VARIATION OF ANTICYCLOTOMIC IWASAWA INVARIANTSIN HIDA FAMILIES

FRANCESC CASTELLA, CHAN-HO KIM, AND MATTEO LONGO

Abstract. In this paper, using the construction of big Heegner points [LV11] in the definitequaternionic setting and their relation to special values of L-functions [CL14], we obtainanticyclotomic analogs of the results of Emerton–Pollack–Weston [EPW06] on the variationof Iwasawa invariants in Hida families.

Contents

Introduction 11. Hida theory 32. Big Heegner points 63. Anticyclotomic p-adic L-functions 84. Anticyclotomic Selmer groups 155. Applications to the main conjecture 17References 19

Introduction

The purpose of this paper is the study of anticyclotomic analogs of the results of [EPW06] onthe variation of Iwasawa invariants in Hida families. Let ⇢ : GQ := Gal(Q/Q)! GL2(F) bean odd and absolutely irreducible Galois representation over a finite field F of characteristic p.After the celebrated proof of Serre’s conjecture [KW09], we know that ⇢ is modular. Let H(⇢)denote the set of all p-ordinary and p-stabilized newforms with mod p Galois representationisomorphic to ⇢.

Let K be an imaginary quadratic field of discriminant prime to p. Let N� be a square-freeproduct of an odd number of primes, each inert in K, containing all such primes at which ⇢is ramified. As in [PW11], we say that (⇢, N�) satisfies condition (CR) if the following hold:

Assumption (CR). (1) ⇢ is irreducible, and surjective if F = F5.(2) If `|N� and ` ⌘ ±1 (mod p), then ⇢ is ramified at `.

Let � be the Galois group of the anticyclotomic Zp-extension K1/K. Associated with eachf 2 H(⇢) there is a p-adic L-function Lp(f/K) 2 ⇤ := O[[�]], where O is the ring of integers ofa finite extension F of Qp over which f is defined, characterised by an interpolation propertyof the form

�(Lp(f/K)) = Cp(f,�) · Ep(f,�) ·L(f,�, k/2)

⌦f,N�

as � runs over the p-adic characters of � corresponding to certain algebraic Hecke charactersof K, where Cp(f,�) is an explicit nonzero constant, Ep(f,�) is a p-adic multiplier, and ⌦f,N�

is a complex period (specified up to a p-adic unit) making the above ratio algebraic.

Date: April 30, 2015.

1

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

The anticyclotomic Iwasawa main conjecture gives an arithmetic interpretation of Lp(f/K).More precisely, let

⇢f : GQ �! AutF (Vf ) ' GL2(F )

be a self-dual twist of the p-adic Galois representation associated to f , fix an O-stable latticeTf ⇢ Vf , and set Af = Vf/Tf . Since f is p-ordinary, there is a unique one-dimensional GQp-

invariant subspace F+p Vf ⇢ Vf where the inertia group at p acts via "k/2cyc , where "cyc is the

p-adic cyclotomic character and is of finite order. Let F+p Af be the image of F+

p Vf in Af

and set F�p Af = Af/F

+p Af . Define the minimal Selmer group of f by

Sel(K1, f) := ker

8<

:H1(K1, Af ) �!Y

w-pH1(K1,w, Af )⇥

Y

w|p

H1(K1,w, F�p Af )

9=

;

where w runs over the places of K1. By standard arguments (see [Gre89], for example), oneknows that the Pontryagin dual of Sel(K1, f) is finitely generated over ⇤. The anticyclotomicmain conjecture is then the following.

Conjecture 1. Sel(K1, f)_ is ⇤-torsion, and

Ch⇤(Sel(K1, f)_) = (Lp(f/K)).

For f corresponding to p-ordinary elliptic curves, and under rather stringent assumptionson ⇢f which were later relaxed by Pollack–Weston [PW11], one of the divisibilities predicted byConjecture 1 was obtained by Bertolini–Darmon [BD05] using Heegner points and Kolyvagin’smethod of Euler systems. More recently, after the work of Chida–Hsieh [CH15] the divisibility

Ch⇤(Sel(K1, f)_) ◆ (Lp(f/K))

is known for all newforms f 2 H(⇢) of weight k p� 2 and trivial nebentypus, provided thepair (⇢, N�

f ) satisfies a mild strengthening of condition (CR). Here, N�f denotes as usual the

product of the prime factors of Nf which are inert in K.The restriction to weights k p� 2 in [CH15] comes from the use of the version of Ihara’s

lemma proved in [DT94]. While it seems di�cult to directly extend their arguments to higherweights, it might be possible to obtain the above divisibility for all weights by adapting thestrategy of Bertolini–Darmon [BD05] to the setting of Heegner points in Hida families [LV11].In fact, the results of this paper complete the proof of many new cases of Conjecture 1 usingbig Heegner points, but by a rather di↵erent approach, as we now explain.

Associated with every f 2 H(⇢) there are anticyclotomic Iwasawa invariants µan(K1, f),�an(K1, f), µalg(K1, f), and �alg(K1, f). The analytic (resp. algebraic) �-invariants are thenumber of zeros of Lp(f/K) (resp. of a generator of the characteristic ideal of Sel(K1, f)_),while the µ-invariants are defined as the exponent of the highest power of p dividing the sameobjects. In this paper we study the behavior of these invariants as f varies over the subsetH(⇢) of H(⇢) consisting of newforms with N�

f = N�. Our main result is then the following.

Theorem 2. Suppose that ⇢ is p-ordinary, p-distinguished, and ramified at all `|N�, and fix⇤ 2 {alg, an}.

(1) For all f 2 H(⇢), we haveµ⇤(K1, f) = 0.

(2) Let f1, f2 2 H(⇢) lie on the branches T(a1), T(a2), respectively. Then

�⇤(K1, f1)� �⇤(K1, f2) =X

`|N+1 N+

2

e`(a2)� e`(a1)

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 `.

VARIATION OF ANTICYCLOTOMIC IWASAWA INVARIANTS 3

Provided p splits in K, and under the same assumptions on ⇢ as in Theorem 2, the deepwork of Skinner–Urban [SU14] establishes one of the divisibilities in a related “three-variable”Iwasawa main conjecture. Combining their work with the main result of this paper, we deducethe following.

Corollary 3. Let ⇢ be as in Theorem 2 and suppose that p splits in K. If the anticyclotomicmain conjecture holds for some newform f0 2 H(⇢) of weight k0 ⌘ 2 (mod p� 1) and trivialnebentypus, then it holds for all newforms f 2 H(⇢) of weight k ⌘ 2 (mod p� 1) and trivialnebentypus.

As hinted at above, the proof of our main results closely follows the methods of [EPW06].In fact, on the algebraic side the arguments of loc.cit. apply in our context almost verbatim,and the main contribution of this paper is the development of anticyclotomic analogs of theirresults on the analytic side. Indeed, the proof of the analytic parts of [EPW06] is based onthe study of certain variants of the two-variable p-adic L-functions of Mazur–Kitagawa, whoseconstruction relies on the theory of modular symbols on classical modular curves. In contrast,by our assumptions on N� we are led to work on a family of Shimura curves associated with a(definite) quaternion algebra over Q of discriminant N� > 1, and these curves are well-knownto have no cusps.

In the cyclotomic case, modular symbols are useful two ways: They yield a concrete realiza-tion of the degree one compactly supported cohomology of open modular curves, and providea powerful tool for studying the arithmetic properties of critical values of Hecke L-functions.Our basic observation is that in the present anticyclotomic setting, Heegner points on definiteShimura curves provide a similarly convenient way of describing the central critical values ofthe Rankin L-series L(f/K,�, s).

