VARIATION OF ANTICYCLOTOMIC IWASAWA INVARIANTS
IN HIDA FAMILIESFRANCESC CASTELLA, CHAN-HO KIM, AND MATTEO LONGO

Abstract. Building on the construction of big Heegner points in the quaternionic setting[LV11], and their relation to special values of Rankin–Selberg L-functions [CL16], we obtain
anticyclotomic analogues of the results of Emerton–Pollack–Weston [EPW06] on the variationof Iwasawa invariants in Hida families. In particular, combined with the known cases of the
anticyclotomic Iwasawa main conjecture in weight 2, our results yield a proof of the main
conjecture for p-ordinary newforms of higher weights and trivial nebentypus.

Contents
Introduction
1. Hida theory
2. Big Heegner points
3. Anticyclotomic p-adic L-functions
4. Anticyclotomic Selmer groups
5. Applications to the main conjecture
References

1
4
7
10
17
19
22

Introduction
In a remarkable paper [EPW06], Emerton–Pollack–Weston obtained striking results on the
behaviour of the cyclotomic Iwasawa invariants attached to p-ordinary modular forms as theyvary in Hida families. In particular, combined with Greenberg’s conjecture on the vanishingof the µ-invariant, their main result reduces the proof of the main conjecture to the weight twocase. In this paper, we develop analogous results for newforms based-changed to imaginary
quadratic fields in the definite anticyclotomic setting. In particular, combined with Vatsal’s
result on the vanishing of the anticyclotomic µ-invariant [Vat03], and the known cases of the
anticyclotomic main conjecture in weight 2 (thanks to the works of Bertolini–Darmon [BD05],
Pollack–Weston [PW11], and Skinner–Urban [SU14]), our results yield a proof of Iwasawa’s
main conjecture for p-ordinary modular forms of higher weights k > 2 and trivial nebentypus
in the anticyclotomic setting.
Let us begin by recalling the setup of [EPW06], but adapted to the context at hand. Letρ̄ : GQ := Gal(Q/Q) −→ GL2 (F)be a continuous Galois representation defined over a finite field F of characteristic p > 3,
and assume that ρ̄ is odd and irreducible. After the proof of Serre’s conjecture [KW09], weknow that ρ̄ is modular, meaning that ρ̄ is isomorphic to the mod p Galois representation ρ̄f0
associated to an elliptic newform f0 . Throughout this paper, it will be assumed that ρ̄ ' ρ̄f0
for some newform f0 of weight 2 and trivial nebentypus.
2010 Mathematics Subject Classification. 11R23 (Primary); 11F33 (Secondary).
1

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F. CASTELLA, C.-H. KIM, AND M. LONGOLet N (ρ̄) be the tame conductor of ρ̄, and let K/Q be an imaginary quadratic field of
discriminant prime −DK < 0 to pN (ρ̄). The field K then determines a decomposition
N (ρ̄) = N (ρ̄)+ · N (ρ̄)−
with N (ρ̄)+ (resp. N (ρ̄)− ) only divisible by primes which are split (resp. inert) in K. We
similarly define the decomposition M = M + · M − for any positive integer M prime to DK .
As in [PW11], we consider the following conditions on a pair (ρ̄, N − ), where N − is a fixedsquare-free product of an odd number of primes inert in K:
Assumption (CR).
(1) ρ̄ is irreducible;(2) N (ρ̄)− | N − ;(3) ρ̄ is ramified at every prime ` | N − such that ` ≡ ±1 (mod p).
Let H(ρ̄) be the set of all p-ordinary and p-stabilized newforms with mod p Galois representation isomorphic to ρ̄, and let Γ := Gal(K∞ /K) denote the Galois group of the anticyclotomic
Zp -extension of K. Associated with each f ∈ H(ρ̄) of tame level Nf with Nf− = N − , definedover say a finite extension F/Qp with ring of integers O, there is a p-adic L-function
Lp (f /K) ∈ O[[Γ]]constructed by Bertolini–Darmon [BD96] in weight two, and by Chida–Hsieh [CH16] for higher
weights. The p-adic L-function Lp (f /K) is characterized, as χ runs over the p-adic charactersof Γ corresponding to certain algebraic Hecke characters of K, by an interpolation propertyof the form
L(f, χ, k/2)
χ(Lp (f /K)) = Cp (f, χ) · Ep (f, χ) ·,
Ωf,N −
where Cp (f, χ) is an explicit nonzero constant, Ep (f, χ) is a p-adic multiplier, and Ωf,N − isa complex period making the above ratio algebraic. (Of course, implicit in all the above is aι∞ιpfixed choice of complex and p-adic embeddings C ←- Q ,→ Qp .)The anticyclotomic Iwasawa main conjecture gives an arithmetic interpretation of Lp (f /K).
More precisely, letρf : GQ −→ AutF (Vf ) ' GL2 (F )be a self-dual twist of the p-adic Galois representation associated to f , fix an O-stable lattice
Tf ⊆ Vf , and set Af := Vf /Tf . Let Dp ⊆ GQ be the decomposition group corresponding toour fixed embedding ιp , and let εcyc be the p-adic cyclotomic character. Since f is p-ordinary,
there is a unique one-dimensional Dp -invariant subspace Fp+ Vf ⊆ Vf where the inertia groupk/2

at p acts via εcyc ψ, with ψ a finite order character. Let Fp+ Af be the image of Fp+ Vf in Af
and set Fp− Af := Af /Fp+ Af . Following the terminology in [PW11], the minimal Selmer groupof f is defined by




Y
Y
Sel(K∞ , f ) := ker H 1 (K∞ , Af ) −→H 1 (K∞,w , Af ) ×
H 1 (K∞,w , Fp− Af ) ,


w-p

w|p

where w runs over the places of K∞ . By standard arguments (see [Gre89], for example), one
knows that the Pontryagin dual of Sel(K∞ , f ) is finitely generated over the anticyclotomic
Iwasawa algebra Λ := O[[Γ]]. The anticyclotomic main conjecture is then the following:
Conjecture 1. The Pontryagin dual Sel(K∞ , f )∨ is Λ-torsion, andChΛ (Sel(K∞ , f )∨ ) = (Lp (f /K)).

ANTICYCLOTOMIC IWASAWA INVARIANTS IN HIDA FAMILIES3

For newforms f of weight 2 corresponding to elliptic curves E/Q with ordinary reduction atp, and under rather stringent assumptions on ρ̄f which were later relaxed by Pollack–Weston[PW11], one of the divisibilities predicted by Conjecture 1 was obtained by Bertolini–Darmon[BD05] using Heegner points and Kolyvagin’s method of Euler systems. More recently, after
the work of Chida–Hsieh [CH15] the divisibilityChΛ (Sel(K∞ , f )∨ ) ⊇ (Lp (f /K))
is known for newforms f of weight k 6 p−2 and trivial nebentypus, provided the pair (ρ̄f , Nf− )satisfies a mild strengthening of Hypotheses (CR). This restriction to small weights comes fromthe use of Ihara’s lemma [DT94], and it seems difficult to directly extend their arguments in[CH15] to higher weights. Instead, as we shall explain in the following paragraphs, in thispaper we will complete the proof of Conjecture 1 to all weights k ≡ 2 (mod p − 1) by adifferent approach, using Howard’s big Heegner points in Hida families [How07], as extended
by Longo–Vigni [LV11] to quaternionic Shimura curves.
Associated with every f ∈ H(ρ̄) there are anticyclotomic Iwasawa invariants µan (K∞ , f ),λan (K∞ , f ), µalg (K∞ , f ), and λalg (K∞ , f ). The analytic (resp. algebraic) λ-invariants are thenumber of zeros of Lp (f /K) (resp. of a generator of the characteristic ideal of Sel(K∞ , f )∨ ),while the µ-invariants are defined as the exponent of the highest power of $ (with $ ∈ O any
uniformizer) dividing the same objects. Our main results on the variation of these invariants
are summarized in the following. (Recall that we assume ρ̄ ' ρ̄f0 for some newform f0 ofweight 2 and trivial nebentypus.)
Theorem 2. Assume in addition that:• ρ̄ is irreducible;• ρ̄ is p-ordinary, “non-anomalous” and p-distinguished:ρ̄|Dp ' 0ε̄ ∗δ̄with ε̄, δ̄ : Dp → F× characters such that δ̄ is unramified, δ̄(Frobp ) 6= ±1 and δ̄ 6= ε̄;• N (ρ̄)− is the square-free product of an odd number of primes.
−

Let H− (ρ̄) := HN (ρ̄) (ρ̄) consist of all newforms f ∈ H(ρ̄) with Nf− = N (ρ̄)− , and fix ∗ ∈{alg, an}. Then the following hold:(1) For all f ∈ H− (ρ̄), we haveµ∗ (K∞ , f ) = 0.
(2) Let f1 , f2 ∈ H− (ρ̄) lie on the branches T(a1 ), T(a2 ) (defined in §1.4), respectively.
ThenXλ∗ (K∞ , f1 ) − λ∗ (K∞ , f2 ) =e` (a2 ) − e` (a1 ),`|Nf+ Nf+12

where the sum is over the split primes in K which divide the tame level of f1 or f2 ,and e` (aj ) is an explicit non-negative invariant of the branch T(aj ) and the prime `.
Provided that p splits in K, and under the same hypotheses on ρ̄ as in Theorem 2, thework of Skinner–Urban [SU14] establishes one of the divisibilities in a related “three-variable”
Iwasawa main conjecture. Combining their work with our Theorem 2, and making use of theaforementioned results of Bertolini–Darmon [BD05] and Pollack–Weston [PW11] in weight 2,
we obtain many new cases of Conjecture 1 (cf. Corollary 5.5):
Corollary 3. Suppose that ρ̄ is as in Theorem 2 and that p splits in K. Then the anticyclotomic Iwasawa main conjecture holds for every f ∈ H− (ρ̄) of weight k ≡ 2 (mod p − 1) andtrivial nebentypus.

