel( K n , A m ) (cid:1) ≃ Sel( K ∞ , A ) Γ , where the subscript denotes Γ -coinvariants.Proof. The proof is identical to the proof of [Ber01, Lemma 6.5]. (cid:3)
We define the universal norm submodule of Sel(
K, T ) . by U S ( K, T ) = \ n ≥ cores K n /K Sel( K n , T ) . Lemma 5.9.
We have an identification lim ←− n lim ←− m ˆ H (cid:0) G n , Sel( K n , A m ) (cid:1) ≃ Sel(
K, T ) /U S ( K, T ) . Proof.
The proof of [Ber01, Lemma 6.6] holds verbatim. (cid:3)
Putting this all together, we obtain the following.
Theorem 5.10.
There exists a perfect pairing hh− , −ii : Sel( K, T ) /U S ( K, T ) × Sel( K ∞ , A ) Γ → Γ ⊗ O F / O . As a consequence of this theorem, we obtain the following useful corollary.
Corollary 5.11.
The Λ -module Sel( K ∞ , A ) admits no proper Λ -submodule of finiteindex if and only if Sel(
K, T ) /U S ( K, T ) is O -torsion-free.Proof. See [Ber01, Corollary 6.2]. (cid:3)
NTICYCLOTOMIC SELMER GROUPS OF POSITIVE CORANKS – II 13
Computation of universal norms.
In this section we will prove Theo-rem 5.1, which will follow from Corollary 5.11 if we can show that Sel(
K, T ) /U S ( K, T )is torsion-free.Recall that R n = R ,n = ( O /̟ O )[ G n ], that A n = A ,n , and that A = A [ ̟ ]. Lemma 5.12. If A n = 0 , then Sel( K n , A ) admits a free R n -submodule U n suchthat A n ⊂ U n .Proof. This is proved in exactly the same way as [HLV20, Lemma 5.5]; one mustsimply ignore the ± symbols and replace E by A . The only detail which will benecessary later is that U n is actually defined as an appropriate submodule of A n +1 which is invariant under Gal( K n +1 /K n ). (cid:3) Let us now recall the definition of the Shafarevich-Tate groups in our setting, aswell as the definition of relative Shafarevich-Tate groups.
Definition 5.13.
Let n ∈ N ∪ {∞} . Write D ( K n , A ) for the maximal divisiblesubgroup of Sel( K n , A ) . Then we define the Shafarevich-Tate group of f over K n by X ( K n , A ) = Sel( K n , A ) / D ( K n , A ) . Consider the tautological exact sequence0 → D ( K n , A ) → Sel( K n , A ) → X ( K n , A ) → . Since D ( K n , A ) is divisible, we have D ( K n , A ) /̟ m = 0 for any m ≥
1, so taking ̟ m -torsion induces a short exact sequence0 → D ( K n , A )[ ̟ m ] → Sel( K n , A m ) → X ( K n , A )[ ̟ m ] → , where the middle term is identified via Theorem 4.4. We will simplify notation andinstead write 0 → D ( K n , A m ) → Sel( K n , A m ) → X ( K n , A m ) → . Recall from Theorem 4.4 that we have an injection Sel( K n , A m ) ֒ → Sel( K n +1 , A m ),hence we also have an injection D ( K n , A m ) ֒ → D ( K n +1 , A m ). Thus, the naturalrestriction map H ( K n , A m ) → H ( K n +1 , A m ) induces a map X ( K n , A m ) → X ( K n +1 , A m ) . Definition 5.14.
We define the relative Tate-Shafarevich groups X ( K n +1 /K n , A m ) = ker ( X ( K n , A m ) → X ( K n +1 , A m )) where the map is induced by restriction. Following Bertolini [Ber01], we impose the following assumption. (Sha)
For all n ≥
0, we assume X ( K n +1 /K n , A ) = 0.Under this assumption, we may show that each submodule U n defined in Lemma 5.12is contained in D ( K n , A ). For n ∈ N , write G n for Gal( K n +1 /K n ). Proposition 5.15.
Suppose X ( K n +1 /K n , A ) = 0 and A n = 0 . Then U n iscontained in D ( K n , A ) . Proof.
Consider the commutative diagram with exact rows0 −−−−→ D ( K n , A ) −−−−→ Sel( K n , A ) −−−−→ X ( K n , A ) −−−−→ y y y −−−−→ D ( K n +1 , A ) G n −−−−→ Sel( K n +1 , A ) G n −−−−→ X ( K n +1 , A ) G n By Theorem 4.4, the middle vertical map is an isomorphism, while the kernel ofthe right vertical map is 0 by assumption, so by the snake lemma the left verticalmap is also an isomorphism, so we have(5.2) D ( K n , A ) ≃ D ( K n +1 , A ) G n . But U n is defined as a submodule of A G n n +1 , so by Lemma 5.12, U n is containedin the right-hand side of (5.2), hence also the left-hand side, which completes theproof. (cid:3) The utility of this last proposition comes from our ability to describe Sel(
K, T )in terms of the modules D ( K n , A m ). Proposition 5.16.
We have an identification
Sel(
K, T ) = lim ←− m D ( K, A m ) . Proof.
For each m ≥ → D ( K, A m ) → Sel(
K, T m ) → X ( K, A m ) → T m ≃ A m and that theirSelmer groups agree at finite level. Each of these groups is finite, so projectivelimits will preserve the exact sequence, and the order of X ( K, A m ) is boundedindependent of m , so taking lim ←− m proves the claim. (cid:3) The following proposition completes the proof of Theorem 5.1 via Corollary 5.11.
Proposition 5.17.
Assume ( f, K, p ) is admissible in the sense of § and thathypotheses (non. triv.) and (Sha) hold. Then Sel(
K, T ) /U S ( K, T ) is a torsion-free O -module.Proof. We begin by noting that Sel(
K, T ) is a torsion-free Λ-module of rank 1; thisis [How04, Proposition 3.4.3] when k = 2 and [LV19a, Theorem 3.5] when k ≥ Λ X ( f ) = 1, so arguing as in [Ber95, § O U S ( K ∞ , T ) = rank Λ ˆ S ( K ∞ , T ) = rank Λ X ( f ) = 1 . Thus, if we can show that
U S ( K, T ) contains a nontrivial element of Sel(
K, T )not divisible by ̟ , then U S ( K, T ) ≃ O and Sel( K, T ) /U S ( K, T ) is a torsion-free O -module as desired.Let us write T D n = lim ←− m D ( K n , A m ) for the Tate module of D ( K n , A ). Thenby Proposition 5.15, we have an inclusion U n ⊂ D ( K n , A ) ≃ T D n /̟T D n . Recall that U n is a free R n -module. Let ˜ U n be a free Z p [ G n ]-submodule of T D n of rank one lifting U n modulo ̟ , generated by an element v n . Then the fact that NTICYCLOTOMIC SELMER GROUPS OF POSITIVE CORANKS – II 15 ˜ U n is free implies that cor K n /K ( v n ) is not divisible by ̟ . On the other hand,Lemma 5.16 tells us thatcor K n /K ( v n ) ∈ T D = U S ( K, T ) . By compactness, we may find a subsequence (cid:0) cor K n /K ( v n i ) (cid:1) i ≥ converging to anelement of Sel( K, T ) that lies in
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Department of Mathematics, Union College, Bailey Hall 202, Schenectady,NY 12308, USA
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