Also fundamental for the method of [EPW06] is the possibility to “deform” modular symbolsin Hida families. In our anticyclotomic context, the construction of big Heegner points in Hidafamilies was obtained in the work [LV11] of the third named author in collaboration with Vigni,following an original construction due to Howard [How07], while the relation between thesepoints and classical L-values was established in the work [CL14] by the first and third namedauthors. With these key results at hand, and working over appropriate quotients of the Heckealgebras considered in [EPW06] via the Jaquet–Langlands correspondence, we are then ableto adapt the arguments of loc.cit. to our setting, making use of the ramification hypotheseson ⇢ to ensure a multiplicity one property of certain Hecke modules (among other uses).

We conclude this introduction with the following overview of the contents of this paper. Inthe next section, we briefly recall the Hida theory used in this paper, following the expositionin [EPW06, §1] for the most part. In Section 2, we describe an extension of the constructionof big Heegner points to “imprimitive” branches of the Hida family, an extension necessaryfor the purposes of this paper. In Section 3, we construct two-variable p-adic L-functionsattached to a Hida family and to each of its irreducible components (or branches), and proveTheorem 3.9 relating the two. This theorem is the key technical result of this paper, and theanalytic part of Theorem 2 follows easily from this. In Section 4, we deduce the algebraic partof Theorem 2 using the residual Selmer groups studied in [PW11, §3.2]. Finally, in Section 5we give the applications of our results to the anticyclotomic main conjecture.

1. Hida theory

1.1. Hecke algebras. Fix a positive integer N admitting a factorization N = N+N� with(N+, N�) = 1 and N� square-free, and fix a prime p - N .

For each integer k � 2, denote by hN,r,k the Zp-algebra generated by the Hecke operators T`

for ` - Np, the operators U` for `|Np, and the diamond operators hai for a 2 (Z/prZ)⇥, actingon the space Sk(�0,1(N, pr),Qp) of cusp forms of weight k on �0,1(N, pr) := �0(N) \ �1(pr).For k = 2, we abbreviate hN,r := hN,r,2.

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

Let eord := limn!1 Un!p be Hida’s ordinary projector, and define

hordN,r,k := eordhN,r,k hordN,r := eordhN,r hordN := lim �r

hordN,r

where the limit is over the projections induced by the natural restriction maps.Let TN�

N,r,k be the quotient of hordN,r,k acting faithfully on the subspace of eordSk(�0,1(N, pr),Qp)

consisting of forms which are new at all primes dividing N�. Set TN�N,r := TN�

N,r,2 and define

TN�N := lim �

r

TN�N,r .

Each of these Hecke algebras are equipped with natural Zp[[Z⇥p ]]-algebra structures via the

diamond operators, and by a well-known result of Hida, hordN is finite and flat over Zp[[1+pZp]].

1.2. Galois representations on Hecke algebras. For each positive integer M |N we mayconsider the new quotient Tnew

M of hordM , and the Galois representation

⇢M : GQ �! GL2(TnewM ⌦ 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

projection hordN ! TN�N of the Hecke operators of level prime toN . As in [EPW06, Prop. 2.3.2],

one can show that the canonical map

T0N �!

Y

M

TnewM

where the product is over all integers M � 1 with N�|M |N , becomes an isomorphism aftertensoring with L. Taking the product of the Galois representations ⇢M we thus obtain

⇢ : GQ �! GL2(T0N ⌦ L).

For any maximal ideal m of TN , let (T0N )m denote the localization of T0

N at m and let

⇢m : GQ �! GL2�(T0

N )m ⌦ L�

be the resulting Galois representation. If the residual representation ⇢m is irreducible, then⇢m admits an integral model (still denoted in the same manner)

⇢m : GQ �! GL2�(T0

N )m�

which is unique up to isomorphism.

1.3. Residual representations. Let ⇢ : GQ ! GL2(F) be an odd irreducible Galois repre-sentation defined over a finite field F of characteristic p. By [KW09], ⇢ is modular, meaningthat it arises as the residual representation associated with a modular form of some weightand level defined over Qp. Consider three more conditions we may impose on ⇢, where N� isa fixed square-free product of an odd number of primes.

Assumption (SU). (1) ⇢ is p-ordinary : the restriction of ⇢ to a decomposition groupGp ⇢ GQ at p has a one-dimensional unramified quotient over F.

(2) ⇢ is p-distinguished : ⇢|Gp ⇠�" ⇤0 �

�with " 6= �.

(3) ⇢ is ramified at all primes `|N�.

Fix once and for all a representation ⇢ as above satisfying Assumption (SU), together with ap-stabilization of ⇢ in the sense of [EPW06, Def. 2.2.10]. Let V be a two-dimensional F-vectorspace which a↵ords ⇢, and for any finite set of primes ⌃ that does not contain p, define

(1) N(⌃) := N(⇢)Y

`2⌃`m`

VARIATION OF ANTICYCLOTOMIC IWASAWA INVARIANTS 5

where N(⇢) is the tame conductor of ⇢, and m` := dimF V I` .Combining [EPW06, Thm. 2.4.1] and [EPW06, Prop. 2.4.2] with the fact that ⇢ is ramified

at the primes dividing N�, one can see that there exist unique maximal ideals n and m ofTN�

N(⌃) and T0N(⌃), respectively, such that n lifts m, (T0

N(⌃))m ' (TN�

N(⌃))n, and ⇢m ' ⇢. Definethe ordinary Hecke algebra T⌃ attached to ⇢ and ⌃ by

T⌃ := (T0N(⌃))m.

Thus T⌃ is a local factor of T0N(⌃), and we let

⇢⌃ : GQ �! GL2 (T⌃)

denote the Galois representation deduced from ⇢m.Following the terminology of [EPW06, §2.4], we shall refer to Spec(T⌃) as “the Hida family”

H(⇢) attached to ⇢ (and our chosen p-stabilization) that is minimally ramified outside ⌃.

1.4. Branches of the Hida family. If a is a minimal prime of T⌃ (for a finite set of primes⌃ as above), we put T(a) := T⌃/a and let

⇢(a) : GQ �! GL2(T(a))be the Galois representation induced by ⇢⌃. As in [EPW06, Prop. 2.5.2], one can show thatthere is a unique divisor N(a) of N(⌃) and a unique minimal prime a0 ⇢ Tnew

N(a) above a suchthat the diagram

T⌃//

✏✏

T0N(⌃)

//Q

N�|M |N(⌃) TnewM

✏✏T⌃/a

= // T(a) // TnewN(a)/a

0

commutes. We call N(a) the tame conductor of a and set

T(a)� := TnewN(a)/a

0.

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

1.5. Arithmetic specializations. For any finite Zp[[1+pZp]]-algebra T, we say that a heightone prime } of T is an arithmetic prime of T if } is the kernel of a Zp-algebra homomorphismT! Qp such that the composite map

1 + pZp ! Zp[[1 + pZp]]⇥ ! T⇥ ! Q

⇥p

is given by � 7! �k�2 on some open subgroup of 1+ pZp, for some integer k � 2. We then saythat } has weight k.

Let a ⇢ T⌃ be a minimal prime as above. For each n � 1, let an 2 T(a)� be the image ofTn under the natural projection hordN(⌃) ! T(a)�, and form the q-expansion

f(a) =X

n�1

anqn 2 T(a)�[[q]].

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 that

f}(a) :=X

n�1

(an mod }0)qn 2 O�}[[q]]

is the q-expansion a p-ordinary eigenform f} of weight k and tame level N(a), where O�} :=

T(a)�/}0 (see [EPW06, Prop. 2.5.6]).

