Comptes Rendus Mathématique

. Let K be an imaginary quadratic ﬁeld where p splits. We study signed Selmer groups for non-ordinary modular forms over the anticyclotomic (cid:90) p -extension of K , showing that one inclusion of an Iwasawa main conjecture involving the p -adic L -function of Bertolini–Darmon–Prasanna implies that their µ -invariants vanish. This gives an alternative method to reprove a recent result of Matar on the vanishing of the µ -invariants of plus and minus signed Selmer groups for elliptic curves


Introduction
Let E /Q be an elliptic curve with good ordinary reduction at a prime p, and suppose E [p] is irreducible as a Gal(Q/Q)-module. Associated to E and the cyclotomic Z p -extension of Q is a p-primary Selmer group Sel(Q cyc , E ). Mazur [27] studied Iwasawa-theoretic properties of E over Q cyc by showing that Sel(Q cyc , E ) satisfies the so-called control theorem. This allows us to pass between arithmetic properties of E over Q cyc and those over intermediate finite subextensions of Q cyc . A famous conjecture due to Ralph Greenberg asserts that the Iwasawa µ-invariant of Sel(Q cyc , E ) is zero [10,Conjecture 1.11], meaning that its Pontryagin dual is a finitely generated Z p -module.
When E has good supersingular reduction at p, Sel(Q cyc , E ) is less suitable for studying the Iwasawa theory of E since it does not satisfy a nice control theorem. One can instead study refined Selmer groups Sel ± (Q cyc , E ) defined by Kobayashi [21] or Sel #/ (Q cyc , E ) defined by Sprung [33] depending on whether a p (E ) = 0 or not. As E [p] is always irreducible in this case, one conjectures that the µ-invariants of these refined Selmer groups vanish. Indeed, extended numerical calculations carried out by Pollack [30] confirm this speculation.
Some progress has been made on these conjectures; for instance, it is known that if E 1 and E 2 are elliptic curves such that E 1 [p] E 2 [p], then the vanishing of the µ-invariant of the Selmer group of one curve is equivalent to the vanishing of that of the other curve [11,19] (see also [12] and [32] where these results have been generalized to the settings of modular forms and abelian varieties at non-ordinary primes).
While these conjectures are interesting in their own rights, they have important applications in the study of the Iwasawa main conjecture (which relates Selmer groups to analytic p-adic L-functions). This was one of the primary motivations of Greenberg and Vatsal in [11]. More generally, given a p-ordinary modular form f , Emerton, Pollack, and Weston showed that if the µ-invariant of both the Selmer group Sel(Q cyc , f ) and the analytic p-adic L-function L p ( f ) vanish, then the main conjecture holds for every modular form in the Hida family to which f belongs [9, Corollary 1].
While one may confirm the vanishing of µ(Sel(Q cyc , E )) for particular elliptic curves, general vanishing results remain elusive. If we instead fix an imaginary quadratic field K /Q which satisfies a Heegner hypothesis and consider the corresponding anticyclotomic Z p -extension, much more is known. 1 For instance, in the p-ordinary case, the authors proved a general vanishing result for µ-invariants associated to modular forms [14,Theorem 4.5]. In the supersingular setting, a recent result of Matar gives a vanishing result for anticyclotomic µ-invariants associated to an elliptic curve E with a p (E ) = 0, see [26,Theorem B].
The goal of this paper is to offer a method for extending these results to non-ordinary modular forms, conditional on one inclusion of an anticyclotomic Iwasawa main conjecture (see hypothesis (H− ⊆) in Section 4). We work with the signed #/ -Selmer groups defined in [6], whose construction generalizes that of the ±-Selmer groups in the elliptic curve case. See Theorem 14 for our main result. Our approach is similar to the one employed in [14] and gives a new proof of [26,Theorem B] under a different set of hypothesis (see Remark 15). In addition to the Iwasawa main conjecture, we rely on the study of auxiliary Selmer groups and a result on the mod p non-vanishing of generalized Heegner cycles by Hsieh [18].
On combining our results from [13,14], we expect that Theorem 14 will have applications in anticyclotomic Iwasawa main conjectures in the non-ordinary case. In order to obtain results in this direction, one will have to prove that the signed Selmer groups do not admit non-trivial submodules of finite index, generalizing our previous joint work with Vigni [15]. We plan to study these questions in the future.
We assume that p > 2r ; we assume f to be non-ordinary at p, and so it follows from a result of Fontaine and Edixhoven that any twist of the associated residual representation is absolutely irreducible. Let K be an imaginary quadratic extension of Q with discriminant d K coprime to N p in which p = pp and every prime dividing N splits. We assume K has unit group {±1} and that d k is odd or divisible by 8.
If all of these conditions are met, we say that ( f , K , p) is admissible.

