Kida’s Formula and Congruences

To John Coates, for his 60^{th} birthday

Robert Pollack and Tom Weston

Received: August 30, 2005 Revised: June 21, 2006

Abstract. We consider a generalization of a result of Kida in classical Iwasawa theory which relates Iwasawa invariants of p-extensions of number fields. In this paper, we consider Selmer groups of a general class of Galois representations which includes the case of p-ordinary Hilbert modular forms and p-supersingular modular forms.

2000 Mathematics Subject Classification: 11R23

Keywords and Phrases: Iwasawa theory, Kida’s formula

1. Introduction

Let f be a modular eigenform of weight at least two and let F be a finite abelian extension of Q. Fix an odd prime p at which f is ordinary in the sense that the p^{th} Fourier coefficient of f is not divisible by p. In Iwasawa theory, one associates two objects to f over the cyclotomic Z_{p}-extension F_{∞} of F: a Selmer group Sel(F_{∞}, A_{f}) (where A_{f} denotes the divisible version of the two-dimensional Galois representation attached to f) and a p-adic L-function L_{p}(F_{∞}, f). In this paper we prove a formula, generalizing work of Kida and Hachimori–Matsuno, relating the Iwasawa invariants of these objects over F with their Iwasawa invariants over p-extensions of F. For Selmer groups our results are significantly more general. Let T be a lattice in a nearly ordinary p-adic Galois representation V; set A = V/T. When Sel(F_{∞}, A) is a cotorsion Iwasawa module, its Iwasawa μ-invariant μ^{alg}(F_{∞}, A) is said to vanish if Sel(F_{∞}, A) is cofinitely generated and its λinvariant λ^{alg}(F_{∞}, A) is simply its p-adic corank. We prove the following result relating these invariants in a p-extension.

Theorem 1. Let F′/F be a finite Galois p-extension that is unramified at all places dividing p. Assume that T satisfies the technical assumptions (1)–(5) of Section 2. If Sel(F_{∞}, A) is Λ-cotorsion with μ^{alg}(F_{∞}, A) = 0, then Sel(F′_{∞}, A) is Λ-cotorsion with μ^{alg}(F′_{∞}, A) = 0. Moreover, in this case

λ^{alg}(F′_{∞}, A) = [F′_{∞} : F_{∞}] · λ^{alg}(F_{∞}, A) + ∑_{w′} m(F′_{∞,w′}/F_{∞,w}, V)

where the sum extends over places w′ of F′_{∞} which are ramified in F′_{∞}/F_{∞}. If V is associated to a cuspform f and F′ is an abelian extension of Q, then the same results hold for the analytic Iwasawa invariants of f.

Here m(F′_{∞,w′}/F_{∞,w}, V) is a certain difference of local multiplicities defined in Section 2.1. In the case of Galois representations associated to Hilbert modular forms, these local factors can be made quite explicit; see Section 4.1 for details. It follows from Theorem 1 and work of Kato that if the p-adic main conjecture holds for a modular form f over Q, then it holds for f over all abelian pextensions of Q; see Section 4.2 for details. These Riemann-Hurwitz type formulas were first discovered by Kida [5] in the context of λ-invariants of CM fields. More precisely, when F′/F is a pextension of CM fields and μ^{−}(F_{∞}/F) = 0, Kida gave a precise formula for λ^{−}(F′_{∞}/F′) in terms of λ^{−}(F_{∞}/F) and local data involving the primes that ramify in F′/F. (See also [4] for a representation theoretic interpretation of Kida’s result.) A similar formula in a somewhat different setting was given for elliptic curves with complex multiplication at ordinary primes by Wingberg [12]; Hachimori–Matsuno [3] established the cyclotomic version in general. The analytic analogue was first established for ideal class groups by Sinnott [10] and for elliptic curves by Matsuno [7]. Our proof is most closely related to the arguments in [10] and [7] where congruences implicitly played a large role in their study of analytic λ-invariants. In this paper, we make the role of congruences more explicit and apply these methods to study both algebraic and analytic λ-invariants. As is usual, we first reduce to the case where F′/F is abelian. (Some care is required to show that our local factors are well behaved in towers of fields; this is discussed in Section 2.1.) In this case, the λ-invariant of V over F′ can be expressed as the sum of the λ-invariants of twists of V by characters of Gal(F′/F). The key observation (already visible in both [10] and [7]) is that since Gal(F′/F) is a p-group, all of its characters are trivial modulo a prime over p and, thus, the twisted Galois representations are all congruent to V modulo a prime over p. The algebraic case of Theorem 1 then follows from the results of [11] which gives a precise local formula for the difference between λ-invariants of congruent Galois representations. The analytic case is handled similarly using the results of [1]. The basic principle behind this argument is that a formula relating the Iwasawa invariants of congruent Galois representations should imply of a transition formula for these invariants in p-extensions. As an example of this, in Section 4.3, we use results of [2] to prove a Kida formula for the Iwasawa invariants (in the sense of [8, 6, 9]) of weight 2 modular forms at supersingular primes.