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F. CASTELLA, C.-H. KIM, AND M. LONGO

Let us briefly explain the new ingredients in the proof of Theorem 2. As it will be clear
to the reader, the results contained in Theorem 2 are anticyclotomic analogues of the resultsof Emerton–Pollack–Weston [EPW06] in the cyclotomic setting. In fact, on the algebraicside the arguments of loc.cit. carry over almost verbatim, and our main innovations in thispaper are in the development of anticyclotomic analogous of their results on the analytic
side. Indeed, the analytic results of [EPW06] are based on the study of certain two-variable
p-adic L-functions à la Mazur–Kitagawa, whose construction relies on the theory of modular
symbols on classical modular curves. In contrast, we need to work on a family of Shimura
curves associated with definite quaternion algebras, for which cusps are not available. In thecyclotomic case, modular symbols are useful two ways: They yield a concrete realization of thedegree one compactly supported cohomology of open modular curves, and provide a powerfultool for studying the arithmetic properties of critical values of the L-functions attached tomodular forms. Our basic observation is that in the present anticyclotomic setting, Heegner
points on definite Shimura curves provide a similarly convenient way of describing the central
critical values of the Rankin L-series L(f /K, χ, s).
Also fundamental for the method of [EPW06] is the possibility to “deform” modular symbols
in Hida families. In our anticyclotomic context, the construction of big Heegner points in Hida
families was obtained in the work [LV11] of the third-named author in collaboration with
Vigni, while the relation between these points and Rankin–Selberg L-values was establishedin the work [CL16] by the first- and third-named authors. With these key results at hand,
and working over appropriate quotients of the Hecke algebras considered in [EPW06] via the
Jaquet–Langlands correspondence, we are then able to develop analogues of the arguments ofloc.cit. in our setting, making use of the ramification hypotheses on ρ̄ to ensure a multiplicity
one property of certain Hecke modules, similarly as in the works of Pollack–Weston [PW11]
and the second-named author [Kim17].
We conclude this Introduction with an overview of the contents of the paper. In the next
section, we briefly recall the Hida theory that we need, following the exposition in [EPW06, §1]
for the most part. In Section 2, we describe a key extension of the construction of big Heegner
points of [LV11] to “imprimitive” branches of the Hida family. In Section 3, we construct twovariable p-adic L-functions attached to a Hida family and to each of its irreducible components(or branches), and prove Theorem 3.10 relating the two. This theorem is the key technicalresult of this paper, and the analytic part of Theorem 2 follows easily from this. In Section 4,
we deduce the algebraic part of Theorem 2 using the residual Selmer groups studied in [PW11,§3.2]. Finally, in Section 5 we give the applications of our results to the anticyclotomic Iwasawa
main conjecture.
Acknowledgements. During the preparation of this paper, F.C. was partially supported by
the European Research Council (ERC) under the European Union’s Horizon 2020 researchand innovation programme (grant agreement No. 682152); C.K. was partially supported by
an AMS–Simons Travel Grant; M.L. was partially supported by PRIN 2010-11 “Arithmetic
Algebraic Geometry and Number Theory” and by PRAT 2013 “Arithmetic of Varieties overNumber Fields”.
1. Hida theory
Throughout this chapter, we fix a positive integer N admitting a factorization
N = N +N −with (N + , N − ) = 1 and N − equal to the square-free product of an odd number of primes.
We also fix a prime p - 6N .

ANTICYCLOTOMIC IWASAWA INVARIANTS IN HIDA FAMILIES

5

1.1. Hecke algebras. For each integer k > 2, denote by hN,r,k the Zp -algebra generated by
the Hecke operators T` for ` - N p, the operators U` for ` | N p, and the diamond operators
hai for a ∈ (Z/pr Z)× , acting on the space Sk (Γ0,1 (N, pr ), Qp ) of cusp forms of weight k on
Γ0,1 (N, pr ) := Γ0 (N ) ∩ Γ1 (pr ). For k = 2, we abbreviate hN,r := hN,r,2 .
Let eord := limn→∞ Upn! be Hida’s ordinary projector, and define
ordhordhN,r,k ,N,r,k := e

ord
hord
hN,r ,
N,r := ehordlim hord,
N := ←− N,rr

where the limit is over the projections induced by the natural restriction maps.
−

ord
ord S (Γ (N, pr ), Q )Let TN
k 0,1
p
N,r,k be the quotient of hN,r,k acting faithfully on the subspace of e−−N
consisting of forms which are new at all primes dividing N − . Set TN
N,r := TN,r,2 and define−−TN
lim TN.
N := ←− N,rr

Each of these Hecke algebras are equipped with natural Zp [[Z×p ]]-algebra structures via theorddiamond operators, and by a well-known result of Hida, hN is finite and flat over Zp [[1+pZp ]].
1.2. Galois representations on Hecke algebras. For each positive integer M | N we may
ord
consider the new quotient Tnew
M of hM , and the Galois representation
ρM : GQ −→ GL2 (Tnew
M ⊗ L)
described in [EPW06, Thm. 2.2.1], where L denotes the fraction field of Zp [[1 + pZp ]].
−Let T0N be the Zp [[1 + pZp ]]-subalgebra of TN
N generated by the image under the natural−N
ordprojection hN → TN of the Hecke operators of level prime to N . As in [EPW06, Prop. 2.3.2],
one can show that the canonical map
Y
T0N −→Tnew
M ,M

where the product is over all integers M > 1 with N − | M | N , becomes an isomorphism
after tensoring with L. Taking the product of the Galois representations ρM we thus obtain
ρ : GQ −→ GL2 (T0N ⊗ L).
For any maximal ideal m of T0N , let (T0N )m denote the localization of T0N at m and letρm : GQ −→ GL2 (T0N )m ⊗ L
be the resulting Galois representation. If the residual representation ρ̄m is irreducible, thenρm admits an integral model (still denoted in the same manner)ρm : GQ −→ GL2 (T0N )mwhich is unique up to isomorphism.
1.3. Residual representations. Let ρ̄ : GQ → GL2 (F) be an odd irreducible Galois representation defined over a finite field F of characteristic p > 3. As in the Introduction, we assume
that ρ̄ ' ρ̄f0 for some newform f0 of weight 2, level N , and trivial nebentypus. Consider thefollowing three conditions we may impose on the pair (ρ̄, N − ):Assumption (SU).
(1) ρ̄ is p-ordinary: the restriction of ρ̄ to a decomposition group Dp ⊆ GQ at p has a
one-dimensional unramified quotient
over F;(2) ρ̄ is p-distinguished : ρ̄|Dp ∼ 0ε̄ ∗δ̄ with ε̄ 6= δ̄;(3) ρ̄ is ramified at every prime ` | N − .

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F. CASTELLA, C.-H. KIM, AND M. LONGO

Fix once and for all a representation ρ̄ satisfying Assumption (SU), together with a pstabilization of ρ̄ in the sense of [EPW06, Def. 2.2.10]. Let V be the two-dimensional F-vector
space which affords ρ̄, and for any finite set of primes Σ that does not contain p or any factorof N − , defineY(1)
N (Σ) := N (ρ̄)
`m` ,
`∈Σ

where N (ρ̄) is the tame conductor of ρ̄, and m` := dimF V I` .
Remark 1.1. By Assumption (SU) we have the divisibility N − | N (ρ̄); we will further assume
that (N − , N (ρ̄)/N − ) = 1.
Combining [EPW06, Thm. 2.4.1] and [EPW06, Prop. 2.4.2] with the fact that ρ̄ is ramified
at the primes dividing N − , one can see that there exist unique maximal ideals n and m of−0TN
N (Σ) and TN (Σ) , respectively, such that• n ∩ T0N (Σ) = m;−

• (T0N (Σ) )m ' (TNN (Σ) )n by the natural map on localizations;• ρ̄m ' ρ̄.
Define the ordinary Hecke algebra TΣ attached to ρ̄ and Σ byTΣ := (T0N (Σ) )m .
Thus TΣ is a local factor of T0N (Σ) , and we letρΣ : GQ −→ GL2 (TΣ )denote the Galois representation deduced from ρm .
Adopting the terminology of [EPW06, §2.4], we shall refer to Spec(TΣ ) as “the Hida family”
H− (ρ̄) attached to ρ̄ (and our chosen p-stabilization) that is minimally ramified outside Σ.
Remark 1.2. Note that by Assumption (SU), all the p-stabilized newforms in H− (ρ̄) havetame level divisible by N − .
1.4. Branches of the Hida family. If a is a minimal prime of TΣ (for a finite set of primes
Σ as above), we put T(a) := TΣ /a and let
ρ(a) : GQ −→ GL2 (T(a))
be the Galois representation induced by ρΣ . As in [EPW06, Prop. 2.5.2], one can show that
there is a unique divisor N (a) of N (Σ) and a unique minimal prime a0 ⊆ Tnew
N (a) above a suchthat the diagram
Q
new
/ T0
/
TΣ
N − |M |N (Σ) TM
N (Σ)


TΣ /a

=

/ T(a)

/ Tnew /a0
N (a)commutes. We call N (a) the tame conductor of a and set
0T(a)◦ := Tnew
N (a) /a .