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

2. Big Heegner points

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,let K be an imaginary quadratic field of discriminant �DK < 0 prime to Np and such thatevery 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, wecan extend the constructions of loc.cit. to branches of the Hida family which are not necessarilyprimitive (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 discrim-inant N�. We fix once and for all an embedding of Q-algebras K ,! B, and use it to identityK with a subalgebra of B. Denote by z 7! z the nontrivial automorphism of K, and choosea basis {1, j} of B over K with

• j2 = � 2 Q⇥ with � < 0,• jt = tj for all t 2 K,• � 2 (Z⇥

q )2 for q | pN+, and � 2 Z⇥

q for q | DK .

Fix a square-root �K =p�DK , and define ✓ 2 K by

✓ :=D0 + �K

2, where D0 :=

⇢DK if 2 - DK ,DK/2 if 2|DK .

For each prime q | pN+, define iq : Bq := B ⌦Q Qq ' M2(Qq) by

iq(✓) =

✓Tr(✓) �Nm(✓)1 0

◆; iq(j) =

p�

✓�1 Tr(✓)0 1

◆

where Tr and Nm are the reduced trace and reduced norm maps on B, respectively. On theother hand, for each prime q - Np we fix any isomorphism iq : Bq ' M2(Qq) with the propertythat iq(OK ⌦Z Zq) ⇢ M2(Zq).

For each r � 0, let RN+,r be the Eichler order of B of level N+pr with respect to the above

isomorphisms {iq : Bq ' M2(Qq)}q-N� , and let UN+,r be the compact open subgroup of bR⇥N+,r

defined by

UN+,r :=

⇢(xq)q 2 bR⇥

N+,r| ip(xp) ⌘

✓1 ⇤0 ⇤

◆(mod pr)

�.

Consider the double coset spaces

(2) eXN+,r = B⇥✏�HomQ(K,B)⇥ bB⇥��UN+,r

where b 2 B⇥ acts on ( , g) 2 HomQ(K,B)⇥ bB⇥ by

b · ( , g) = (b b�1, bg)

and UN+,r acts on bB⇥ by right multiplication.

As explained in [LV11, §2.1], eXN+,r is naturally identified with the set of K-rational pointsof certain genus zero curves defined over Q. Nonetheless, there is a nontrivial Galois actionon eXN+,r defined as follows: If � 2 Gal(Kab/K) and P 2 eXN+,r is the class of a pair ( , g),then

�P := [( , b (a)g)]where a 2 K⇥\ bK⇥ is chosen so that recK(a) = �. It will be convenient to extend this actionto an action of GK := Gal(Q/K) by letting � 2 GK act on eXN+,r as �|Kab . Since Gal(Kab/K)is obviously abelian, we will set P � := �P for the ease of notation.

VARIATION OF ANTICYCLOTOMIC IWASAWA INVARIANTS 7

Finally, we note that eXN+,r is also equipped with standard actions of Up, Hecke operatorsT` for ` - Np, and diamond operators hdi for d 2 (Z/prZ)⇥ (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.

Definition 2.1. We say that P 2 eXN+,r is a Heegner point of conductor c if P is the class ofa pair ( , g) with

(Oc) = (K) \ (B \ g bRN+,rg�1)

and

p((Oc ⌦ Zp)⇥ \ (1 + prOK ⌦ Zp)

⇥) = p((Oc ⌦ Zp)⇥) \ gpUN+,r,pg

�1p

where UN+,r,p denotes the p-component of UN+,r.

Fix a decomposition N+OK = N+N+, and define, for each prime q 6= p,

• &q = 1, if q - N+,

• &q = ��1K

✓✓ ✓1 1

◆2 GL2(Kq) = GL2(Qq), if q = qq splits with q|N+,

and for each s � 0,

• &(s)p =

✓✓ �11 0

◆✓ps 00 1

◆2 GL2(Kp) = GL2(Qp), if p = pp splits in K,

• &(s)p =

✓0 1�1 0

◆✓ps 00 1

◆, if p is inert in K.

Set &(s) := &(s)p

Qq 6=p &q 2 bB⇥, and let ıK : K ,! B be the inclusion. For all n, r � 0, it is

easy to see that the pointePpn,r := [(ıK , &(n+r))] 2 eXN+,r

is a Heegner point of conductor pn+r. Moreover, one can show that the points ePpn,r enjoy thefollowing properties:

• Field of definition: ePpn,r 2 H0(Lpn,r, eXN+,r), where Lpn,r := Hpn+r(µpr) and Hc isthe ring class field of K of conductor c.

• Galois equivariance: For all � 2 Gal(Lpn,r/Hpn+r),

eP �pn,r = h#(�)i · ePpn,r

where # : Gal(Lpn,r/Hpn+r)! Z⇥p /{±1} is such that #2 = "cyc.

• Horizontal compatibility : If r > 1, thenX

�2Gal(Lpn,r/Lpn�1,r)

e↵r( eP �pn,r) = Up · ePpn,r�1

where e↵r : eXN+,r ! eXN+,r�1 is the map induced by the inclusion UN+,r ⇢ UN+,r�1.• Vertical Compatibility : If n > 0, then

X

�2Gal(Lpn,r/Lpn�1,r)

eP �pn,r = Up · ePpn�1,r.

(See [CL14, Thm. 1.2] and the references therein.)

Remark 2.2. Even though it is not reflected in the notation, the points ePpn,r clearly dependon N+ (and the discriminant N� of the quaternion algebra B). In all the constructions inthis paper we will keep N� fixed, but it will be important to consider di↵erent values of N+.

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

2.3. Critical character. Factor the p-adic cyclotomic character as

"cyc = "tame · "wild : GQ �! Z⇥p ' µp�1 ⇥ (1 + pZp)

and define the critical character ⇥ : GQ ! Zp[[1 + pZp]]⇥ by

(3) ⇥(�) = ["1/2wild(�)]

where "1/2wild is the unique square-root of "wild taking values in 1 + pZp, and [·] : 1 + pZp !Zp[[1 + pZp]]⇥ is the map given by the inclusion as group-like elements.

2.4. Big Heegner points. Recall the Shimura curves eXN+,pr from Section 2.1, and set

DN+,r := eord(Div( eXN+,r)⌦Z Zp);

by the Jacquet–Langlands correspondence, DN+,r is naturally endowed with an action of the

Hecke algebra TN�N,r . Let (TN�

N,r)† be the free TN�

N,r-module of rank one equipped with the Galoisaction via the inverse of the critical character ⇥, and set

D†N+,r

:= DN+,r ⌦TN�N,r

(TN�N,r)

†.

Let ePpn,r 2 eXN+,r be the system of Heegner points of Section 2.2, and denote by Ppn,r the

image of eord ePpn,r in DN+,r. By the Galois equivariance of ePpn,r (see [LV11, §7.1]), we have

P�pn,r = ⇥(�) · Ppn,r

for all � 2 Gal(Lpn,r/Hpn+r), and hence Ppn,r defines an element

(4) Ppn,r ⌦ ⇣r 2 H0(Hpn+r ,D†N+,r

).

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

3. Anticyclotomic p-adic L-functions

3.1. Multiplicity one. Keep the notations introduced in Section 2. For each integer k � 2,denote by Lk(R) the set of polynomials of degree less than or equal to k� 2 with coe�cientsin a ring R, and define

JN+,r,k := eordH0( eXN+,r,Lk(Zp))

where Lk(Zp) is the local system on eXN+,r associated with Lk(Zp). Note that JN+,r,k is

naturally a module over the Hecke algebra TN�N,r,k.