Acknowledgments: We would like to thank the anonymous referee for several helpful comments and for pointing out some errors in an earlier draft of this paper.

2. Algebraic invariants

2.1. Local preliminaries. We begin by studying the local terms that appear in our results. Fix distinct primes ℓ and p and let L denote a finite extension of the cyclotomic Z_{p}-extension of Q_{ℓ}. Fix a field K of characteristic zero and a finite-dimensional K-vector space V endowed with a continuous K-linear action of the absolute Galois group G_{L} of L. Set

m_{L}(V) := dim_{K} (V_{I_{L}})^{G_{L}} ,

the multiplicity of the trivial representation in the I_{L}-coinvariants of V. Note that this multiplicity is invariant under extension of scalars, so that we can enlarge K as necessary. Let L′ be a finite Galois p-extension of L. Note that L′ must be cyclic and totally ramified since L contains the Z_{p}-extension of Q_{ℓ}. Let G denote the Galois group of L′/L. Assuming that K contains all [L′ : L]^{th} roots of unity, for a character χ : G → K^{×} of G, we set V_{χ} = V ⊗_{K} K(χ) with K(χ) a one-dimensional K-vector space on which G acts via χ. We define

m(L′/L, V) := ∑_{χ∈G^{∨}} m_{L}(V) − m_{L}(V_{χ})

where G^{∨} denotes the K-dual of G. The next result shows how these invariants behave in towers of fields.

Lemma 2.1. Let L′′ be a finite Galois p-extension of L and let L′ be a Galois extension of L contained in L′′. Assume that K contains all [L′′ : L]^{th} roots of unity. Then

m(L′′/L, V) = [L′′ : L′] · m(L′/L, V) + m(L′′/L′, V).

Proof. Set G = Gal(L′′/L) and H = Gal(L′′/L′). Consider the Galois group G_{L}/I_{L′′} over L of the maximal unramified extension of L′′. It sits in an exact sequence

(1)  0 → G_{L′′}/I_{L′′} → G_{L}/I_{L′′} → G → 0

which is in fact split since the maximal unramified extensions of both L and L′′ are obtained by adjoining all prime-to-p roots of unity.

Fix a character χ ∈ G^{∨}. We compute

m_{L}(V_{χ}) = dim_{K}((V_{χ})_{I_{L}})^{G_{L}}
= dim_{K}((((V_{χ})_{I_{L′′}})_{G})^{G_{L′′}} )^{G}
= dim_{K}((((V_{χ})_{I_{L′′}})^{G_{L′′}} )_{G})^{G} since (1) is split
= dim_{K}(((V_{χ})_{I_{L′′}} )^{G_{L′′}} )^{G} since G is finite cyclic
= dim_{K}((V_{I_{L′′}})^{G_{L′′}} ⊗ χ)^{G} since χ is trivial on G_{L′′}.

The lemma thus follows from the following purely group-theoretical statement applied with W = (V_{I_{L′′}})^{G_{L′′}}: for a finite dimensional representation W of a finite abelian group G over a field of characteristic zero containing μ_{#G}, we have

∑_{χ∈G^{∨}}(⟨W, 1⟩_{G} − ⟨W, χ⟩_{G}) =

#H · ∑_{χ∈(G/H)^{∨}}(⟨W, 1⟩_{G} − ⟨W, χ⟩_{G}) + ∑_{χ∈H^{∨}}(⟨W, 1⟩_{H} − ⟨W, χ⟩_{H})

for any subgroup H of G; here ⟨W, χ⟩_{G} (resp. ⟨W, χ⟩_{H}) is the multiplicity of the character χ in W regarded as a representation of G (resp. H). To prove this, we compute

∑_{χ∈G^{∨}}( ⟨W, 1⟩_{G} − ⟨W, χ⟩_{G})
= #G · ⟨W, 1⟩_{G} − ⟨W, Ind_{1}^{G} 1⟩_{G}
= #G · ⟨W, 1⟩_{G} − #H · ⟨W, Ind_{H}^{G} 1⟩_{G} + #H · ⟨W, Ind_{H}^{G} 1⟩_{G} − ⟨W, Ind_{1}^{G} 1⟩_{G}
= #H · ∑_{χ∈(G/H)^{∨}}(⟨W, 1⟩_{G} − ⟨W, χ⟩_{G}) + ∑_{χ∈H^{∨}} (⟨W, Ind_{H}^{G} 1⟩_{G} − ⟨W, Ind_{H}^{G} χ⟩_{G})
= #H · ∑_{χ∈(G/H)^{∨}}(⟨W, 1⟩_{G} − ⟨W, χ⟩_{G}) + ∑_{χ∈H^{∨}}(⟨W, 1⟩_{H} − ⟨W, χ⟩_{H})

by Frobenius reciprocity. □

2.2. Global preliminaries. Fix a number field F; for simplicity we assume that F is either totally real or totally imaginary. Fix also an odd prime p and a finite extension K of Q_{p}; we write O for the ring of integers of K, π for a fixed choice of uniformizer of O, and k = O/π for the residue field of O. Let T be a nearly ordinary Galois representation over F with coefficients in O; that is, T is a free O-module of some rank n endowed with an O-linear action of the absolute Galois group G_{F}, together with a choice for each place v of F dividing p of a complete flag