Running notation
Fix an embedding K → C as well as an embedding Q → C p such that p lands inside the maximal denote the finite extension of Q p (inside of C p ) generated by the Fourier coefficients of f , and let O denote its valuation ring with a fixed uniformizer . As stated above, we assume that f is non-ordinary at p, in the sense that a p ( f ) is not a unit in O .
Let W f be a realization of Deligne's 2-dimensional F-representation associated to f . If we normalize such that the cyclotomic character has Hodge-Tate weight +1, then W f has Hodge- and where ι denotes the involution on Λ sending a group-like element to its inverse.

Signed anticyclotomic Selmer groups
In this section we briefly recall the construction and properties of some signed anticyclotomic Selmer groups associated to our modular form f . For proofs and more details, the reader is referred to [6, § 2 and § 4].

Remark 1.
In [6], the authors work with the Galois representation attached to f twisted by a pdistinguished (in particular nontrivial) ray class character over K . However, this condition is only used in the study of Euler systems in § 3 of loc. cit., which we will not need for our purposes. In the present paper, we will set the ray class character to be the trivial character. Note that the signed Selmer groups defined in loc. cit. with trivial ray class character and r = 1 have also been studied in [8].
In [6], Sections 2.3 and 2.5 are devoted to defining #/ -Coleman maps associated to T f , which are used to define signed Selmer groups in the non-ordinary setting. In particular, for each q ∈ {p,p}, Shapiro's lemma gives an identification By decomposing Perrin-Riou's big logarithm map via the theory of Wach modules, for each q ∈ {p,p} and • ∈ {#, } we obtain maps In fact, two-variable Coleman maps over the Iwasawa algebra of the Z 2 p -extension of K have been constructed in [6]. For our purposes, we are taking the anticyclotomic specializations of those maps.

Remark 2.
In general, one may define a pair of Coleman maps with respect to any Wach module basis satisfying [23,Theorem 3.5]. The Coleman maps defined in [5] are built out of the Wach module basis constructed in [2, Proposition V.2.3], which only depends on p, k and a p ( f ).
The relevant local Selmer structures are defined as follows.
Definition 3. For each q ∈ {p,p} and L q ∈ {0, , #, }, define the local structure For any place v p of K ∞ , we write In what follows, Σ is any finite set of places of K containing those dividing N p∞, and K Σ is the maximal extension of K unramified outside of Σ. .
For each place λ, let us write When λ p or L λ ∉ {0, }, we obtain a dual Selmer structure for A f at λ by taking When λ | p and L λ ∈ {0, }, we set Definition 5. For L = (L p , Lp) ∈ {0, , #, } 2 , we define the Selmer groups of A f over K ∞ by We will be interested in the Pontryagin duals of these Selmer groups, so we set the following notation.

Remark 7.
To motivate the study of these Selmer groups, we mention that the maps Col #/ ,q defined using Wach modules are generalizations of the Coleman maps studied by Kobayashi [21] and Sprung [33], which were constructed using Honda theory of formal groups. In particular, when f corresponds to an elliptic curve defined over Q and a p ( f ) = 0, we recover the ±-Selmer groups that have generated much study in the literature. See [23, § 5.2] where the comparison between the two approaches (over the cyclotomic extension of Q) is discussed. Since the twovariable Coleman maps in [5] are defined via inverse limits of the one-variable cyclotomic Coleman maps over unramified extensions over Q p , it can be checked that the plus and minus conditions over the Z 2 p -extensions are given by the "jumping conditions" of Kobayashi and Kim in [20,21]. In particular, the resulting conditions over the anticyclotomic Z p -extension of K also agree.

Remark 8.
In the proof of Lemma 9 below, we will need to consider the Selmer groups Sel L (K ∞ , A f (ψ)), where A f (ψ) is the twist of A f by a finite order character ψ of Γ. These Selmer groups are defined by modifying Definition 5 in the obvious way.

Some preliminary lemmas
We will now collect several preliminary lemmas which will be necessary for the proof of Theorem 14 in the next section.
The first lemma we prove is a generalization of [7, Lemma 2.3].
The following statements are true.