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 (TN�N,r,k)m. In partic-

ular, there is a (TN�N,r,k)m-module isomorphism

(JN+,r,k)m↵N,r,k' (TN�

N,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. ⇤

Associated with any N�-new eigenform f 2 Sk(�0,1(N, pr)) whose associated maximal ideal

in TN�N,r,k is m, there is a Zp-algebra homomorphism (TN�

N,r,k)m ! O, where O is the ring ofintegers of a finite extension F/Qp generated by the Hecke eigenvalues of f . Composing witha fixed isomorphism ↵N,r,k as in Theorem 3.1, we thus obtain the functional

�f : (JN+,r,k)m �! O.

VARIATION OF ANTICYCLOTOMIC IWASAWA INVARIANTS 9

On the other hand, if �f 2 Sk( eXN+,r) is a p-adically normalised Jacquet–Langlands transfer(in the sense of [CH13, §4.1]) of f , then by evaluation �f defines another O-valued functional

�f : (JN+,r,k)m �! O.

By the multiplicity one theorem, �f and �f di↵er by a nonzero constant �f 2 F⇥ which iseasily seen to be necessarily a p-adic unit. Since both �f and �f are themselves defined up toa p-adic unit, we may assume �f = �f , as we shall do in the following.

If f is in fact a newform, following [PW11, §2.1] and [CH13, §4.1] we define Gross period

(5) ⌦f,N� :=(f, f)�0(N)

⇠f (N+, N�)

where ⇠f (N+, N�) is the self-product of �f with respect to a certain “intersection” pairing(see [CH13, Eq.(3.9)]). In [loc.cit., §5.4], it is shown that a certain p-adic L-function Lp(f/K)normalized by the complex period ⌦f,N� has vanishing µ-invariant. The preceding descriptionof �f in terms of �f will thus allow us to show that this property is preserved over the Hidafamily.

3.2. One-variable p-adic L-functions. Denote by � the Galois group of the anticyclotomicZp-extension K1/K. For each n, let Kn ⇢ K1 be defined by Gal(Kn/K) ' Z/pnZ and let�n be the subgroup of � such that �/�n ' Gal(Kn/K).

Let Ppn+1,r ⌦ ⇣r 2 H0(Hpn+1+r ,D†N+,r

) be the Heegner point of conductor pn+1, and define

(6) Qn,r := CorHpn+1+r/Kn(Ppn+1,r ⌦ ⇣r) 2 H0(Kn,D

†N+,r

);

with a slight abuse of notation, we will still denote byQn,r the its image under the natural map

H0(Kn,D†N+,r

) ⇢ DN+,r ! JN+,r composed with localization at m, where JN+,r := JN+,r,2.

Definition 3.2. For any open subset ��n of �, define

µr(��n) := U�np · Qn,r 2 (JN+,r)m.

Proposition 3.3. The rule µr is a measure on �.

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 �r

(JN+,r)m

which is naturally equipped with an action of the big Hecke algebra TN�N = lim �r

TN�N,r .

Theorem 3.4. 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 (TN�N )m. In particular,

there is a (TN�N )m-module isomorphism

(JN+)m↵N' (TN�

N )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.

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

Proof of Theorem 3.1. Let }N,r,k be the product of all the arithmetic primes of TN�N of weight

k which become trivial upon restriction to 1 + prZp. By [Hid88, Thm. 9.4], we then have

(7) (JN+)m ⌦ TN�N /}N,r,k ' (JN+,r,k)mr,k

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

TN�N by Theorem 3.4, it follows that (JN+,r,k)mr,k is free of rank one over TN�

N,r,k ' TN�N /}N,r,k,

as was to be shown. ⇤Remark 3.5. 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 num-ber 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 Heegnerpoints, the points

U�rp · Qn,r 2 (JN+,r)m

are compatible for varying r, thus defining an element

Qn := lim �r

U�rp · Qn,r 2 (JN+)m.

Definition 3.6. For any open subset ��n of �, define

µ(��n) := U�np · Qn 2 (JN+)m.

Proposition 3.7. The rule µ is a measure on �.

Proof. This follows immediately from the “horizontal compatibility” of Heegner points. ⇤Upon the choice of an isomorphism ↵N as in Theorem 3.4, we may regard µ as an element

L(m, N) 2 (TN�N )m⌦ZpZp[[�]].

Denoting by L(m, N)⇤ the image of L(m, N) under the involution induced by � 7! ��1

on group-like elements, we set L(m, N) := L(m, N) · L(m, N)⇤, to which we will refer as thetwo-variable p-adic L-function attached to (TN�

N )m.

3.5. Two-variable p-adic L-functions on branches of the Hida family. Let F be a finitefield of characteristic p, let ⇢ : GQ ! GL2(F) be an odd irreducible (and hence modular!)Galois representation satisfying Assumption (SU), and let T⌃ be the universal ordinary Heckealgebra

(8) T⌃ := (T0N(⌃))m ' (TN�

N(⌃))n

associated with ⇢ and a finite set of primes ⌃ as described in Section 1.3.

Remark 3.8. Note that N�|N(⇢) by hypothesis. Throughout the following, it will be assumedthat N� contains all prime factors of N(⇢) which are inert in K and at which ⇢ is ramified,and that every prime factor of N(⌃)/N� splits in K. In particular, every prime ` 2 ⌃ splitsin K.

The construction of the preceding section produces a two-variable p-adic L-function

L(n, N(⌃)) 2 (TN�

N(⌃))n⌦ZpZp[[�]]

which combined with the isomorphism (8) yields an element

L⌃(⇢) 2 T⌃⌦ZpZp[[�]].

VARIATION OF ANTICYCLOTOMIC IWASAWA INVARIANTS 11

If a is a minimal prime of T⌃, we thus obtain an element

L⌃(⇢, a) 2 T(a)�⌦ZpZp[[�]]

by reducing L⌃(⇢) 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

(9) TN�

N(a) �! TnewN(a) �! Tnew

N(a)/a0 = T(a)�,

the construction of the preceding section yields an L-function

L(m, N(a)) 2 (TN�

N(a))m⌦ZpZp[[�]]

giving rise, via (9), to a second element

L(⇢, a) 2 T(a)�⌦ZpZp[[�]].

It is natural to compare L⌃(⇢, a) and L(⇢, 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 ⌃ = {`1, . . . , `n} and for each ` = `i 2 ⌃, let e` be the valuation ofN(⌃)/N(a) at `, and define the reciprocal Euler factor E`(a, X) 2 T(a)�[X] by

E`(a, X) :=

8><

>:

1 if e` = 0

1� (T` mod a0)⇥�1(`)X if e` = 1

1� (T` mod a0)⇥�1(`)X + `X2 if e` = 2.

Also, writing ` = ll, define E`(a) 2 T(a)�⌦ZpZp[[�]] by

(10) E`(a) := E`(a, `�1�l) · E`(a, `

�1�l)

where �l, �l are arithmetic Frobenii at l, l in �, respectively, and put E⌃(a) :=Q

`2⌃E`(a).Recall that N�|N(a)|N(⌃) and set

N(a)+ := N(a)/N�; N(⌃)+ := N(⌃)/N�

both of which consist entirely of prime factors which split in K.The purpose of this section to prove the following result.

Theorem 3.9. There is an isomorphism of T(a)�-modules

T(a)� ⌦(TN�

N(⌃))n(JN(⌃)+)n ' T(a)� ⌦

(TN�N(a))m

(JN(a)+)m

such that the map induced on the corresponding spaces of measures valued in these modulessends L⌃(⇢, a) to E⌃(a) · L(⇢, a).