0 = T_{v}^{0} ⊂ T_{v}^{1} ⊂ ⋯ ⊂ T_{v}^{n} = T

stable under the action of the decomposition group G_{v} ⊆ G_{F} of v. We make the following assumptions on T:

(1) For each place v dividing p we have

(T_{v}^{i}/T_{v}^{i−1}) ⊗ k ≅ (T_{v}^{j}/T_{v}^{j−1}) ⊗ k

as k[G_{v}]-modules for all i ≠ j;

(2) If F is totally real, then rank T^{c_{v}=1} is independent of the archimedean place v (here c_{v} is a complex conjugation at v);

(3) If F is totally imaginary, then n is even.

Remark 2.2. The conditions above are significantly more restrictive than are actually required to apply the results of [11]. As our main interest is in abelian (and thus necessarily Galois) extensions of Q, we have chosen to include the assumptions (2) and (3) to simplify the exposition. The assumption (1) is also stronger than necessary: all that is actually needed is that the centralizer of T ⊗ k consists entirely of scalars and that 𝔤𝔩_{n}/𝔟_{v} has trivial adjoint G_{v}- invariants for all places v dividing p; here 𝔤𝔩_{n} denotes the p-adic Lie algebra of GL_{n} and 𝔟_{v} denotes the p-adic Lie algebra of the Borel subgroup associated to the complete flag at v. In particular, when T has rank 2, we may still allow the case that T ⊗ k has the form

( \begin{matrix}χ & ∗ \\ 0 & χ \end{matrix} )

so long as ∗ is non-trivial. (Equivalently, if T is associated to a modular form f, the required assumption is that f is p-distinguished.)

Set A = T ⊗_{O} K/O; it is a cofree O-module of corank n with an O-linear action of G_{F}. Let c equal the rank of T_{v}^{c_{v}=1} (resp. n/2) if F is totally real (resp. totally imaginary) and set

A_{v}^{cr} := im(T_{v}^{c} ⊗_{O} K ↪ T ⊗_{O} K ↠ A).

We define the Selmer group of A over the cyclotomic Z_{p}-extension F_{∞} of F by

Sel(F_{∞}, A) = ker (H^{1}(F_{∞}, A) → ( ⊕_{w∤p}H^{1}(F_{∞,w}, A)) × ( ⊕_{w|p}H^{1}(F_{∞,w}, A/A_{v}^{cr}))) .

The Selmer group Sel(F_{∞}, A) is naturally a module for the Iwasawa algebra Λ_{O} := O[[Gal(F_{∞}/F)]]. If Sel(F_{∞}, A) is Λ_{O}-cotorsion (that is, if the dual of Sel(F_{∞}, A) is a torsion Λ_{O}-module), then we write μ^{alg}(F_{∞}, A) and λ^{alg}(F_{∞}, A) for its Iwasawa invariants; in particular, μ^{alg}(F_{∞}, A) = 0 if and only if Sel(F_{∞}, A) is a cofinitely generated O-module, while λ^{alg}(F_{∞}, A) is the O-corank of Sel(F_{∞}, A).

Remark 2.3. In the case that T is in fact an ordinary Galois representation (meaning that the action of inertia on each T_{v}^{i}/T_{v}^{i−1} is by an integer power e_{i} (independent of v) of the cyclotomic character such that e_{1} > e_{2} > … > e_{n}), then our Selmer group Sel(F_{∞}, A) is simply the Selmer group in the sense of Greenberg of a twist of A; see [11, Section 1.3] for details.

2.3. Extensions. Let F′ be a finite Galois extension of F with degree equal to a power of p. We write F′_{∞} for the cyclotomic Z_{p}-extension of F′ and set G = Gal(F′_{∞}/F_{∞}). Note that T satisfies hypotheses (1)–(3) over F′ as well, so that we may define Sel(F′_{∞}, A) analogously to Sel(F_{∞}, A). (For (1) this follows from the fact that G_{v} acts on (T_{v}^{i}/T_{v}^{i−1})⊗k by a character of prime-to-p order; for (2) and (3) it follows from the fact that p is assumed to be odd.)

Lemma 2.4. The restriction map

(2)  Sel(F_{∞}, A) → Sel(F′_{∞}, A)^{G}

has finite kernel and cokernel.

Proof. This is straightforward from the definitions and the fact that G is finite and A is cofinitely generated; see [3, Lemma 3.3] for details. □

We can use Lemma 2.4 to relate the μ-invariants of A over F_{∞} and F′_{∞}.