In particular, note that N − | N (a) by construction, and that the natural map T(a) → T(a)◦is an embedding of local domains.

ANTICYCLOTOMIC IWASAWA INVARIANTS IN HIDA FAMILIES

7

1.5. Arithmetic specializations. For any finite Zp [[1+pZp ]]-algebra T, we say that a height
one prime ℘ of T is an arithmetic prime of T if ℘ is the kernel of a Zp -algebra homomorphism
T → Qp such that the composite map
×

1 + pZp −→ Zp [[1 + pZp ]]× −→ T× −→ Qp

is given by γ 7→ γ k−2 on some open subgroup of 1 + pZp , for some integer k > 2. We then saythat ℘ has weight k.
Let a ⊆ TΣ be a minimal prime as above. For each n > 1, let an ∈ T(a)◦ be the image of◦Tn under the natural projection hord
N (Σ) → T(a) , and form the q-expansionX
an q n ∈ T(a)◦ [[q]].
f (a) =n>1

By [Hid86, Thm. 1.2], if ℘ is an arithmetic prime of T(a) of weight k, then there is a uniqueheight one prime ℘0 of T(a)◦ such thatXf℘ (a) :=(an mod ℘0 )q n ∈ O℘◦ [[q]],n>1where O℘◦ := T(a)◦ /℘0 , is the q-expansion a p-ordinary eigenform f℘ of weight k and tamelevel N (a) (see [EPW06, Prop. 2.5.6]).
2. Big Heegner points
As in Chapter 1, we fix an integer N > 1 admitting a factorization N = N + N − with
(N + , N − ) = 1 and N − equal to the square-free product of an odd number of primes, and fix
a prime p - 6N . Also, we let K/Q be an imaginary quadratic field of discriminant −DK < 0prime to N p and such that every prime factor of N + (resp. N − ) splits (resp. is inert) in K.
In this section we describe a mild extension of the construction in [LV11] (following [How07])of big Heegner points attached to K. Indeed, using the results from the preceding section, we
can extend the constructions of loc.cit. to branches of the Hida family which are not necessarily
primitive (in the sense of [Hid86, §1]). The availability of such extension is fundamental forthe purposes of this paper.
2.1. Definite Shimura curves. Let B be the definite quaternion algebra over Q of discriminant N − . We fix once and for all an embedding of Q-algebras K ,→ B, and use it to identity
K with a subalgebra of B. Denote by z 7→ z the nontrivial automorphism of K, and choosea basis {1, j} of B over K such that• j 2 = β ∈ Q× with β < 0;
• jt = t̄j for all t ∈ K;
2+×
• β ∈ (Z×q ) for q | pN , and β ∈ Zq for q | DK .
√Fix a square-root δK = −DK , and define θ ∈ K by
D 0 + δK
DK
if 2 - DK ;0
θ :=, where D :=DK /2 if 2 | DK .
2
Note that OK = Z + Zθ, and for every prime q | pN + , define iq : Bq := B ⊗Q Qq ' M2 (Qq )
by




p
Tr(θ) −Nm(θ)−1 Tr(θ)
iq (θ) =,iq (j) = β
,
1
0
0
1
where Tr and Nm are the reduced trace and reduced norm maps on B, respectively. On theother hand, for each prime q - N p we fix any isomorphism iq : Bq ' M2 (Qq ) with the property
that iq (OK ⊗Z Zq ) ⊂ M2 (Zq ).

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F. CASTELLA, C.-H. KIM, AND M. LONGO

For each r > 0, let RN + ,r be the Eichler order of B of level N + pr with respect to the aboveb× +isomorphisms {iq : Bq ' M2 (Qq )}q-N − , and let UN + ,r be the compact open subgroup of R
N ,rdefined byb× + | ip (xp ) ≡ 1 ∗UN + ,r := (xq )q ∈ R(mod pr ) .
N ,r0 ∗Consider the double coset spaceseN + ,r = B × HomQ (K, B) × B
b × UN + ,r ,(2)Xb × bywhere b ∈ B × acts on (Ψ, g) ∈ HomQ (K, B) × B
b · (Ψ, g) = (bΨb−1 , bg)
b × by right multiplication. As is well-known (see e.g. [LV11, §2.1]), X
eN + ,rand UN + ,r acts on B
may be naturally identified with the set of K-rational points of certain genus zero curves
eN + ,r defined as follows:
defined over Q. Nonetheless, there is a nontrivial Galois action on X
eN + ,r is the class of a pair (Ψ, g), thenIf σ ∈ Gal(K ab /K) and P ∈ Xb
P σ := [(Ψ, Ψ(a)g)],b × is chosen so that recK (a) = σ. It will be convenient to extend this action
where a ∈ K × \K
to an action of GK := Gal(Q/K) in the obvious manner.
eN + ,r is also equipped with standard actions of Up , Hecke operators
Finally, we note that X
T` for ` - N p, and diamond operators hdi for d ∈ (Z/pr Z)× (see [LV11, §2.4], for example).
2.2. Compatible systems of Heegner Points. For each integer c > 1, let Oc = Z + cOK
be the order of K of conductor c.
eN + ,r is a Heegner point of conductor c if P is theDefinition 2.1. We say that a point P ∈ X
class of a pair (Ψ, g) withbN + ,r g −1 )Ψ(Oc ) = Ψ(K) ∩ (B ∩ g R
andΨp ((Oc ⊗ Zp )× ∩ (1 + pr OK ⊗ Zp )× ) = Ψp ((Oc ⊗ Zp )× ) ∩ gp UN + ,r,p gp−1 ,where UN + ,r,p denotes the p-component of UN + ,r .
Fix a decomposition N + OK = N+ N+ , and for each prime q 6= p define• ςq = 1, ifq - N +;−1 θ θ• ςq = δ K∈ GL2 (Kq ) = GL2 (Qq ), if q = qq splits with q | N+ ,1 1
and for each s > 0, let s 
θ −1
p 0(s)• ςp =∈ GL2 (Kp ) = GL2 (Qp ), if p = pp splits in K;
1 0
0 1

 s 
0 1
p 0(s)• ςp =, if p is inert in K.
−1 0
0 1
(s) Q
b × via the isomorphisms {iq : Bq 'Set ς (s) := ςpq6=p ςq , viewed as an element in B
M2 (Qq )}q-N − introduced in Section 2.1. Let ıK : K ,→ B be the inclusion. Then one easily
checks (see [CL16, Thm. 1.2] and the references therein) that for all n, r > 0 the points
eN + ,rPepn ,r := [(ıK , ς (n+r) )] ∈ X
are Heegner point of conductor pn+r with the following properties:

ANTICYCLOTOMIC IWASAWA INVARIANTS IN HIDA FAMILIES

9

eN + ,r ), where Lpn ,r := Hpn+r (µpr ) and Hc is• Field of definition: Pepn ,r ∈ H 0 (Lpn ,r , X
the ring class field of K of conductor c.
• Galois equivariance: For all σ ∈ Gal(Lpn ,r /Hpn+r ), we havePepσn ,r = hϑ(σ)i · Pepn ,r ,2where ϑ : Gal(Lpn ,r /Hpn+r ) → Z×
p /{±1} is such that ϑ = εcyc .
• Horizontal compatibility: If r > 1, thenXαer (Pepσn ,r ) = Up · Pepn ,r−1 ,σ∈Gal(Lpn ,r /Lpn−1 ,r )eN + ,r → X
eN + ,r−1 is the map induced by the inclusion UN + ,r ⊆ UN + ,r−1 .
where αer : X• Vertical Compatibility: If n > 0, thenX
Pepσn ,r = Up · Pepn−1 ,r .
σ∈Gal(Lpn ,r /Lpn−1 ,r )Remark 2.2. We will only consider the points Pepn ,r for a fixed a value of N − (which amounts
to fixing the quaternion algebra B/Q), but it will be fundamental to consider different valuesof N + , and the relations between the corresponding Pepn ,r (which clearly depend on N + ) undervarious degeneracy maps.
2.3. Critical character. Factor the p-adic cyclotomic character asεcyc = εtame · εwild : GQ −→ Z×
p ' µp−1 × (1 + pZp )and define the critical character Θ : GQ → Zp [[1 + pZp ]]× by
1/2