Proof. The proof follows very closely the constructions and arguments given in [EPW06, §3.8].Let r � 1. If M is any positive integer with (M,pN�) = 1, and d0|d are divisors of M , we

have degeneracy maps

Bd,d0 : eXM,r �! eXM/d,r

induced by ( , g) 7! ( ,⇡d0g), where ⇡d0 2 bB⇥ has local component� 1 00 `val`(d

0)

�at every prime

`|d0 and 1 outside d0. We thus obtain a map on homology

(Bd,d0)⇤ : eordH0( eXM,r,Zp) �! eordH0( eXM/d,r,Zp)

and we may define

(11) ✏r : eordH0( eXN(⌃)+,r,Zp) �! eordH0( eXN(a)+,r,Zp)

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

by ✏r := ✏(`n) � · · · � ✏(`1), where for every ` = `i 2 ⌃ we put

✏(`) :=

8><

>:

1 if e` = 0

(B`,1)⇤ � (B`,`)⇤`�1T` if e` = 1

(B`2,1)⇤ � (B`2,`)⇤`�1T` + (B`2,`2)⇤`

�1h`iN(a)p if e` = 2.

As before, let M be a positive integer with (M,pN�) = 1 all of whose prime factors split inK, and let ` - Mp be a prime which also splits in K. We shall adopt the following simplifyingnotations for the system of points ePpn,r 2 eXN+,r constructed in Section 2.2:

P := eP (M)pn,r 2 eXM,r, P (`) := eP (M`)

pn,r 2 eXM`,r, P (`2) := eP (M`2)pn,r 2 eXM`2,r.

It is easy to check that for a suitable factorization ` = ll we then have the following relations:

• (B`,1)⇤(P (`)) = P

• (B`,`)⇤(P (`)) = P �l

• (B`2,1)⇤(P(`2)) = P

• (B`2,`)⇤(P(`2)) = P �l

• (B`2,`2)⇤(P(`2)) = P �2

l

in eXM,r, where �l 2 Gal(Lpn,r/K) is a Frobenius element at l. Letting P denote the image of

eordP in DM,r, and defining P(`) 2 DM`,r and P(`2) 2 DM`2,r similarly, it follows that

• (B`,1)⇤(P(`) ⌦ ⇣r) = P ⌦ ⇣r• (B`,`)⇤(P(`) ⌦ ⇣r) = P�l ⌦ ⇣r = ⇥�1(�l) · (P ⌦ ⇣r)�l

• (B`2,1)⇤(P(`2) ⌦ ⇣r) = P ⌦ ⇣r• (B`2,`)⇤(P(`2) ⌦ ⇣r) = P�l ⌦ ⇣r = ⇥�1(�l) · (P ⌦ ⇣r)�l

• (B`2,`2)⇤(P(`2) ⌦ ⇣r) = P�2l ⌦ ⇣r = ⇥�2(�l) · (P ⌦ ⇣r)�l

as elements in D†M,r. Finally, setting Q := CorHpn+1+r/Kn

(P) 2 H0(Kn,D†M,r), and defining

Q(`) 2 H0(Kn,D†M`,r) and Q(`2) 2 H0(Kn,D

†M`2,r

) similarly, we see that

• (B`,1)⇤(Q(`)) = Q• (B`,`)⇤(Q(`)) = ⇥�1(�l) · Q�l

• (B`2,1)⇤(Q(`2)) = Q• (B`2,`)⇤(Q(`2)) = ⇥�1(�l) · Q�l

• (B`2,`2)⇤(Q(`2)) = ⇥�2(�l) · Q�2l

in H0(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(`) ⌦ ⇣r)⌘

= (B`,`)⇤

0

@⇣ X

�2Gal(Hpn+1+r/Kn)

⇥(��1) · (P(`))�⌘⌦ ⇣r

1

A

=X

�2Gal(Hpn+1+r/Kn)

⇥(��1) · (B`,`)⇤((P(`))� ⌦ ⇣r)

=X

�2Gal(Hpn+1+r/Kn)

⇥(��1)⇥�1(�l) · (P � ⌦ ⇣r)�l

= ⇥�1(�l) · Q�l .

VARIATION OF ANTICYCLOTOMIC IWASAWA INVARIANTS 13

Now let Qn,r 2 JN(⌃)+,r be as in (6) with N = N(⌃). Using the above formulae, we easilysee that of any finite order character � of � of conductor pn, the e↵ect of ✏r on the elementP

�2�/�n�(�)Q�

n,r is given by multiplication byY

`i : e`i=1

(1� (�⇥)�1(�li)`�1i T`i)

Y

`i : e`i=2

(1� (�⇥)�1(�li)`�1i T`i + (�⇥)�2(�li)`

�1i h`iiN(a)p).

Similarly, we see that ✏r has the e↵ect of multiplying the elementP

�2�/�n��1(�)Q�

n,r byY

`i : e`i=1

(1�(��1⇥)�1(�li)`�1i T`i)

Y

`i : e`i=2

(1�(��1⇥)�1(�li)`�1i T`i+(��1⇥)�2(�li)`

�1i h`iiN(a)p).

Hence, using the relations

�(�li) = ��1(�li); ⇥(�li) = ⇥(�li) = ✓(`i); ✓2(`i) = h`iiN(a)p

it follows that the e↵ect of ✏r on the product of the above two elements is given by multipli-cation by

Y

li|`i : e`i=1

(1� �(�li)✓�1(`i)`�1i T`i)

Y

li|`i : e`i=2

(1� �(�li)✓�1(`i)`�1i T`i + �2(�li)`

�1i ).

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

(12) T(a)� ⌦(TN�

N(⌃))n(JN(⌃)+)n �! T(a)� ⌦

(TN�N(a))m

(JN(a)+)m

having as e↵ect on the corresponding measures as stated in Theorem 3.9. Hence to concludethe proof it remains to show that (12) is an isomorphism.

By Theorem 3.4, 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 injectivityof the dual map modulo p:

H0( eXN(a)+,1;Fp)ord[m] �! (TN�

N(a)/m)⌦TN�N(⌃)/n

(H0( eXN(a)+,1;Fp)ord[m0])(13)

�! (TN�

N(a)/m)⌦TN�N(⌃)/n

(H0( eXN(⌃)+,1;Fp)ord[m0])

�! (TN�

N(a)/m)⌦TN�N(⌃)/n

(H0( eXN(⌃)+,1;Fp)ord[n]);

or equivalently (by a version of [EPW06, Lemma 3.8.1]), to showing that the composite ofthe first two arrows in (13) is injective.

In turn, the latter injectivity follows from Lemma 3.10, 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 towhether or not ` divides M+, N+ := `n`M+, and

(14) ✏⇤` : H0( eXM+,1;Fp)

ord[m] �! (TN�

M+N�/m)⌦T0N+N�/m0 (H0( eXN+,1;Fp)

ord[m0])

is the map defined by

✏⇤` :=

⇢B⇤

`,1 �B⇤`,``

�1T` if n` = 1B⇤

`2,1 �B⇤`2,``

�1T` +B⇤`2,`2`

�1h`iN(a)p if n` = 2.

Lemma 3.10. The map (14) is injective.

Proof of Lemma 3.10. As in the proof of the analogous result [EPW06, Lemma 3.8.2] in themodular curve case, it su�ces to show the injectivity of the map

(H0( eXM+,1;F)ord[mF])

n`+1 �`��! H0( eXN+,1;F)ord[m0

F]

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

defined by

�` :=

⇢B⇤

`,1⇡1 +B⇤`,`⇡2 if n` = 1

B⇤`2,1⇡1 +B⇤

`2,`⇡2 +B⇤`2,`2⇡3 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 3.10 to the primes in ⌃, the proof of Theorem 3.9 follows. ⇤

3.7. Analytic Iwasawa invariants. Upon the choice of an isomorphism

Zp[[�]] ' Zp[[T ]]

we may regard the p-adic L-functions L⌃(⇢, a) and L(⇢, a), as well as the Euler factor E⌃(⇢, a),as elements in T(a)�[[T ]]. In this section we apply the main result of the preceding section tostudy the variation of the Iwasawa invariants attached to the anticyclotomic p-adic L-functionsof p-ordinary modular forms.