Corollary 2.5. If Sel(F_{∞}, A) is Λ-cotorsion with μ^{alg}(F_{∞}, A) = 0, then Sel(F′_{∞}, A) is Λ-cotorsion with μ^{alg}(F′_{∞}, A) = 0.

Proof. This is a straightforward argument using Lemma 2.4 and Nakayama’s lemma for compact local rings; see [3, Corollary 3.4] for details. □

Fix a finite extension K′ of K containing all [F′ : F]^{th} roots of unity. Consider a character χ : G → O′^{×} taking values in the ring of integers O′ of K′; note that χ is necessarily even since [F′ : F] is odd. We set

A_{χ} = A ⊗_{O} O′(χ)

where O′(χ) is a free O′-module of rank one with G_{F_{∞}}-action given by χ. If we give A_{χ} the induced complete flags at places dividing p, then A_{χ} satisfies hypotheses (1)–(3) and we have

A_{χ,v}^{cr} = A_{v}^{cr} ⊗_{O} O′(χ) ⊆ A_{χ}

for each place v dividing p. We write Sel(F_{∞}, A_{χ}) for the corresponding Selmer group, regarded as a Λ_{O′}-module; in particular, by λ^{alg}(F_{∞}, A_{χ}) we mean the O′-corank of Sel(F_{∞}, A_{χ}), rather than the O-corank. We write G^{∨} for the set of all characters χ : G → O′^{×}.

Proposition 2.6. Assume that Sel(F_{∞}, A) is Λ-cotorsion with μ^{alg}(F_{∞}, A) = 0. If G is an abelian group, then there is a natural map

⊕_{χ∈G^{∨}} Sel(F_{∞}, A_{χ}) → Sel(F′_{∞}, A) ⊗_{O} O′

with finite kernel and cokernel.

Proof. First note that as O′[[G_{F′}]]-modules we have

A ⊗_{O} O′ ≅ A_{χ}

from which it easily follows that

(3)  (Sel(F′_{∞}, A) ⊗_{O} O′(χ))^{G} = Sel(F′_{∞}, A_{χ})^{G}.

Also, for any cofinitely generated O[G]-module S, the natural map

(4)  ⊕_{χ∈G^{∨}} (S ⊗ O′(χ))^{G} → S ⊗ O′

has finite kernel and cokernel. Since we are assuming that μ^{alg}(F_{∞}, A) = 0, we may take S = Sel(F′_{∞}, A) in (4); combining this with (3) yields a map

⊕_{χ∈G^{∨}} (Sel(F′_{∞}, A_{χ}))^{G} → Sel(F′_{∞}, A_{χ}) ⊗ O′

with finite kernel and cokernel. Now applying Lemma 2.4 for each twist A_{χ}, we obtain our proposition. □

As an immediate corollary, we have the following.

Corollary 2.7. If Sel(F_{∞}, A) is Λ-cotorsion with μ^{alg}(F_{∞}, A) = 0, then each group Sel(F_{∞}, A_{χ}) is Λ_{O′}-cotorsion with μ^{alg}(F_{∞}, A_{χ}) = 0. Moreover, if G is abelian, then

λ^{alg}(F′_{∞}, A) = ∑_{χ∈G^{∨}} λ^{alg}(F_{∞}, A_{χ}).

2.4. Algebraic transition formula. We continue with the notation of the previous section. We write R(F′_{∞}/F_{∞}) for the set of prime-to-p places of F′_{∞} which are ramified in F′_{∞}/F_{∞}. For a place w′ ∈ R(F′_{∞}/F_{∞}), we write w for its restriction to F_{∞}.

Theorem 2.8. Let F′/F be a finite Galois p-extension with Galois group G which is unramified at all places dividing p. Let T be a nearly ordinary Galois representation over F with coefficients in O satisfying (1)–(3). Set A = T ⊗ K/O and assume that:

(4) H^{0}(F, A[π]) = H^{0}(F, Hom(A[π], μ_{p})) = 0;

(5) H^{0}(I_{v}, A/A_{v}^{cr}) is O-divisible for all v dividing p.

If Sel(F_{∞}, A) is Λ-cotorsion with μ^{alg}(F_{∞}, A) = 0, then Sel(F′_{∞}, A) is Λcotorsion with μ^{alg}(F′_{∞}, A) = 0. Moreover, in this case,

λ^{alg}(F′_{∞}, A) = [F′_{∞} : F_{∞}] · λ^{alg}(F_{∞}, A) + ∑_{w′∈R(F′_{∞}/F_{∞})}m(F′_{∞,w′}/F_{∞,w}, V)

with V = T ⊗ K and m(F′_{∞,w′}/F_{∞,w}, V) as in Section 2.1.

