THE VANISHING OF ANTICYCLOTOMIC μ-INVARIANTS FOR NON-ORDINARY MODULAR FORMS

Let K be an imaginary quadratic field where p splits. We study signed Selmer groups for non-ordinary modular forms over the anticyclotomic Zp-extension of K, showing that their μ-invariants vanish. This generalizes and gives a new proof of 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 [Maz72] 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 [Gre99, 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 [Kob03] or Sel #/♭ (Q cyc , E) defined by Sprung [Spr12] 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 [Pol03] 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 [GV00,Kim09] (see also [HL19a] and [Pon20] 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 [GV00]. 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 [EPW06, 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 [HL21,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 [Mat21,Theorem B].
The goal of this paper is to extend these results to non-ordinary modular forms. We work with the signed #/♭-Selmer groups defined in [BL21], whose construction generalizes that of the ±-Selmer groups in the elliptic curve case. See Theorem 5.1 for our main result. Our approach is similar to the one employed in [HL21] and is entirely different from the proof given in [Mat21]. We rely on the study of auxiliary Selmer groups, a result on the mod p non-vanishing of generalized Heegner cycles by Hsieh [Hsi14], and recent progress on anticyclotomic Iwasawa Main Conjectures made by Kobayashi-Ota [KO20].
On combining our results from [HL19b,HL21], we expect that Theorem 5.1 will have applications in anticyclotomic Iwasawa main conjectures in the non-ordinary case. In order to do 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 [HLV21]. We plan to study these questions in the future.

Hypotheses and notation
2.1. Running hypotheses. Throughout this paper, p ≥ 5 denotes a fixed odd prime. We will consider a normalized eigen-newform of level Γ 0 (N ) and weight 2r, where N > 3 is squarefree, p ∤ N , and r ≥ 1 is an integer. We assume that p > 2r; since f is assumed non-ordinary at p, 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.
2.2. Running notation. Fix an embedding K ֒→ C as well as an embedding Q ֒→ C p such that p lands inside the maximal ideal of O Cp , the ring of integers of C p . Let F = Q p ({a n (f )}) 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 ̟. We will 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, 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 [BL21, §2 and §4].
Remark 3.1. In [BL21], the authors work with the Galois representation attached to f twisted by a p-distinguished (in particular nontrivial) ray class character over K. However, this condition is only used in the study of an Euler system 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 [CÇ SS18].
In [BL21], 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 Col •,q : H 1 (K q , T) → Λ (In fact, two-variable Coleman maps over the Iwasawa algebra of the Z 2 p -extension of K have been constructed in [BL21]. For our purposes, we are taking the anticyclotomic specializations of those maps.) The relevant local Selmer structures are defined as follows.
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. Definition 3.3. For L = (L p , Lp) ∈ {0, ∅, #, ♭} 2 , we define the (compact) Selmer groups .
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 3.4. For L = (L p , Lp) ∈ {0, ∅, #, ♭} 2 , we define the Selmer groups of We will be interested in the Pontryagin duals of these Selmer groups, so we set the following notation. Remark 3.6. 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 [Kob03] and Sprung [Spr12], 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 [LLZ10,§5.2] where the comparison between the two approaches (over the cyclotomic extension of Q) is discussed. Since the two-variable Coleman maps in [BL18] are defined via inverse limits of the onevariable 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 [Kob03,Kim13]. In particular, the resulting conditions over the anticyclotomic Z p -extension of K also agree.
Remark 3.7. In the proof of Lemma 4.1 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 3.4 in the obvious way.

Some preliminary lemmas
We will now collect several preliminary lemmas which will be necessary for the proof of Theorem 5.1 in the next section. The first lemma we prove is a generalization of [Cas17, Lemma 2.3].
(2) rank Λ X (⋆,∅) (f ) = 1 (3) We have an equality of Λ-ideals (1) follows. Using this fact, statements (2) Note that the ∅ and 0 local conditions which we defined for T are dual to each other under (3.2). Since T f /̟ i T f ≃ A f [̟ i ] for all i ≥ 0, we may apply [MR04, Lemma 3.5.3 and Theorem 4.1.13] to obtain an isomorphism 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 [MR04, Definition 4.1.11]. In particular, by Wiles' formula [MR04, 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 [BL21, 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 (4.1) 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 [Cas17, Lemma 2.3].
We now introduce a technical assumption, the validity of which is discussed below in Remark 4.3.
where D is a torsion Λ-module.

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 Our focus of study in this paper is the µ-invariant. For any set of local conditions L as in Section 3, we write The following is our main result. Proof. Let L(f /K) denote the p-adic L-function attached to f over K defined by Brakočević and Bertolini-Darmon-Prasanna in [Bra11,BDP13]. By [KO20, Theorem 1.5], which is valid regardless of the (good) reduction type at p, we have that X (∅,0) (f ) is a torsion Λ-module and We know from [Hsi14, Theorem B], which is equally valid in the supersingular setting, that L(f /K) ∈ Λ \ ̟Λ, hence µ (∅,0) (f ) = 0. Since X (⋆,0) (f ) is a quotient of X (∅,0) (f ), there is a short exact sequence 0 → C → X (∅,0) (f ) ։ X (⋆,0) (f ) → 0.
As X (∅,0) (f ) is a finitely generated torsion Λ-module by [KO20, Theorem 1.5], we see that C must also be torsion over Λ. By [HL19b, Proposition 2.1], µ-invariants are additive in short exact sequences of Λ-modules where the first term is torsion.
Remark 5.2. As mentioned in Remark 3.6, when f is associated to an elliptic curve over Q with a p (f ) = 0, the #/♭-Selmer groups coincide with the ±-Selmer groups. Remark 5.3. Write N = N − N + , where N − (resp. N + ) is divisible only by primes which are inert (resp. split) in K. In this paper we have assumed N − = 1, since this hypothesis is present in the work of Kobayashi and Ota [KO20], which plays a crucial role in the proof of Theorem 5.1. This assumption also allows us to apply a theorem of Hsieh [Hsi14,Theorem B] in the same proof without also assuming "Hypothesis A" of loc. cit.