(3)

Θ(σ) = [εwild (σ)],
1/2where εwild is the unique square-root of εwild taking values in 1 + pZp , and [·] : 1 + pZp →Zp [[1 + pZp ]]× is the map given by the inclusion as group-like elements.
eN + ,pr from Section 2.1, and set
2.4. Big Heegner points. Recall the Shimura curves X
eN + ,r ) ⊗Z Zp ).
DN + ,r := eord (Div(X
By the Jacquet–Langlands correspondence, DN + ,r is naturally endowed with an action of the
−
N− †
N−
Hecke algebra TN
N,r . Let (TN,r ) be the free TN,r -module of rank one equipped with the Galois
action via the inverse of the critical character Θ, and set−†
D†N + ,r := DN + ,r ⊗TN − (TNN,r ) .
N,reN + ,r be the system of Heegner points of Section 2.2, and denote by Ppn ,r the
Let Pepn ,r ∈ X
image of eord Pepn ,r in DN + ,r . By the Galois equivariance of Pepn ,r (see [LV11, §7.1]), we havePpσn ,r = Θ(σ) · Ppn ,rfor all σ ∈ Gal(Lpn ,r /Hpn+r ), and hence Ppn ,r defines an element(4)

Ppn ,r ⊗ ζr ∈ H 0 (Hpn+r , D†N + ,r ).

In the next section we shall see how this system of points, for varying n and r, can be used
to construct various anticyclotomic p-adic L-functions.

10

F. CASTELLA, C.-H. KIM, AND M. LONGO

3. Anticyclotomic p-adic L-functions
3.1. Multiplicity one. Keep the notations introduced in Chapter 2. For each integer k > 2,
denote by Lk (R) the set of polynomials of degree less than or equal to k − 2 with coefficientsin a ring R, and define
eN + ,r , Lk (Zp )),JN + ,r,k := eord H0 (XeN + ,r associated with Lk (Zp ). The module JN + ,r,k iswhere Lk (Zp ) is the local system on X−

endowed with an action of the Hecke algebra TN
N,r,k and with perfect “intersection pairing”:
h , ik : JN + ,r,k × JN + ,r,k −→ Zp(5)(see [CH16, Eq. (3.9)]) with respect to which the Hecke operators are self-adjoint.
−

Theorem 3.1. Let m be a maximal ideal of TN
N,r,k whose residual representation is irreducible−

and satisfies Assumption (SU). Then (JN + ,r,k )m is free of rank one over (TNN,r,k )m . In partic−

ular, there is a (TNN,r,k )m -module isomorphism(JN + ,r,k )mαN,r,k−' (TNN,r,k )m .

Proof. If k = 2 and r = 1, this follows by combining [PW11, Thm. 6.2] and [loc.cit., Prop. 6.5].
The general case will be deduced from this case in Section 3.3 using Hida theory.
Let f ∈ Sk (Γ0,1 (N, pr )) be an N − -new eigenform, and suppose that m is the maximal ideal−of TN
N,r,k containing the kernel of the associated Zp -algebra homomorphism−

πf : (TNN,r,k )m −→ O,
where O is the the finite extension of Zp generated by the Fourier coefficients of f . Composing
πf with an isomorphism αN,r,k as in Theorem 3.1, we obtain an O-valued functionalψf : (JN + ,r,k )m −→ O.
By the duality (5), the map ψf corresponds to a generator gf of the πf -isotypical componentof JN + ,r,k , and following [PW11, §2.1] and [CH16, §4.1] we define the Gross period Ωf,N −attached to f by
(6)

Ωf,N − :=(f, f )Γ0 (N ).
hgf , gf ik

Remark 3.2. By Vatsal’s work [Vat03] (see also [PW11, Thm. 2.3] and [CH16, §5.4]), theanticyclotomic p-adic L-functions Lp (f /K) in Theorem 3.14 below (normalized by the complex
period Ωf,N − ) have vanishing µ-invariant. The preceding uniform description of ψf for all fwith a common maximal ideal m will allow us to show that this property is preserved in Hida
families.
3.2. One-variable p-adic L-functions. Denote by Γ the Galois group of the anticyclotomic
Zp -extension K∞ /K. For each n, let Kn ⊂ K∞ be defined by Gal(Kn /K) ' Z/pn Z and let
Γn be the subgroup of Γ such that Γ/Γn ' Gal(Kn /K).
Let Ppn+1 ,r ⊗ ζr ∈ H 0 (Hpn+1+r , D†N + ,r ) be the Heegner point of conductor pn+1 , and define(7)

Qn,r := CorHpn+1+r /Kn (Ppn+1 ,r ⊗ ζr ) ∈ H 0 (Kn , D†N + ,r );with a slight abuse of notation, we also denote by Qn,r its image under the natural map⊆

H 0 (Kn , D†N + ,r ) −→ DN + ,r −→ JN + ,rcomposed with localization at m, where JN + ,r := JN + ,r,2 .

ANTICYCLOTOMIC IWASAWA INVARIANTS IN HIDA FAMILIES

11

Definition 3.3. For any open subset σΓn of Γ, defineµr (σΓn ) := Up−n · Qσn,r ∈ (JN + ,r )m .
Proposition 3.4. The rule µr is a measure on Γ.
Proof. This follows immediately from the “horizontal compatibility” of Heegner points.



3.3. Gross periods in Hida families. Keep the notations of Section 3.1, and let(JN + )m := lim(JN + ,r )m
←−
r

−

−

which is naturally equipped with an action of the big Hecke algebra TN
limr TN
.
N =←
− N,r
−

Theorem 3.5. Let m be a maximal ideal of TN
N whose residual representation is irreducible−
and satisfies Assumption (SU). Then (JN + )m is free of rank one over (TNN )m . In particular,−
there is a (TNN )m -module isomorphismαN−(JN + )m ' (TNN )m .
Proof. As in [EPW06, Prop. 3.3.1]. Note that the version of Hida’s control theorem in ourcontext is provided by [Hid88, Thm. 9.4].

We can now conclude the proof of Theorem 3.1 just as in [EPW06, §3.3]. For the convenienceof the reader, we include here the argument.
−

Proof of Theorem 3.1. Let ℘N,r,k be the product of all the arithmetic primes of TN
N of weightk which become trivial upon restriction to 1 + pr Zp . By [Hid88, Thm. 9.4], we then have−

(JN + )m ⊗ TN
N /℘N,r,k ' (JN + ,r,k )mr,k(8)−

where mr,k is the maximal ideal of TN
N,r,k induced by m. Since (JN + )m is free of rank one over
−

−

−

N
N
TN
N by Theorem 3.5, it follows that (JN + ,r,k )mr,k is free of rank one over TN /℘N,r,k ' TN,r,k ,as was to be shown.
Remark 3.6. In the above proofs we made crucial use of [Hid88, Thm. 9.4], which is statedin the context of totally definite quaternion algebras which are unramified at all finite places,
since this is the only relevant case for the study of Hilbert modular forms over totally real number fields of even degree. However, the proofs immediately extend to the (simpler) situationof definite quaternion algebras over Q.
3.4. Two-variable p-adic L-functions. By the “vertical compatibility” satisfied by Heegner
points, the points
Up−r · Qn,r ∈ (JN + ,r )mare compatible for varying r, thus defining an element
Qn := lim Up−r · Qn,r ∈ (JN + )m .
←−r

Definition 3.7. For any open subset σΓn of Γ, defineµ(σΓn ) := Up−n · Qσn ∈ (JN + )m .
Proposition 3.8. The rule µ is a measure on Γ.
Proof. This follows immediately from the “horizontal compatibility” of Heegner points.