Note that m(F′_{∞,w′}/F_{∞,w}, V) in fact depends only on w and not on w′. The hypotheses (4) and (5) are needed to apply the results of [11]; they will not otherwise appear in the proof below. We note that the assumption that F′/F is unramified at p is primarily needed to assure that the condition (5) holds for twists of A as well. Since p-groups are solvable and the only simple p-group is cyclic, the next lemma shows that it suffices to consider the case of Z/pZ-extensions.

Lemma 2.9. Let F′′/F be a Galois p-extension of number fields and let F′ be an intermediate extension which is Galois over F. Let T be as above. If Theorem 2.8 holds for T with respect to any two of the three field extensions F′′/F′, F′/F and F′′/F, then it holds for T with respect to the third extension.

Proof. This is clear from Corollary 2.5 except for the λ-invariant formula. Substituting the formula for λ(F′_{∞}, A) in terms of λ(F_{∞}, A) into the formula for λ(F′′_{∞}, A) in terms of λ(F′_{∞}, A), one finds that it suffices to show that

∑_{w′′∈R(F′′_{∞}/F_{∞})} m(F′′_{∞,w′′}/F_{∞,w}, V) =

[F′′_{∞} : F′_{∞}] · ∑_{w′∈R(F′_{∞}/F_{∞})} m(F′_{∞,w′}/F_{∞,w}, V)

+ ∑_{w′′∈R(F′′_{∞}/F′_{∞})} m(F′′_{∞,w′′}/F′_{∞,w′}, V).

This formula follows upon summing the formula of Lemma 2.1 over all w′′ ∈ R(F′′_{∞}/F_{∞}) and using the two facts:

• [F′′_{∞} : F′_{∞}]/[F′′_{∞,w′′} : F′_{∞,w′}] equals the number of places of F′′_{∞} lying over w′ (since the residue field of F_{∞,w} has no p-extensions);

• m(F′′_{∞,w′′}/F′_{∞,w′}, V) = 0 for any w′′ ∈ R(F′′_{∞}/F_{∞}) − R(F′′_{∞}/F′_{∞}).

□

Proof of Theorem 2.8. By Lemma 2.9 and the preceding remark, we may assume that F′_{∞}/F_{∞} is a cyclic extension of degree p. The fact that Sel(F′_{∞}, A) is cotorsion with trivial μ-invariant is simply Corollary 2.5. Furthermore, by Corollary 2.7, we have

λ^{alg}(F′_{∞}, A) = ∑_{χ∈G^{∨}} λ^{alg}(F_{∞}, A_{χ}).

For χ ∈ G^{∨}, note that χ is trivial modulo a uniformizer π′ of O′ as it takes values in μ_{p}. In particular, the residual representations A_{χ}[π′] and A[π] are isomorphic. Under the hypotheses (1)–(5), the result [11, Theorem 1] gives a precise formula for the relation between λ-invariants of congruent Galois representations. In the present case it takes the form:

λ^{alg}(F_{∞}, A_{χ}) = λ^{alg}(F_{∞}, A) + ∑_{w′∤p}(m_{F_{∞,w}}(V ⊗ ω^{−1}) − m_{F_{∞,w}}(V_{χ} ⊗ ω^{−1}))

where the sum is over all prime-to-p places w′ of F′_{∞}, w denotes the place of F_{∞} lying under w′ and ω is the mod p cyclotomic character. The only non-zero terms in this sum are those for which w′ is ramified in F′_{∞}/F_{∞}. For any such w′, we have μ_{p} ⊆ F_{∞,w} by local class field theory so that ω is in fact trivial at w; thus

λ^{alg}(F_{∞}, A_{χ}) = λ^{alg}(F_{∞}, A) + ∑_{w′∈R(F′_{∞}/F_{∞})}(m_{F_{∞,w}}(V) − m_{F_{∞,w}}(V_{χ})).

Summing over all χ ∈ G^{∨} then yields

λ^{alg}(F′_{∞}, A) = [F′_{∞} : F_{∞}] · λ^{alg}(F_{∞}, A) + ∑_{w′∈R(F′_{∞}/F_{∞})}m(F′_{∞,w′}/F_{∞,w}, V)

which completes the proof. □

3. Analytic invariants

3.1. Definitions. Let f = ∑ a_{n}q^{n} be a modular eigenform of weight k ≥ 2, level N and character ε. Let K denote the finite extension of Q_{p} generated by the Fourier coefficients of f (under some fixed embedding Q̄ ↪ Q̄_{p}), let O denote the ring of integers of K and let k denote the residue field of O. Let V_{f} denote a two-dimensional K-vector space with Galois action associated to f in the usual way; thus the characteristic polynomial of a Frobenius element at a prime ℓ ∤ Np is

x^{2} − a_{ℓ}x + ℓ^{k−1}ε(ℓ).