For any power series f(T ) 2 R[[T ]] with coe�cients in a ring R, recall that the contentof f(T ) is defined to be the ideal I(f(T )) ✓ R generated by the coe�cients of f(T ). If }is a height one prime of T⌃ belonging to the branch T(a) (in the sense that a is the uniqueminimal prime of T⌃ contained in }), we denote by L(⇢, a)(}) the element of O}[[�]] obtainedfrom L(⇢, a) via reduction modulo }. In particular, we note that L(⇢, a)(}) has unit contentif and only if its µ-invariant vanishes.

Theorem 3.11. The following are equivalent:

(1) µ(L(⇢, a)(})) = 0 for some newform f} in H(⇢).(2) µ(L(⇢, a)(})) = 0 for every newform f} in H(⇢).(3) L(⇢, a) has unit content for some irreducible component T(a) of H(⇢).(4) L(⇢, a) has unit content for every irreducible component T(a) of H(⇢).

Proof. The argument in [EPW06, Thm 3.7.5] applies verbatim, replacing the appeal to [loc.cit.,Cor. 3.6.3] by our Theorem 3.9 above. ⇤

When any of the conditions in Theorem 3.11 hold, we shall write

µan(⇢) = 0.

For a power series f(T ) with unit content and coe�cients in a local ring R, recall that the�-invariant �(f(T )) is defined to be the smallest degree in which f(T ) has a unit coe�cient.

Theorem 3.12. Assume that µan(⇢) = 0.

(1) Let T(a) be an irreducible component of H(⇢). As } varies over the arithmetic primesof T(a), the �-invariant �(L(⇢, a)(})) takes on a constant value, denoted �an(⇢, a).

(2) For any two irreducible components T(a1),T(a2) of H(⇢), we have that

�an(⇢, a1)� �an(⇢, a2) =X

`6=p

e`(a2)� e`(a1)

where e`(a) = �(E`(a)).

Proof. The first part follows immediately from the definitions. For the second part, the argu-ment in [EPW06, Thm. 3.7.7] applies verbatim, replacing their appeal to [loc.cit., Cor. 3.6.3]by our Theorem 3.9 above. ⇤

By Theorem 3.11 and Theorem 3.12, the Iwasawa invariants of L(⇢, a)(}) are well-behavedas } varies. However, for the applications of these results to the Iwasawa main conjecture itis of course necessary to relate L(⇢, a)(}) to p-adic L-functions defined by the interpolationof special values of L-functions. This question was addressed in [CL14], as we now recall.

VARIATION OF ANTICYCLOTOMIC IWASAWA INVARIANTS 15

Theorem 3.13. If } is the arithmetic prime of T(a) corresponding to a p-ordinary p-stabilizednewform f} of weight k � 2 and trivial nebentypus, then

L(⇢, a)(}) = Lp(f}/K)

where Lp(f}/K) is the p-adic L-function of Chida–Hsieh [CH13]. In particular, if � : �! 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(⇢, a)(}) interpolates the central critical values

L(f}/K,�, k/2)

⌦f},N�

as � varies, where ⌦f},N� is the complex period (5).

Proof. This is a reformulation of the main result of [CL14]. (Note that the constant �} 2 F⇥}

in [CL14], Thm. 4.6] is not needed here, since the specialization map of [loc.cit., §3.1] is beingreplaced by the map (JN+)m ! (JN+,r,k)mr,k induced by the isomorphism (7), which preservesintegrality.) ⇤

Corollary 3.14. Let f1, f2 2 H(⇢) be newforms with trivial nebentypus lying in the branchesT(a1), T(a2), respectively. Then µan(⇢) = 0 and

�(Lp(f1/K))� �(Lp(f2/K)) =X

6=p

e`(a2)� e`(a1)

where e`(aj) = �(E`(aj)).

Proof. By [CH13, Thm. 5.7] (extending Vatsal’s result [Vat03] to higher weights), if f 2 H(⇢)has weight k p + 1 and trivial nebentypus, then µ(Lp(f/K)) = 0. By Theorem 3.11 andTheorem 3.13, this implies µan(⇢) = 0. The result thus follows from Theorem 3.12, usingagain Theorem 3.13 to replace �an(⇢, aj) by �(Lp(fj/K)). ⇤

4. Anticyclotomic Selmer groups

We continue with the notation of the previous sections. In particular, ⇢ : GQ ! GL2(F) isan odd irreducible Galois representation satisfying (SU), H(⇢) is the associated Hida family,and ⌃ is a finite set of primes split in the imaginary quadratic field K.

For each f 2 H(⇢), let Vf denote the self-dual Tate twist of the p-adic Galois representationassociated 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 F

+p Vf ⇢ Vf where the inertia group

at p acts via "k/2cyc , with of finite order. Let F+p Af be the image of F+

p Vf in Af , and definethe minimal Selmer group of f by

Sel(K1, f) := ker

8<

:H1(K1, Af ) �!Y

w-pH1(K1,w, Af )⇥

Y

w|p

H1(K1,w, F�p Af )

9=

;

where w runs over the places of K1 and we set F�p Af := Af/F

+p Af .

It is well-known that Sel(K1, f) is cofinitely generated over ⇤. When it is also ⇤-cotorsion,we define the µ-invariant µ(Sel(K1, f)) (resp. �-invariant �(Sel(K1, f))) to the largest powerof $ dividing (resp. the number of zeros of) the characteristic power series of the Pontryagindual of Sel(K1, f). The same remarks and definitions apply to Sel(K1, f).

A distinguishing feature of the anticyclotomic setting (in comparison with cyclotomic Iwa-sawa theory) is the presence of primes which split infinitely in the corresponding Zp-extension.

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

Indeed, being inert in K, all primes `|N� are infinitely split in K1/K. As a result, the aboveSelmer group di↵ers in general from the Greenberg Selmer group of f , which is defined by

Sel(K1, f) := ker

8<

:H1(K1, Af ) �!Y

w-pH1(I1,w, Af )⇥

Y

w|p

H1(K1,w, F�p Af )

9=

;

where I1,w ⇢ GK1 denotes the inertia group at w.If S is a finite set of primes in K, we let SelS(K1, f) and SelS(K1, f) be the “S-primitive”

Selmer groups defined as above by omitting the local conditions at the primes in S (exceptthose 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 sequence

(15) 0 �! Sel(K1, f) �! Sel(K1, f) �!Y

`|N�

Hun`

where

Hun` := ker

8<

:Y

w|`

H1(K1,w, Af ) �!Y

w|`

H1(I1,w, Af )

9=

;

is the set of unramified cocycles. In [PW11], Pollack and Weston carried out a careful analysisof the di↵erence between Sel(K1, f) and Sel(K1, f). Even though in loc.cit. they focusedon the case where f is associated with an elliptic curve, many of their arguments apply moregenerally. In fact, the next result follows essentially from their work.

Theorem 4.1. Assume that ⇢ satisfies (SU). Then the following are equivalent:

(1) Sel(K1, f0) is ⇤-cotorsion with µ-invariant zero for some newform f0 2 H(⇢).(2) Sel(K1, f) is ⇤-cotorsion with µ-invariant zero for all newforms f 2 H(⇢).(3) Sel(K1, f) is ⇤-cotorsion with µ-invariant zero for all newforms f 2 H(⇢).