12

F. CASTELLA, C.-H. KIM, AND M. LONGO

Upon the choice of an isomorphism αN as in Theorem 3.5, we may regard µ as an element−ˆL(m, N ) ∈ (TNN )m ⊗Zp Zp [[Γ]].
Denoting by L(m, N )∗ the image of L(m, N ) under the involution induced by γ 7→ γ −1 ongroup-like elements, we set
L(m, N ) := L(m, N ) · L(m, N )∗ ,−to which we will refer as the two-variable p-adic L-function attached to (TNN )m .
3.5. Two-variable p-adic L-functions on branches of the Hida family. Let TΣ be theuniversal p-ordinary Hecke algebra−TΣ := (T0N (Σ) )m ' (TNN (Σ) )n(9)

associated with a mod p representation ρ̄ and a finite set of primes Σ as in Section 1.3.
Remark 3.9. Recall that N − | N (ρ̄) by Assumption (SU). Throughout the following, it will befurther assumed that every prime factor of N (Σ)/N − splits in K. In particular, every prime` ∈ Σ splits in K, and any f ∈ H− (ρ̄) = Spec(TΣ ) has tame level Nf with
Nf− = N (ρ̄)− = N − .
The construction of the preceding section produces a two-variable p-adic L-function−ˆL(n, N (Σ)) ∈ (TNN (Σ) )n ⊗Zp Zp [[Γ]]which combined with the isomorphism (9) yields an element
ˆ Zp Zp [[Γ]].
LΣ (ρ̄) ∈ TΣ ⊗If a is a minimal prime of TΣ , we thus obtain an element
ˆ Zp Zp [[Γ]]LΣ (ρ̄, a) ∈ T(a)◦ ⊗by reducing LΣ (ρ̄) mod a (see §1.4). On the other hand, if we let m denote the inverse imageof the maximal ideal of T(a)◦ under the composite surjection
(10)

−

new
new
0
◦
TN
N (a) −→ TN (a) −→ TN (a) /a = T(a) ,the construction of the preceding section yields an L-function−ˆL(m, N (a)) ∈ (TNN (a) )m ⊗Zp Zp [[Γ]]giving rise, via (10), to a second element
ˆ Zp Zp [[Γ]].
L(ρ̄, a) ∈ T(a)◦ ⊗It is natural to compare LΣ (ρ̄, a) and L(ρ̄, a), a task that is carried out in the next section,and provides the key for understanding the variation of analytic Iwasawa invariants.
3.6. Comparison. Write Σ = {`1 , . . . , `n } and for each ` = `i ∈ Σ, let e` be the valuation of
N (Σ)/N (a) at `, and define the reciprocal Euler factor E` (a, X) ∈ T(a)◦ [X] by
if e` = 0;10−1E` (a, X) := 1 − (T` mod a )Θ (`)X
if e` = 1;


0
−1
2
1 − (T` mod a )Θ (`)X + `X
if e` = 2.
ˆ Zp Zp [[Γ]] by
Also, writing ` = ll̄, define E` (a) ∈ T(a)◦ ⊗(11)E` (a) := E` (a, `−1 γl ) · E` (a, `−1 γl̄ ),where γl , γl̄ are arithmetic Frobenii at l, l̄ in Γ, respectively, and put EΣ (a) :=Q`∈Σ E` (a).

ANTICYCLOTOMIC IWASAWA INVARIANTS IN HIDA FAMILIES

13

Recall that N − | N (a) | N (Σ) and set
N (a)+ := N (a)/N − ,N (Σ)+ := N (Σ)/N − ,both of which consist entirely of prime factors which split in K. The purpose of this section
is to prove the following result.
Theorem 3.10. There is an isomorphism of T(a)◦ -modules
T(a)◦ ⊗(TN − ) (JN (Σ)+ )n ' T(a)◦ ⊗(TN − ) (JN (a)+ )m
N (Σ) nN (a) msuch that the map induced on the corresponding spaces of measures valued in these modules
sends LΣ (ρ̄, a) to EΣ (a) · L(ρ̄, a).
Proof. The proof follows closely the constructions and arguments in [EPW06, §3.8], suitably
adapted to the quaternionic setting at hand. Let r > 1. If M is any positive integer with(M, pN − ) = 1, and d0 | d are divisors of M , we have degeneracy maps
eM,r −→ X
eM/d,rBd,d0 : X1
00)
at every prime
val
(d
0` `

b × has local component
induced by (Ψ, g) 7→ (Ψ, πd0 g), where πd0 ∈ B
00` | d and 1 outside d . We thus obtain a map on homology

eM,r , Zp ) −→ eord H0 (XeM/d,r , Zp )(Bd,d0 )∗ : eord H0 (Xand we may defineeN (Σ)+ ,r , Zp ) −→ eord H0 (XeN (a)+ ,r , Zp )r : eord H0 (X(12)

by r := (`n ) ◦ · · · ◦ (`1 ), where for every ` = `i ∈ Σ we put1(`) := (B`,1 )∗ − (B`,` )∗ `−1 T`(B`2 ,1 )∗ − (B`2 ,` )∗ `−1 T` + (B`2 ,`2 )∗ `−1 h`iN (a)p

if e` = 0;
if e` = 1;
if e` = 2.

As before, let M be a positive integer with (M, pN − ) = 1 all of whose prime factors split inK, and let ` - M p be a prime which also splits in K. We shall adopt the following simplifying
eN + ,r constructed in Section 2.2:
notations for the system of points Pepn ,r ∈ X(M )eM,r ,
P := Pepn ,r ∈ X(M `)eM `,r ,P (`) := Pepn ,r ∈ X

2(M `2 )P (` ) := Pepn ,reM `2 ,r .
∈X

eM,r :
It is easy to check that we have the following relations in X
• (B`,1 )∗ (P (`) ) = P• (B`,` )∗ (P (`) ) = P σl2
• (B`2 ,1 )∗ (P (` ) ) = P
2• (B`2 ,` )∗ (P (` ) ) = P σl22
• (B`2 ,`2 )∗ (P (` ) ) = P σlwhere σl ∈ Gal(Lpn ,r /K) is a Frobenius element at a prime l | `. Letting P denote the image
2of eord P in DM,r , and defining P (`) ∈ DM `,r and P (` ) ∈ DM `2 ,r similarly, it follows that
• (B`,1 )∗ (P (`) ⊗ ζr ) = P ⊗ ζr
• (B`,` )∗ (P (`) ⊗ ζr ) = P σl ⊗ ζr = Θ−1 (σl ) · (P ⊗ ζr )σl2• (B`2 ,1 )∗ (P (` ) ⊗ ζr ) = P ⊗ ζr
2
• (B`2 ,` )∗ (P (` ) ⊗ ζr ) = P σl ⊗ ζr = Θ−1 (σl ) · (P ⊗ ζr )σl22• (B`2 ,`2 )∗ (P (` ) ⊗ ζr ) = P σl ⊗ ζr = Θ−2 (σl ) · (P ⊗ ζr )σl14

F. CASTELLA, C.-H. KIM, AND M. LONGO

as elements in D†M,r . Finally, setting Q := CorHpn+1+r /Kn (P) ∈ H 0 (Kn , D†M,r ), and defining
Q(`) ∈ H 0 (Kn , D†M `,r ) and Q(` ) ∈ H 0 (Kn , D†M `2 ,r ) similarly, we see that2

• (B`,1 )∗ (Q(`) ) = Q• (B`,` )∗ (Q(`) ) = Θ−1 (σl ) · Qσl2• (B`2 ,1 )∗ (Q(` ) ) = Q
2• (B`2 ,` )∗ (Q(` ) ) = Θ−1 (σl ) · Qσl22• (B`2 ,`2 )∗ (Q(` ) ) = Θ−2 (σl ) · Qσlin H 0 (Kn , D†M,r ). Each of these equalities is checked by an explicit calculation. For example,
for the second one:(B`,` )∗ (Q(`) ) = (B`,` )∗ CorHpn+1+r /Kn (P (`) ⊗ ζr )




X
Θ(σ̃ −1 ) · (P (`) )σ̃ ⊗ ζr 
= (B`,` )∗ σ∈Gal(Hpn+1+r /Kn )X=

Θ(σ̃ −1 ) · (B`,` )∗ ((P (`) )σ̃ ⊗ ζr )σ∈Gal(Hpn+1+r /Kn )X=

Θ(σ̃ −1 )Θ−1 (σl ) · (P σ̃ ⊗ ζr )σlσ∈Gal(Hpn+1+r /Kn )= Θ−1 (σl ) · Qσl .
Now let Qn,r ∈ JN (Σ)+ ,r be as in (7) with N = N (Σ). Using the above formulae, we easilynsee
P that of any σfinite order character χ of Γ of conductor p , the effect of r on the elementσ∈Γ/Γn χ(σ)Qn,r is given by multiplication by
Y
Y
−1
−2
(1 − (χΘ)−1 (σli )`−1
(1 − (χΘ)−1 (σli )`−1i T`i + (χΘ) (σli )`i h`i iN (a)p ).
i T`i )e`i =2e`i =1

P
Similarly, we see that r has the effect of multiplying the element σ∈Γ/Γn χ−1 (σ)Qσn,r by
Y
Y
−1
−1
−2
T
)
(1 − (χ−1 Θ)−1 (σli )`−1
(1 − (χ−1 Θ)−1 (σli )`−1`i
i
i T`i + (χ Θ) (σli )`i h`i iN (a)p ).
e`i =2e`i =1Hence, using the relations
χ(σl̄i ) = χ−1 (σli ),θ2 (`i ) = h`i iN (a)p ,Θ(σli ) = Θ(σl̄i ) = θ(`i ),it follows that the effect of r on the product of the above two elements is given by multiplication byYY−1
2(1 − χ(σli )θ−1 (`i )`−1
(1 − χ(σli )θ−1 (`i )`−1i T`i )i T`i + χ (σli )`i ).
li |`ie`i =1li |`ie`i =2