Fix a Galois stable O-lattice T_{f} in V_{f}. We assume that T_{f} ⊗ k is an irreducible Galois representation; in this case T_{f} is uniquely determined up to scaling. Set A_{f} = T_{f} ⊗ K/O. Assuming that f is p-ordinary (in the sense that a_{p} is relatively prime to p) and fixing a canonical period for f, one can associate to f a p-adic Lfunction L_{p}(Q_{∞}/Q, f) which lies in Λ_{O}. This is well-defined up to a p-adic unit (depending upon the choice of a canonical period) and thus has well-defined Iwasawa invariants. Let F/Q be a finite abelian extension and let F_{∞} denote the cyclotomic Z_{p}- extension of F. For a character χ of Gal(F/Q), we denote by f_{χ} the modular eigenform ∑ a_{n}χ(n)q^{n} obtained from f by twisting by χ (viewed as a Dirichlet character). If f is p-ordinary and F/Q is unramified at p, then f_{χ} is again p-ordinary and we define

L_{p}(F_{∞}/F, f) = ∏_{χ∈Gal(F/Q)^{∨}} L_{p}(Q_{∞}/Q, f_{χ}).

If F/Q is ramified at p, it is still possible to define L_{p}(F_{∞}/F, f); see [7, pg. 5], for example. If F_{1} and F_{2} are two distinct number fields whose cyclotomic Z_{p}-extensions agree, the corresponding p-adic L-functions of f over F_{1} and F_{2} need not agree. However, it is easy to check that the λ-invariants of these two power series are equal while their μ-invariants differ by a factor of a power of p. As we are only interested in the case of vanishing μ-invariants, we will abuse notation somewhat and simply denote the Iwasawa invariants of L_{p}(F_{∞}/F, f) by μ^{an}(F_{∞}, f) and λ^{an}(F_{∞}, f).

3.2. Analytic transition formula. Let F/Q be a finite abelian extension of Q and let F′ be a finite p-extension of F such that F′/Q is abelian. As always, let F_{∞} and F′_{∞} denote the cyclotomic Z_{p}-extensions of F and F′. As before, we write R(F′_{∞}/F_{∞}) for the set of prime-to-p places of F′_{∞} which are ramified in F′_{∞}/F_{∞}.

Theorem 3.1. Let f be a p-ordinary modular form such that T_{f} ⊗ k is irreducible and p-distinguished. If μ^{an}(F_{∞}, f) = 0, then μ^{an}(F′_{∞}, f) = 0. Moreover, if this is the case, then

λ^{an}(F′_{∞}, f) = [F′_{∞} : F_{∞}] · λ^{an}(F_{∞}, f) + ∑_{w′∈R(F′_{∞}/F_{∞})}m(F′_{∞,w′}/F_{∞,w}, V_{f}).

Proof. By Lemma 2.9, we may assume [F : Q] is prime-to-p. Indeed, let F_{0} be the maximal subfield of F of prime-to-p degree over Q. By Lemma 2.9, knowledge of the theorem for the two extensions F′/F_{0} and F/F_{0} would then imply it for F′/F as well. Furthermore, replacing F (resp. F′) by the maximal tamely ramified subextension of F_{∞} (resp. F′_{∞}), we may assume that every character of Gal(F/Q) and Gal(F′/Q) is the product of a power of the mod p cyclotomic character and a character unramified at p. After making these reductions, we let M denote the (unique) p-extension of Q inside of F′ such that MF = F′. Set G = Gal(F/Q) and H = Gal(M/Q), so that Gal(F′/Q) ≅ G × H. We have

(5)  μ^{an}(F_{∞}, f) = ∑_{ψ∈Gal(F/Q)^{∨}} μ^{an}(Q_{∞}, f_{ψ})

and

(6) μ^{an}(F′_{∞}, f) = ∑_{ψ∈Gal(F′/Q)^{∨}} μ^{an}(Q_{∞}, f_{ψ}) = ∑_{ψ∈G^{∨}} ∑_{χ∈H^{∨}} μ^{an}(Q_{∞}, f_{ψχ}).

Since we are assuming that μ^{an}(F_{∞}, f) = 0 and since these μ-invariants are nonnegative, from (5) it follows that μ^{an}(Q_{∞}, f_{ψ}) = 0 for each ψ ∈ Gal(F/Q)^{∨}. Fix ψ ∈ G^{∨}. For any χ ∈ H^{∨}, ψχ is congruent to ψ modulo any prime over p and thus f_{χ} and f_{ψχ} are congruent modulo any prime over p. Then, since μ^{an}(Q_{∞}, f_{ψ}) = 0, by [1, Theorem 3.7.5] it follows that μ^{an}(Q_{∞}, f_{ψχ}) = 0 for each χ ∈ H^{∨}. (Note that the results of [1] apply to twists of p-ordinary forms by powers of the mod p cyclotomic character; this is why the reduction to the tamely ramified case is necessary for this argument.) Therefore, by (6) we have that μ^{an}(F′_{∞}, f) = 0 proving the first part of the theorem. For λ-invariants, we again have

λ^{an}(F_{∞}, f) = ∑_{ψ∈Gal(F/Q)^{∨}} λ^{an}(Q_{∞}, f_{ψ}).

and

(7)  λ^{an}(F′_{∞}, f) = ∑_{ψ∈G^{∨}} ∑_{χ∈H^{∨}} λ^{an}(Q_{∞}, f_{ψχ}).