Moreover, in that case Sel(K1, f) ' Sel(K1, f).

Proof. Assume f0 is a newform in H(⇢) for which Sel(K1, f0) is ⇤-cotorsion with µ-invariantzero, and set N+ := N(⌃)/N�. By [PW11, Prop. 5.1], we then have the exact sequences

0 �! Sel(K1, f0) �! SelN+(K1, f0) �!

Y

`|N+

H` �! 0(16)

0 �! Sel(K1, f0) �! SelN+(K1, f0) �!

Y

`|N+

H` �! 0(17)

where H` is the product of H1(K1,w, Af0) over the places w|` in K1. Since every prime `|N+

splits in K (see Remark 3.8), the ⇤-cotorsionness and the vanishing of the µ-invariant of H`

can be deduced from [GV00, Prop. 2.4]. Since Sel(K1, f0)[$] is finite by assumption, it thus

follows from (16) that SelN+(K1, f0)[$] is finite. Combined with (15) and [PW11, Cor. 5.2],

the same argument using (17) shows that then SelN+(K1, f0)[$] is also finite.

On the other hand, following the arguments in the proof [PW11, Prop. 3.6] we see that forany f 2 H(⇢) we have

SelN+(K1, ⇢) ' SelN

+(K1, f)[$]

SelN+(K1, ⇢) ' SelN

+(K1, f)[$].

As a result, the argument in the previous paragraph implies that, for any newform f 2 H(⇢),

both SelN+(K1, f)[$] and SelN

+(K1, f)[$] are finite , from where (using (16) and (17) with

f in place of f0) the ⇤-cotorsionness and the vanishing of both the µ-invariant of Sel(K1, f)and of Sel(K1, f) follows. In view of (15) and [PW11, Lemma 3.4], the resulf follows. ⇤

VARIATION OF ANTICYCLOTOMIC IWASAWA INVARIANTS 17

Let w be a prime of K1 above ` 6= p and denote by Gw ⇢ GK1 its decomposition group.Let T(a) be the irreducible component of T⌃ passing through f , and define

�w(a) := dimFAGwf /$.

(Note that this is well-defined by [EPW06, Lemma 4.3.1].) Assume ` = ll splits in K and put

(18) �`(a) :=X

w|`

�w(a)

where the sum is over the (finitely many) primes w of K1 above `.In view of Theorem 4.1, we write µalg(⇢) = 0 whenever any of the µ-invariants appearing in

that result vanish. In that case, for any newform f in H(⇢) we may consider the �-invariants�(Sel(K1, f)) = �(Sel(K1, f)).

Theorem 4.2. Let ⇢ and ⌃ be as above, and assume that µalg(⇢) = 0. If f1 and f2 are anytwo newforms in the Hida family of ⇢ lying in the branches T(a1) and T(a2), respectively, then

�(Sel(K1, f1))� �(Sel(K1, f2)) =X

` 6=p

�`(a1)� �`(a2).

Proof. Since N�|N(ai)|N(⌃) and N(⌃)/N� is 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 Iwasawa invariants. Recall the definition of the analytic invariant e`(a) =�(E`(a)), where E`(a) is the Euler factor from Section 3.6, and of the algebraic invariant �`(a)introduced in (18).

Lemma 5.1. Let a1, a2 be minimal primes of T⌃. For any prime ` 6= p split in K, we have

�`(a1)� �`(a2) = e`(a2)� e`(a1).

Proof. Let a be a minimal prime of T⌃, let f be a newform in the branch T(a), and let }f ⇢ abe the corresponding height one prime. Since ` = ll splits in K, we have

�w|`H1(K1,w, Af ) =

��w|lH

1(K1,w, Af )��⇣�w|lH

1(K1,w, Af )⌘

and [GV00, Prop. 2.4] immediately implies that

Ch⇤��w|`H

1(K1,w, Af )_� = E`(f, `

�1�l) · E`(f, `�1�l)

where E`(f, `�1�l) · E`(f, `�1�l) is the specialization of E`(a) at }f . The result thus followsfrom [EPW06, Lemma 5.1.5]. ⇤Theorem 5.2. Assume that ⇢ satisfies (SU). If for some newform f0 2 H(⇢) we have

µ(Sel(K1, f0)) = µ(Lp(f0/K)) = 0 and �(Sel(K1, f0)) = �(Lp(f0/K))

then

µ(Sel(K1, f)) = µ(Lp(f/K)) = 0 and �(Sel(K1, f)) = �(Lp(f/K))

for all newforms f 2 H(⇢).

Proof. Let f be any newform in H(⇢). Since the µ-invariants of f0 vanish, the vanishing ofµ(Sel(K1, f)) and µ(Lp(f/K)) follows from Theorem 4.1 and Theorem 3.11, respectively.

On the other hand, combining Theorems 3.12 and 4.2, and Lemma 5.1, we see that

�(Sel(K1, f))� �(Sel(K1, f0)) = �(Lp(f/K))� �(Lp(f0/K)),

and hence the equality �(Sel(K1, f0)) = �(Lp(f0/K)) implies the same equality for f . ⇤

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

5.2. Applications to the main conjecture. As an immediate consequence of Weierstrasspreparation theorem, our Theorem 5.2 together with one the divisibilities predicted by theanticyclotomic main conjecture implies the full anticyclotomic main conjecture.

Theorem 5.3 (Skinner–Urban). Let f 2 Sk(�0(N)) be a newform of weight k ⌘ 2 (mod p�1)and trivial nebentypus, and assume that ⇢f satisfies (SU) and that p splits in K. Then

(Lp(f/K)) ◆ Ch⇤(Sel(K1, f)_).

Proof. This follows from specializing the divisibility in [SU14, Thm. 3.26] to the anticyclotomicline. Indeed, let f =

Pn�1 an(f)q

n 2 I[[q]] be the ⇤-adic form with coe�cients in I := T(a)�associated with the branch of the Hida family containing f , let ⌃ be a finite set of primes asin Section 3.5, let ⌃0 ◆ ⌃ be a finite set of primes of K containing ⌃ and all primes dividingpN(a)DK , and assume that ⌃0 contains at least one prime ` 6= p that splits in K. Under theseassumptions, in [SU14, Thm. 3.26] it is shown that

(19) (L⌃0p (f/K)) ◆ Ch⇤f (L1)(Sel⌃

0(L1, Af )

_)

where L1 = K1Kcyc is the Z2p-extension of K, ⇤f (L1) is the three-variable Iwasawa algebra

I[[Gal(L1/K)]], and L⌃0p (f/K) and Sel⌃

0(L1, Af ) are the “⌃0-primitive” p-adic L-function

and Selmer group defined in [SU14, §3.4.5] and [SU14, §§3.1.3,10], respectively.Recall the character ⇥ : GQ ! Zp[[1 + pZp]]⇥ from Section 2.3, regarded as a character on

Gal(L1/K), and letTw⇥�1 : ⇤f (L1) �! ⇤f (L1)

be the I-linear isomorphism induced by Tw⇥�1(g) = ⇥�1(g)g for g 2 Gal(L1/K). Choose atopological generator � 2 Gal(Kcyc/K), and expand

Tw⇥�1(L⌃0p (f/K)) = L⌃0

p,0(f/K) + L⌃0p,1(f/K)(� � 1) + · · ·

with L⌃0p,i(f/K) 2 ⇤f (K1) = I[[�]]. In particular, note that L⌃0

p,0(f/K) is the restriction of the

twisted three-variable p-adic L-function Tw⇥�1(L⌃0p (f/K)) to the “self-dual” plane.