Taking the limit over r, we thus obtain a T(a)◦ -linear map(13)

T(a)◦ ⊗(TN − ) (JN (Σ)+ )n −→ T(a)◦ ⊗(TN − ) (JN (a)+ )m
N (Σ) nN (a) mhaving as effect on the corresponding measures as stated in Theorem 3.10. Hence to conclude
the proof it remains to show that (13) is an isomorphism.
By Theorem 3.5, both the source and the target of this map are free of rank one over T(a)◦ ,and as in [EPW06, p.559] (using [Hid88, Thm. 9.4]), one is reduced to showing the injectivity

ANTICYCLOTOMIC IWASAWA INVARIANTS IN HIDA FAMILIESof the dual map modulo p:eN (a)+ ,1 ; Fp )ord [m] −→ (TN − /m) ⊗ N −
H 0 (XN (a)TN (Σ) /n

15eN (a)+ ,1 ; Fp )ord [m0 ])(H 0 (X

−
0 e
ord 0
[m ])
−→ (TNN (a) /m) ⊗TN − /n (H (XN (Σ)+ ,1 ; Fp )

(14)

N (Σ)

−
0 e
ord
[n]);
−→ (TNN (a) /m) ⊗TN − /n (H (XN (Σ)+ ,1 ; Fp )N (Σ)

or equivalently (by a version of [EPW06, Lemma 3.8.1]), to showing that the composite of
the first two arrows in (14) is injective.
In turn, the latter injectivity follows from Lemma 3.11 below, where the notations are as
follows:• M + is any positive integer with (M + , pN − ) = 1;
• ` 6= p is a prime;• n` = 1 or 2 according to whether or not ` divides M + ;• N + := `n` M + ,
and
0 e
ord 0
eM + ,1 ; Fp )ord [m] −→ (TN −+ − /m) ⊗T0(15)∗ : H 0 (X[m ])/m0 (H (XN + ,1 ; Fp )
`

M N

N+N−

is the map defined by∗` :=

∗ − B ∗ `−1 T
if n` = 1;
B`,1
`
`,`
∗
∗
−1
∗
−1
B`2 ,1 − B`2 ,` ` T` + B`2 ,`2 ` h`iN (a)p if n` = 2.

Lemma 3.11. The map (15) is injective.
Proof. As in the proof of the analogous result [EPW06, Lemma 3.8.2] in the modular curve
case, it suffices to show the injectivity of the mapβ`
eM + ,1 ; F)ord [mF ])n` +1 −−→
eN + ,1 ; F)ord [m0 ](H 0 (X
H 0 (XF

defined byβ` :=∗ π + B∗ π
B`,1
if n` = 1;1`,` 2
B`∗2 ,1 π1 + B`∗2 ,` π2 + B`∗2 ,`2 π3 if n` = 2.

But in our quaternionic setting the proof of this injectivity follows from [SW99, Lemma 3.26]
for n` = 1 and [loc.cit., Lemma 3.28] for n` = 2.
Applying inductively Lemma 3.11 to the primes in Σ, the proof of Theorem 3.10 follows. 
3.7. Analytic Iwasawa invariants. Upon the choice of an isomorphism
Zp [[Γ]] ' Zp [[T ]]we may regard the p-adic L-functions LΣ (ρ̄, a) and L(ρ̄, a), as well as the Euler factor EΣ (ρ̄, a),as elements in T(a)◦ [[T ]]. In this section we apply the main result of the preceding section to
study the variation of the Iwasawa invariants attached to the anticyclotomic p-adic L-functionsof p-ordinary modular forms.
For any power series f (T ) ∈ R[[T ]] with coefficients in a ring R, the content of f (T ) isdefined to be the ideal I(f (T )) ⊆ R generated by the coefficients of f (T ). If ℘ is a height one
prime of TΣ belonging to the branch T(a) (in the sense that a is the unique minimal prime ofTΣ contained in ℘), we denote by L(ρ̄, a)(℘) the element of O℘ [[Γ]] obtained from L(ρ̄, a) via
reduction modulo ℘. In particular, we note that L(ρ̄, a)(℘) has unit content if and only if itsµ-invariant vanishes.
Theorem 3.12. The following are equivalent:(1) µ(L(ρ̄, a)(℘)) = 0 for some newform f℘ in H− (ρ̄).

16

F. CASTELLA, C.-H. KIM, AND M. LONGO(2) µ(L(ρ̄, a)(℘)) = 0 for every newform f℘ in H− (ρ̄).
(3) L(ρ̄, a) has unit content for some irreducible component T(a) of H− (ρ̄).
(4) L(ρ̄, a) has unit content for every irreducible component T(a) of H− (ρ̄).
Proof. The argument in [EPW06, Thm 3.7.5] applies verbatim, replacing the appeal to [loc.cit.,
Cor. 3.6.3] by our Theorem 3.10 above.
When any of the conditions in Theorem 3.12 hold, we shall writeµan (ρ̄) = 0.
For a power series f (T ) with unit content and coefficients in a local ring R, the λ-invariantλ(f (T )) is defined to be the smallest degree in which f (T ) has a unit coefficient.
Theorem 3.13. Assume that µan (ρ̄) = 0.
(1) Let T(a) be an irreducible component of H− (ρ̄). As ℘ varies over the arithmetic primesof T(a), the λ-invariant λ(L(ρ̄, a)(℘)) takes on a constant value, denoted λan (ρ̄, a).
(2) For any two irreducible components T(a1 ), T(a2 ) of H− (ρ̄), we have thatX
λan (ρ̄, a1 ) − λan (ρ̄, a2 ) =e` (a2 ) − e` (a1 ),`6=pwhere e` (a) = λ(E` (a)).
Proof. The first part follows immediately from the definitions. For the second part, the argument in [EPW06, Thm. 3.7.7] applies verbatim, replacing their appeal to [loc.cit., Cor. 3.6.3]
by our Theorem 3.10 above.

By Theorem 3.12 and Theorem 3.13, the Iwasawa invariants of L(ρ̄, a)(℘) are well-behavedas ℘ varies. However, for the applications of these results to the Iwasawa main conjecture it
is of course necessary to relate L(ρ̄, a)(℘) to p-adic L-functions defined by the interpolationof special values of L-functions. This question was addressed in [CL16], as we now recall.
Theorem 3.14. If ℘ is the arithmetic prime of T(a) corresponding to a p-ordinary p-stabilized
newform f℘ of weight k > 2 and trivial nebentypus, thenL(ρ̄, a)(℘) = Lp (f℘ /K),
where Lp (f℘ /K) is the p-adic L-function of Chida–Hsieh [CH16]. In particular, if χ : Γ → C×p
is the p-adic avatar of an anticyclotomic Hecke character of K of infinity type (m, −m) with−k/2 < m < k/2, then L(ρ̄, a)(℘) interpolates the central critical values
L(f℘ /K, χ, k/2)
Ωf℘ ,N −as χ varies, where Ωf℘ ,N − is the complex Gross period (6).
Proof. This is a reformulation of the main result of [CL16]. (Note that the constant λ℘ ∈ F℘×in [CL16], Thm. 4.6] is not needed here, since the specialization map of [loc.cit., §3.1] is being
replaced by the map (JN + )m → (JN + ,r,k )mr,k induced by the isomorphism (8), which preserves
integrality.)
Corollary 3.15. Let f1 , f2 ∈ H− (ρ̄) be newforms with trivial nebentypus lying in the branches
T(a1 ), T(a2 ), respectively. Then µan (ρ̄) = 0 andX
λ(Lp (f1 /K)) − λ(Lp (f2 /K)) =e` (a2 ) − e` (a1 ),`6=pwhere e` (aj ) = λ(E` (aj )).

ANTICYCLOTOMIC IWASAWA INVARIANTS IN HIDA FAMILIES

17

Proof. By [CH16, Thm. 5.7] (extending Vatsal’s result [Vat03] to higher weights), if f ∈ H− (ρ̄)
has weight k 6 p + 1 and trivial nebentypus, then µ(Lp (f /K)) = 0. By Theorem 3.12 andTheorem 3.14, this implies µan (ρ̄) = 0. The result thus follows from Theorem 3.13, using
again Theorem 3.14 to replace λan (ρ̄, aj ) by λ(Lp (fj /K)).