By [1, Theorem 3.7.7] the congruence between f_{χ} and f_{ψχ} implies that

λ^{an}(Q_{∞}, f_{ψχ}) − λ^{an}(Q_{∞}, f_{ψ}) =

∑_{v} (m_{Q_{∞,v}}(V_{f_{ψχ}} ⊗ ω^{−1}) − m_{Q_{∞,v}}(V_{f_{ψ}} ⊗ ω^{−1}))

where the sum is over all places v of Q_{∞} at which χ is ramified. (Note that in [1] the sum extends over all prime-to-p places; however, the terms are trivial unless χ is ramified at v. Also note that the mod p cyclotomic characters that appear are actually trivial since if Q_{∞,v} has a ramified Galois p-extensions for v ∤ p, then μ_{p} ⊆ Q_{∞,v}.) Combining this with (7) and the definition of m(M_{∞,v′}/Q_{∞,v}, V_{f_{ψ}}), we conclude that

λ^{an}(F′_{∞}, f) = ∑_{ψ∈G^{∨}}([F′_{∞} : F_{∞}] · λ^{an}(Q_{∞}, f_{ψ})+

∑_{v′∈R(M_{∞}/Q_{∞})} m(M_{∞,v′}/Q_{∞,v}, V_{f_{ψ}}))

= [F′_{∞} : F_{∞}] · λ^{an}(F_{∞}, f)+

∑_{v′∈R(M_{∞}/Q_{∞})} ∑_{ψ∈G^{∨}} m(M_{∞,v′}/Q_{∞,v}, V_{f_{ψ}})

= [F′_{∞} : F_{∞}] · λ^{an}(F_{∞}, f) + ∑_{v′∈R(M_{∞}/Q_{∞})} g_{v′}(F′_{∞}/M_{∞})·

m(M_{∞,v′}/Q_{∞,v}, Z[Gal(F_{∞,w}/Q_{∞,v})] ⊗ V_{f})

where g_{v′}(F′_{∞}/M_{∞}) denotes the number of places of F′_{∞} above the place v′ of M_{∞}. By Frobenius reciprocity,

m(M_{∞,v′}/Q_{∞,v}, Z[Gal(F_{∞,w}/Q_{∞,v})] ⊗ V_{f}) = m(F′_{∞,w′}/F_{∞,w}, V_{f})

where w′ is the unique place of F′_{∞} above v′ and w. It follows that

λ(F′_{∞}, f) = [F′_{∞} : F_{∞}] · λ^{an}(F_{∞}, f) + ∑_{w′∈R(F′_{∞}/F_{∞})} m(F′_{∞,w′}/F_{∞,w}, V_{f})

as desired. □

4. Additional Results

4.1. Hilbert modular forms. We illustrate our results in the case of the two-dimensional representation V_{f} associated to a Hilbert modular eigenform f over a totally real field F. Although in principle our analytic results should remain true in this context, we focus on the less conjectural algebraic picture. Fix a G_{F} -stable lattice T_{f} ⊆ V_{f} and let A_{f} = T_{f} ⊗ K/O. Let F′ be a finite Galois p-extension of F unramified at all places dividing p; for simplicity we assume also that F′ is linearly disjoint from F_{∞}. Let v be a place of F not dividing p and fix a place v′ of F′ lying over v. For a character φ of G_{v}, we define

h(φ) = \begin{cases}−1 & φ ramified, φ|_{G_{v′}} unramified, and φ ≡ 1 mod π; \\ 0 & φ ≢ 1 mod π or φ|_{G_{v′}} ramified; \\ e_{v}(F′/F) − 1 & φ unramified and φ ≡ 1 mod π \end{cases}

where e_{v}(F′/F) denotes the ramification index of v in F′/F and G_{v′} is the decomposition group at v′. Set

h_{v}(f) = \begin{cases}h(φ_{1}) + h(φ_{2}) & f principal series with characters φ_{1}, φ_{2} at v; \\ h(φ) & f special with character φ at v; \\ 0 & f supercuspidal or extraordinary at v. \end{cases}

For example, if f is unramified principal series at v with Frobenius characteristic polynomial

x^{2} − a_{v}x + c_{v},

then

h_{v}(f) = \begin{cases}2(e_{v}(F′/F) − 1) & a_{v} ≡ 2, c_{v} ≡ 1 mod π \\ e_{v}(F′/F) − 1 & a_{v} ≡ c_{v} + 1 ≢ 2 mod π \\ 0 & otherwise. \end{cases}

Theorem 4.1. Assume that f is ordinary (in the sense that for each place v dividing p the Galois representation V_{f} has a unique one-dimensional quotient unramified at v) and that

H^{0}(F, A_{f}[π]) = H^{0}(F, Hom(A_{f}[π], μ_{p})) = 0.