Proof. Recall from
Note that the and 0 local conditions which we defined for T are dual to each other under (2).
Here we have written H 1 L (K ∞ , A f (ψ j )) for the subgroup of Sel L (K ∞ , A f (ψ j )) consisting of cohomology classes whose restriction to λ | p lies in H 1 L λ (K λ , A f (ψ j )) div , and r is the core rank of the { , } local conditions [28,Definition 4.1.11]. In particular, by Wiles' formula [28, Proposition 2.3.5], r is given by where v denotes the unique archimedean place of K . The first term is equal to 1; this follows from [6, Lemma 2.16], which tells us that the image of Col ,p is of rank 1 over Λ, so the kernel is of rank 1 as well. The second term is equal to 2 by the local Euler characteristic formula, and the third term is visibly 2. Thus r = 1, and letting i → ∞ gives an isomorphism With (3) established, the rest of the arguments are formal and do not rely on any particular details about the local conditions defining the Selmer groups. Thus, the proof concludes in the same way as [7,Lemma 2.3].
We now introduce a technical assumption, the validity of which is discussed below in Remark 11.
(H − ) Lemma 10. Suppose ( f , K , p) is admissible and that (H-) holds. Then there exists a short exact sequence where D is a torsion Λ-module.

Remark 11.
When r = 1, (H − ) is shown to hold in [8,Theorem 5.7]. See also [25,Theorem 1.4] in the case of elliptic curves with a p = 0. The proofs of these results require the machinery of Kolyvagin systems and it goes back to Howard [16,17]. When r > 1 and a p ( f ) = 0, it is expected that (H − ) holds for both # and (see [5,Remark 4.11 and Theorem 4.14 (v)]).
For the rest of the paper, let L(f /K ) ∈ Λ ur denote the p-adic L-function attached to f over K defined by Brakočević and Bertolini-Darmon-Prasanna in [3,4], where Λ ur is the Iwasawa algebra Λ⊗ Z p Z ur p , with Z ur p being the p-adic completion of the ring of integers of the maximal unramified extension of Q p . We note that this p-adic L-function is defined using two periods (a complex period Ω ∞ and a p-adic period Ω p ) which are independent of f (see [4, § 5.2] and [22, § 2.2]).
Indeed, Ω ∞ depends only on K and Ω p is a p-adic unit. In particular, the µ-invariant of L(f /K ) is well-defined.
We assume the following divisibility from the so-called BDP anticyclotomic Iwasawa main conjecture: (H− ⊆) Remark 12. In fact, the BDP anticyclotomic Iwasawa main conjecture predicts that there is an equality When r = 1, Castella-Çiperiani-Skinnner-Sprung gave a proof of this conjecture under certain hypotheses in [8,Theorem 5.8]. For a general r , Kobayashi-Ota showed in [22,Theorem 1.5] that The authors plan to investigate the integrality of this inclusion for general r in future work. We thank the referee for pointing out that the tensor product with Q p creates problems for studying the µ-invariant.
Remark 13. In [14], the authors mistakenly cited the weaker version of (H− ⊆) given in [22,Theorem 1.5] while proving a vanishing µ result for modular forms with good ordinary reduction. In the ordinary case, it is possible to appeal to other sources which do handle the integrality issue, so the validity of [14,Theorem 4.5] is unaffected. We explain this alternate argument here.
In the p-ordinary elliptic curve case, the divisibility of (H− ⊆) has been established by Castella; see displayed equation (A.9) in [7, Proof of Theorem 3.4]. In this argument, Castella refers to Theorem A.5 of op. cit., which is one inclusion of the Heegner point main conjecutre formulated by Perrin-Riou in [29] and it has been proved by [17,Theorem B]. Replacing Howard's result with [24,Theorem 3.5], the argument of Castella carries over to prove (H− ⊆) for general modular forms.

Main result
Recall that for M a finitely-generated Λ-module, there exists a pseudo-isomorphism (that is, a Λ-morphism with finite kernel and cokernel) for suitable integers r, s, t ≥ 0, a i , n j ≥ 1 and irreducible Weierstrass polynomials F j ∈ O [X ]. We then define the µ-invariant and the λ-invariant of M by n j deg F j .
Our focus of study in this paper is the µ-invariant. For any set of local conditions L as in Section 3, we write µ L ( f ) = µ X L f .

Remark 15.
As mentioned in Remark 7, when f is associated to an elliptic curve over Q with a p ( f ) = 0, the #/ -Selmer groups coincide with the ±-Selmer groups.