4. Anticyclotomic Selmer groups
We continue with the notation of the previous sections. In particular, ρ̄ : GQ → GL2 (F)is an odd irreducible Galois representation satisfying Assumption (SU) and isomorphic to ρ̄f0
for some newform f0 of weight 2, H− (ρ̄) is the associated Hida family, and Σ is a finite set ofprimes split in the imaginary quadratic field K.
For each f ∈ H− (ρ̄), let Vf denote the self-dual Tate twist of the p-adic Galois representation
associated to f , fix an O-stable lattice Tf ⊆ Vf , and set Af := Vf /Tf . Since f is p-ordinary,
there is a unique one-dimensional GQp -invariant subspace Fp+ Vf ⊆ Vf where the inertia groupk/2

at p acts via εcyc ψ, with ψ a finite order character. Let Fp+ Af be the image of Fp+ Vf in Af ,and as recalled in the Introduction define the minimal Selmer group of f by




Y
Y
Sel(K∞ , f ) := ker H 1 (K∞ , Af ) −→H 1 (K∞,w , Af ) ×
H 1 (K∞,w , Fp− Af ) ,


w-p

w|p

where w runs over the places of K∞ and we set Fp− Af := Af /Fp+ Af .
It is well-known that Sel(K∞ , f ) is cofinitely generated over Λ. When it is also Λ-cotorsion,
we define the µ-invariant µ(Sel(K∞ , f )) (resp. λ-invariant λ(Sel(K∞ , f ))) to the largest powerof $ dividing (resp. the number of zeros of) the characteristic power series of the Pontryagin
dual of Sel(K∞ , f ).
A distinguishing feature of the anticyclotomic setting (in comparison with cyclotomic Iwasawa theory) is the presence of primes which split infinitely in the corresponding Zp -extension.
Indeed, being inert in K, all primes ` | N − are infinitely split in K∞ /K. As a result, theabove Selmer group differs in general from the Greenberg Selmer group of f , defined as




Y
Y
1
1
1
−
Sel(K∞ , f ) := ker H (K∞ , Af ) −→H (I∞,w , Af ) ×
H (K∞,w , Fp Af ) ,


w-p

w|p

where I∞,w ⊆ GK∞ denotes the inertia group at w.
If S is a finite set of primes in K, we let SelS (K∞ , f ) and SelS (K∞ , f ) be the “S-primitive”
Selmer groups defined as above by omitting the local conditions at the primes in S (except
those above p, when any such prime is in S). Moreover, if S consists of the primes dividing arational integer M , we replace the superscript S by M in the above notation.
Immediately from the definitions, we see that there is as exact sequenceY(16)
0 −→ Sel(K∞ , f ) −→ Sel(K∞ , f ) −→H`un ,`|N −where
H`un := ker


Y


w|`

H 1 (K∞,w , Af ) −→

Y
w|`



H 1 (I∞,w , Af )
is the set of unramified cocycles. In [PW11, §§3, 5], Pollack and Weston carried out a carefulanalysis of the difference between Sel(K∞ , f ) and Sel(K∞ , f ). Even though loc.cit. is mostly

18

F. CASTELLA, C.-H. KIM, AND M. LONGO

concerned with cases in which f is of weight 2, many of their arguments apply more generally.
In fact, the next result follows essentially from their work.
Theorem 4.1. Assume that ρ̄ satisfies Hypotheses (SU). Then the following are equivalent:(1) Sel(K∞ , f0 ) is Λ-cotorsion with µ-invariant zero for some newform f0 ∈ H− (ρ̄).
(2) Sel(K∞ , f ) is Λ-cotorsion with µ-invariant zero for all newforms f ∈ H− (ρ̄).
(3) Sel(K∞ , f ) is Λ-cotorsion with µ-invariant zero for all newforms f ∈ H− (ρ̄).
Moreover, in that case Sel(K∞ , f ) ' Sel(K∞ , f ).
Proof. Assume f0 is a newform in H− (ρ̄) for which Sel(K∞ , f0 ) is Λ-cotorsion with µ-invariant
zero, and set N + := N (Σ)/N − . By [PW11, Prop. 5.1], we then have the exact sequencesY+(17)
0 −→ Sel(K∞ , f0 ) −→ SelN (K∞ , f0 ) −→H` −→ 0;
`|N +

(18)

0 −→ Sel(K∞ , f0 ) −→ Sel

N+(K∞ , f0 ) −→Y

H` −→ 0,`|N +where H` is the product of H 1 (K∞,w , Af0 ) over the places w | ` in K∞ . Since every prime` | N + splits in K (see Remark 3.9), the Λ-cotorsionness and the vanishing of the µ-invariantof H` can be deduced from [GV00, Prop. 2.4]. Since Sel(K∞ , f0 )[$] is finite by assumption,+it thus follows from (17) that SelN (K∞ , f0 )[$] is finite. Combined with (16) and [PW11,+
Cor. 5.2], the same argument using (18) shows that then SelN (K∞ , f0 )[$] is also finite.
On the other hand, following the arguments in the proof [PW11, Prop. 3.6] we see that for
any f ∈ H(ρ̄) we have the isomorphisms
+

+

+

+

SelN (K∞ , ρ̄) ' SelN (K∞ , f )[$];
SelN (K∞ , ρ̄) ' SelN (K∞ , f )[$].
As a result, the argument in the previous paragraph implies that, for any newform f ∈ H− (ρ̄),++
both SelN (K∞ , f )[$] and SelN (K∞ , f )[$] are finite , from where (using (17) and (18) withf in place of f0 ) the Λ-cotorsionness and the vanishing of both the µ-invariant of Sel(K∞ , f )and of Sel(K∞ , f ) follows. In view of (16) and [PW11, Lemma 3.4], the result follows.

Let w be a prime of K∞ above ` 6= p and denote by Gw ⊆ GK∞ its decomposition group.
Let T(a) be the irreducible component of TΣ passing through f , and definewδw (a) := dimF AG
f /$.

(Note that this is well-defined by [EPW06, Lemma 4.3.1].) Assume ` = ll̄ splits in K and put
X(19)δ` (a) :=δw (a),
w|`

where the sum is over the (finitely many) primes w of K∞ above `.
In view of Theorem 4.1, we write µalg (ρ̄) = 0 whenever any of the µ-invariants appearing inthat result vanish. In that case, for any newform f in H− (ρ̄) we may consider the λ-invariants
λ(Sel(K∞ , f )) = λ(Sel(K∞ , f )).
Theorem 4.2. Let ρ̄ and Σ be as above, and assume that µalg (ρ̄) = 0. If f1 and f2 are any
two newforms in H− (ρ̄) lying in the branches T(a1 ) and T(a2 ), respectively, thenX
λ(Sel(K∞ , f1 )) − λ(Sel(K∞ , f2 )) =δ` (a1 ) − δ` (a2 ).
`6=p

ANTICYCLOTOMIC IWASAWA INVARIANTS IN HIDA FAMILIES

19

Proof. Since we have the divisibilities N − | N (ai ) | N (Σ) with the quotient N (Σ)/N − only
divisible by primes that are split in K, the arguments of [EPW06, §4] apply verbatim (cf.
[PW11, Thm. 7.1]).

5. Applications to the main conjecture
5.1. Variation of anticyclotomic Iwasawa invariants. Recall the definition of the analytic invariant e` (a) = λ(E` (a)), where E` (a) is the Euler factor from Section 3.6, and of thealgebraic invariant δ` (a) introduced in (19).
Lemma 5.1. Let a1 , a2 be minimal primes of TΣ . For any prime ` 6= p split in K, we haveδ` (a1 ) − δ` (a2 ) = e` (a2 ) − e` (a1 ).
Proof. Let a be a minimal prime of TΣ , let f be a newform in the branch T(a), and let ℘f ⊆ abe the corresponding height one prime. Since ` = ll̄ splits in K, we have
!
!
M
M
M
1
1
1
H (K∞,w , Af ) =H (K∞,w , Af ) ⊕H (K∞,w , Af )
w|`

w|l

w|l̄

and [GV00, Prop. 2.4] immediately implies that
!
M
1
∨
ChΛ
= E` (f, `−1 γl ) · E` (f, `−1 γl̄ ),H (K∞,w , Af )w|`

where E` (f, `−1 γl ) · E` (f, `−1 γl̄ ) is the specialization of E` (a) at ℘f . The result thus follows
from [EPW06, Lemma 5.1.5].
Theorem 5.2. Suppose that ρ̄ satisfies Assumption (SU). If for some newform f0 ∈ H− (ρ̄)
we have the equalitiesµ(Sel(K∞ , f0 )) = µ(Lp (f0 /K)) = 0andλ(Sel(K∞ , f0 )) = λ(Lp (f0 /K)),andλ(Sel(K∞ , f )) = λ(Lp (f /K))

then the equalitiesµ(Sel(K∞ , f )) = µ(Lp (f /K)) = 0
hold for all newforms f ∈ H− (ρ̄).
Proof. Let f be any newform in H− (ρ̄). Since the algebraic and analytic µ-invariants of
f0 both vanish, the vanishing of µ(Sel(K∞ , f )) and µ(Lp (f /K)) follows from Theorems 4.1
and 3.12, respectively. On the other hand, combining Theorems 3.13 and 4.2, and Lemma 5.1,
we see that
λ(Sel(K∞ , f )) − λ(Sel(K∞ , f0 )) = λ(Lp (f /K)) − λ(Lp (f0 /K)),and hence the equality λ(Sel(K∞ , f0 )) = λ(Lp (f0 /K)) implies the same equality for f .