Because of our assumptions on f , the ⇤-adic form f has trivial tame character, and hencedenoting by Frob` an arithmetic Frobenius at any prime ` - N(a)p, the Galois representation

⇢(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 that

det(X � Frob`|Tf ) = X2 � a`(f)X +⇥2(`)`.

The twist T †f := Tf ⌦⇥�1 is therefore self-dual. Thus combining [Rub00, Lemma 6.1.2] with

a straightforward variant of [SU14, Prop. 3.9] having Gal(K1/K) in place of Gal(Kcyc/K),we see that the divisibility (19) implies that

(20) (L⌃0p,0(f/K)) ◆ Ch⇤f (K1)(Sel⌃

0(K1, A†

f )_).

(Here, as above, Af denotes the Pontryagin dual Tf ⌦I Homcts(I,Qp/Zp), and A†f is the cor-

responding twist.) We next claim that, setting ⌃00 := ⌃0 r ⌃, we have

(21) (L⌃0p,0(f/K)) = (L⌃(⇢, a) ·

Y

v2⌃00v-p

Ev(a))

where L⌃(⇢, a) is the two-variable p-adic L-function constructed in §3.4, and if v lies over therational prime `, Ev(a) is the Euler factor given by

Ev(a) = det(Id� FrobvX|(V †f )Iv)X=`�1Frobv

VARIATION OF ANTICYCLOTOMIC IWASAWA INVARIANTS 19

where Vf := Tf ⌦I Frac(I), and Frobv is an arithmetic Frobenius at v. (Note that for ` = llsplit in K, El(a) ·El(a) is simply the Euler factor (10).) Indeed, combined with Theorem 3.9and Theorem 3.13, the equality (21) specialized to any arithmetic prime } ⇢ T(a) of weight2 is shown in [SU14, (12.3)], from where the claim follows easily from the density of theseprimes. (See also [PW11, Thm. 6.8] for the comparison between the di↵erent periods involvedin the two constructions, which di↵er by a p-adic unit under our assumptions.)

Finally, (20) and (21) combined with Theorem 3.9 and [GV00, Props. 2.3,8] imply that

(L(⇢, a)) ◆ Ch⇤f (K1)(Sel(K1, A†f )

_)

from where the result follows by specializing at }f using Theorem 3.13 and Theorem 4.1. ⇤Corollary 5.4. Suppose that ⇢ satisfies (SU) and that p splits in K. If the anticyclotomicmain conjecture holds for some newform f0 in H(⇢) of weight k0 ⌘ 2 (mod p� 1) and trivialnebentypus, then it holds for all newforms f in H(⇢) of weight k ⌘ 2 (mod p� 1) and trivialnebentypus.

Proof. After Theorem 5.3, to check the anticyclotomic main conjecture for any newform f asin the statement, it su�ces to check that

(22) µ(Sel(K1, f)) = µ(Lp(f/K)) = 0 and �(Sel(K1, f)) = �(Lp(f/K)).

If the anticyclotomic main conjecture holds for some newform f0 as in the statement, thenthe first and third equalities in (22) clearly hold for f0, while the vanishing of µ(Lp(f0/K))follows from Corollary 3.14; by Theorem 5.2, the equalities (22) then also hold f , and hencethe anticyclotomic main conjecture for f follows. ⇤

Acknowledgements. During the preparation of this paper, F.C. was partially supported byGrant MTM20121-34611 and by Prof. Hida’s NSF Research Grant DMS-0753991; C.K. waspartially supported by AMS-Simons Travel Grants; and M.L. was supported by PRIN 2010-11“Arithmetic Algebraic Geometry and Number Theory” and by PRAT 2013 “Arithmetic ofVarieties over Number Fields”.

References

[BD05] M. Bertolini and H. Darmon, Iwasawa’s main conjecture for elliptic curves over anticyclotomic Zp-extensions, Ann. of Math. (2) 162 (2005), no. 1, 1–64. MR 2178960 (2006g:11218)

[CH13] M. Chida and M.-L. Hsieh, Special values of anticyclotomic L-functions for modular forms, preprint,arXiv:1304.331 (2013).

[CH15] , On the anticyclotomic Iwasawa main conjecture for modular forms, Compositio Math., toappear (2015).

[CL14] Francesc Castella and Matteo Longo, Big Heegner points and special values of L-series, preprint,arXiv:1412.7071 (2014).

[DT94] Fred Diamond and Richard Taylor, Nonoptimal levels of mod l modular representations, Invent.Math. 115 (1994), no. 3, 435–462. MR 1262939 (95c:11060)

[EPW06] Matthew Emerton, Robert Pollack, and Tom Weston, Variation of Iwasawa invariants in Hidafamilies, Invent. Math. 163 (2006), no. 3, 523–580. MR 2207234 (2007a:11059)

[Gre89] Ralph Greenberg, Iwasawa theory for p-adic representations, Algebraic number theory, Adv. Stud.Pure Math., vol. 17, Academic Press, Boston, MA, 1989, pp. 97–137. MR 1097613 (92c:11116)

[GV00] Ralph Greenberg and Vinayak Vatsal, On the Iwasawa invariants of elliptic curves, InventionesMathematicae 142 (2000), no. 1, 17–63.

[Hid86] Haruzo Hida, Galois representations into GL2(Zp[[X]]) attached to ordinary cusp forms, Invent.Math. 85 (1986), no. 3, 545–613. MR 848685 (87k:11049)

[Hid88] , On p-adic Hecke algebras for GL2 over totally real fields, Ann. of Math. (2) 128 (1988),no. 2, 295–384. MR 960949 (89m:11046)

[How07] Benjamin Howard, Variation of Heegner points in Hida families, Invent. Math. 167 (2007), no. 1,91–128. MR 2264805 (2007h:11067)

[KW09] Chandrashekhar Khare and Jean-Pierre Wintenberger, Serre’s modularity conjecture. I, Invent.Math. 178 (2009), no. 3, 485–504. MR 2551763 (2010k:11087)

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

[LV11] Matteo Longo and Stefano Vigni, Quaternion algebras, Heegner points and the arithmetic of Hidafamilies, Manuscripta Math. 135 (2011), no. 3-4, 273–328. MR 2813438 (2012g:11114)

[PW11] Robert Pollack and Tom Weston, On anticyclotomic µ-invariants of modular forms, Compos. Math.147 (2011), no. 5, 1353–1381. MR 2834724 (2012k:11174)

[Rub00] Karl Rubin, Euler systems, Annals of Mathematics Studies, vol. 147, Princeton University Press,Princeton, NJ, 2000, Hermann Weyl Lectures. The Institute for Advanced Study. MR 1749177(2001g:11170)

[SU14] Christopher Skinner and Eric Urban, The Iwasawa Main Conjectures for GL2, Invent. Math. 195(2014), no. 1, 1–277. MR 3148103

[SW99] C. M. Skinner and A. J. Wiles, Residually reducible representations and modular forms, Inst. HautesEtudes Sci. Publ. Math. (1999), no. 89, 5–126 (2000). MR 1793414 (2002b:11072)

[Vat03] V. Vatsal, Special values of anticyclotomic L-functions, Duke Math. J. 116 (2003), no. 2, 219–261.MR 1953292 (2004e:11069b)

Department of Mathematics, UCLA, Math Sciences 6363, Los Angeles, 90095 CA, USAE-mail address: castella@math.ucla.edu

Department of Mathematics, UC Irvine, 340 Rowland Hall, Irvine, 92697 CA, USAE-mail address: chanho.math@gmail.com

Dipartimento di Matematica, Universita di Padova, Via Trieste 63, 35121 Padova, ItalyE-mail address: mlongo@math.unipd.it