5.2. Applications to the main conjecture. As an immediate consequence of Weierstrass
preparation theorem, Theorem 5.2 together with one the divisibilities predicted by the anticyclotomic main conjecture implies the full anticyclotomic main conjecture.
Theorem 5.3 (Skinner–Urban). Let f ∈ Sk (Γ0 (N )) be a newform of weight k ≡ 2 (mod p−1)
and trivial nebentypus. Suppose that ρ̄f satisfies Assumption (SU) and that p splits in K.
Then(Lp (f /K)) ⊇ ChΛ (Sel(K∞ , f )∨ ).

20

F. CASTELLA, C.-H. KIM, AND M. LONGOProof. This follows fromP specializing the divisibility in [SU14, Thm. 3.26] to the anticyclotomic
line. Indeed, let f = n>1 an (f )q n ∈ I[[q]] be the Λ-adic form with coefficients in I := T(a)◦associated with the branch of the Hida family containing f , let Σ be a finite set of primes as
in Section 3.5, let Σ0 ⊇ Σ be a finite set of primes of K containing Σ and all primes dividingpN (a)DK , and assume that Σ0 contains at least one prime ` 6= p that splits in K. Under theseassumptions, in [SU14, Thm. 3.26] it is shown that
0

0

Σ
∨
(LΣ
p (f /K)) ⊇ ChΛf (L∞ ) (Sel (L∞ , Af ) ),(20)

where L∞ = K∞ Kcyc is the Z2p -extension of K, Λf (L∞ ) is the three-variable Iwasawa algebra0Σ00I[[Gal(L∞ /K)]], and LΣ
p (f /K) and Sel (L∞ , Af ) are the “Σ -primitive” p-adic L-functionand Selmer group defined in [SU14, §3.4.5] and [SU14, §§3.1.3, 3.1.10], respectively.
Recall the character Θ : GQ → Zp [[1 + pZp ]]× from Section 2.3, regarded as a character on
Gal(L∞ /K), and letTwΘ−1 : Λf (L∞ ) −→ Λf (L∞ )be the I-linear isomorphism induced by TwΘ−1 (g) = Θ−1 (g)g for g ∈ Gal(L∞ /K). Choose a
topological generator γ ∈ Gal(Kcyc /K), and expand
0

0

0

Σ
Σ
TwΘ−1 (LΣp (f /K)) = Lp,0 (f /K) + Lp,1 (f /K)(γ − 1) + · · ·00Σwith LΣp,i (f /K) ∈ Λf (K∞ ) = I[[Γ]]. In particular, note that Lp,0 (f /K) is the restriction of the0twisted three-variable p-adic L-function TwΘ−1 (LΣp (f /K)) to the “self-dual” plane.
Because of our assumptions on f , the Λ-adic form f has trivial tame character, and hence
denoting by Frob` an arithmetic Frobenius at any prime ` - N (a)p, the Galois representation

ρ(a) : GQ −→ GL(Tf ) ' GL2 (T(a)◦ )considered in §1.4 (which is easily seen to agree with the twisted representation considered in[SU14, p.37]) is such thatdet(X − Frob` |Tf ) = X 2 − a` (f )X + Θ2 (`)`.
The twist Tf† := Tf ⊗ Θ−1 is therefore self-dual. Thus combining [Rub00, Lemma 6.1.2] witha straightforward variant of [SU14, Prop. 3.9] having Gal(K∞ /K) in place of Gal(Kcyc /K),
we see that divisibility (20) implies that
(21)

0

0

† ∨
Σ
(LΣ
p,0 (f /K)) ⊇ ChΛf (K∞ ) (Sel (K∞ , Af ) ).

(Here, as above, Af denotes the Pontryagin dual Tf ⊗I Homcts (I, Qp /Zp ), and A†f is the corresponding twist.) We next claim that, setting Σ00 := Σ0 r Σ, we have
Y
0
(22)
(LΣ
(f
/K))
=
(L
(ρ̄,
a)
·
Ev (a)),
Σ
p,0
v∈Σ00
v-p

where LΣ (ρ̄, a) is the two-variable p-adic L-function constructed in Section 3.4, and if v lies
over the rational prime `, Ev (a) is the Euler factor given by
Ev (a) = det(Id − Frobv X|(Vf† )Iv )X=`−1 Frobv ,where Vf := Tf ⊗I Frac(I), and Frobv is an arithmetic Frobenius at v. (Note that for ` = ll̄split in K, El (a) · El̄ (a) is simply the Euler factor (11).) Indeed, combined with Theorem 3.10
and Theorem 3.14, equality (22) specialized to any arithmetic prime ℘ ⊆ T(a) of weight 2 isshown in [SU14, (12.3)], from where the claim follows easily from the density of these primes.
(See also [PW11, Thm. 6.8] for the comparison between the different periods involved in thetwo constructions, which differ by a p-adic unit under our assumptions.)

ANTICYCLOTOMIC IWASAWA INVARIANTS IN HIDA FAMILIES

21

Finally, (21) and (22) combined with Theorem 3.10 and [GV00, Props. 2.3,8] imply that
(L(ρ̄, a)) ⊇ ChΛf (K∞ ) (Sel(K∞ , A†f )∨ ),from where the result follows by specializing at ℘f using Theorem 3.14 and Theorem 4.1. In the opposite direction, we have the following result:PnTheorem 5.4 (Bertolini–Darmon). Let f = ∞n=1 an (f )q be a p-ordinary newform of weight2, level N , and trivial nebentypus. Suppose that ρ̄f satisfies Assumption (CR) and that(PO)

ap (f ) 6≡ ±1(mod p).

Then(Lp (f /K)) ⊆ ChΛ (Sel(K∞ , f )∨ ).
Proof. This is the main result of [BD05], as extended by Pollack–Weston [PW11] to newforms
of weight 2 not necessarily defined over Q and under the stated hypotheses (weaker that in[BD05]) on ρ̄f . See also [KPW17] for a detailed discussion on the additional “non-anomalous”
hypothesis (PO) on f .
Before we combine the previous two theorems with our main results in this paper, we notethat condition (PO) in Theorem 5.4
in terms of the Galois representation ρf
P can be phrased
n be a p-ordinary newform as above, defineda(f)q
associated to f . Indeed, let f = ∞n=1 n
over a finite extension F/Qp with ring of integers O. Then we haveρf |Dp ' ( 0ε ∗δ )on a decomposition group Dp ⊆ GQ at p, with δ : Dp → O× an unramified character sending
Frobp to the unit root αp of X 2 − ap (f )X + p. Since clearly α ≡ ap (f ) (mod p), we see thatcondition (PO) amounts to the requirement that(PO)

δ(Frobp ) 6≡ ±1(mod p).

Now we are finally in a position to prove our main application to the anticyclotomic Iwasawa
main conjecture for p-ordinary newforms.
Corollary 5.5. Suppose that ρ̄ satisfies Assumptions (SU) and (PO) and that p splits in K,
and let f be a newform in H− (ρ̄) of weight k ≡ 2 (mod p − 1) and trivial nebentypus. Thenthe anticyclotomic Iwasawa main conjecture holds for f .
Proof. After Theorem 5.2 and Theorem 5.3, to check the anticyclotomic main conjecture forany newform f as in the statement, it suffices to check the three equalities(23)µ(Sel(K∞ , f0 )) = µ(Lp (f0 /K)) = 0andλ(Sel(K∞ , f0 )) = λ(Lp (f0 /K)).

holds some f0 ∈ H− (ρ̄) of weight k ≡ 2 (mod p − 1) and trivial nebentypus.
Let T(a) be the irreducible component of H− (ρ̄) containing f , and let f0 ∈ S2 (Γ0 (N p))
be the p-stabilized newform corresponding to an arithmetic prime ℘ ⊆ T(a) of weight 2 andtrivial nebentypus. By Assumption (PO), the form f0 is necessarily the p-stabilization of ap-ordinary newform f0] ∈ S2 (Γ0 (N )) (see e.g. [How07, Lemma 2.1.5]). From the combinationof Theorem 5.3 and Theorem 5.4, the anticyclotomic Iwasawa main conjecture holds for f0] ,and since we clearly have
Lp (f0 /K) = Lp (f0] /K)

andSel(K∞ , f0 ) ' Sel(K∞ , f0] )(note that the latter isomorphism relies on the absolute irreducibility of ρ̄), the anticyclotomic
Iwasawa main conjecture holds for f0 as well. In particular, equalities (23) hold for this f0 ,and the result follows.
22

F. CASTELLA, C.-H. KIM, AND M. LONGO

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MR 1953292 (2004e:11069b)
Department of Mathematics, Princeton University, Fine Hall, Princeton, NJ 08544, USA
E-mail address: fcabello@math.princeton.edu
School of Mathematics, Korea Institute for Advanced Study (KIAS), 85 Hoegiro, DongdaemunGu, Seoul 02455, Republic of Korea
E-mail address: chanho.math@gmail.com
Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121 Padova, Italy
E-mail address: mlongo@math.unipd.it