If Sel(F_{∞}, A_{f}) is Λ-cotorsion with μ^{alg}(F_{∞}, A_{f}) = 0, then also Sel(F′_{∞}, A_{f}) is Λ-cotorsion with μ^{alg}(F′_{∞}, A_{f}) = 0 and

λ^{alg}(F′_{∞}, A) = [F′_{∞} : F_{∞}] · λ^{alg}(F_{∞}, A) + ∑_{v} g_{v}(F′_{∞}/F) · h_{v}(f);

here the sum is over the prime-to-p places of F ramified in F′_{∞} and g_{v}(F′_{∞}/F) denotes the number of places of F′_{∞} lying over such a v.

Proof. Fix a place v of F not dividing p and let w denote a place of F_{∞} lying over v. Since there are exactly g_{v}(F_{∞}/F) such places, by Theorem 2.8 it suffices to prove that

(8) h_{v}(f) = m(F′_{∞,w′}/F_{∞,w}, V_{f}) :=

∑_{χ∈Gal(F′_{∞,w′} /F_{∞,w})^{∨}} (m_{F_{∞,w}}(V_{f}) − m_{F_{∞,w}}(V_{f,χ})) .

This is a straightforward case analysis. We will discuss the case that V_{f} is special associated to a character φ at v; the other cases are similar. In the special case, we have

V_{f,χ}|_{I_{F∞,w}} = \begin{cases}K′(χφ) & χφ|_{G_{F∞,w}} unramified; \\ 0 & χφ|_{G_{F∞,w}} ramified. \end{cases}

Since an unramified character has trivial restriction to G_{F_{∞,w}} if and only if it has trivial reduction modulo π, it follows that

m_{F_{∞,w}}(V_{f,χ}) = \begin{cases}1 & φ ≡ 1 mod π and χφ|_{G_{F∞,w}} unramified; \\ 0 & otherwise. \end{cases}

In particular, the sum in (8) is zero if φ ≢ 1 mod π or if φ is ramified when restricted to G_{F′_{∞,w′}} (as then χφ is ramified for all χ ∈ G_{v}^{∨}). If φ ≡ 1 mod π and φ itself is unramified, then m_{F_{∞,w}}(V_{f}) = 1 while m_{F_{∞,w}}(V_{f,χ}) = 0 for χ ≠ 1, so that the sum in (8) is [F′_{∞,w′} : F_{∞,w}]−1 = e_{v}(F′/F)−1, as desired. Finally, if φ ≡ 1 mod π and φ is ramified but becomes unramified when restricted to G_{v′}, then m_{F_{∞,w}}(V_{f}) = 0, while m_{F_{∞,w}}(V_{f,χ}) = 1 for a unique χ, so that the sum is −1. □

Suppose finally that f is in fact the Hilbert modular form associated to an elliptic curve E over F. The only principal series which occur are unramified and we have c_{v} ≡ 1 (mod π) (since the determinant of V_{f} is cyclotomic and F_{∞} has a p-extension (namely, F′_{∞}) ramified at v), so that

h_{v}(f) ≠ 0 ⇔ a_{v} ≡ 2 ⇔ E(F_{v}) has a point of order p

in which case h_{v}(f) = 2(e_{v}(F′/F)−1). The only characters which may occur in a special constituent are trivial or unramified quadratic, and we have h_{v}(f) = e_{v}(F′/F) − 1 or 0 respectively. Thus Theorem 4.1 recovers [3, Theorem 3.1] in this case.

4.2. The main conjecture. Let f be a p-ordinary elliptic modular eigenform of weight at least two and arbitrary level with associated Galois representation V_{f}. Let F be a finite abelian extension of Q with cyclotomic Z_{p}-extension F_{∞}. Recall that the p-adic Iwasawa main conjecture for f over F asserts that the Selmer group Sel(F_{∞}, A_{f}) is Λ-cotorsion and that the characteristic ideal of its dual is generated by the p-adic L-function L_{p}(F_{∞}, f). In fact, when the residual representation of V_{f} is absolutely irreducible, it is known by work of Kato that Sel(F_{∞}, A_{f}) is indeed Λ-cotorsion and that L_{p}(F_{∞}, f) is an element of the characteristic ideal of Sel(F_{∞}, A_{f}). In particular, this reduces the verification of the main conjecture for f over F to the equality of the algebraic and analytic Iwasawa invariants of f over F. The identical transition formulae in Theorems 2.8 and 3.1 thus yield the following immediate application to the main conjecture.

Theorem 4.2. Let F′/F be a finite p-extension with F′ abelian over Q. If the residual representation of V_{f} is absolutely irreducible and p-distinguished, then the main conjecture holds for f over F with μ(F_{∞}, f) = 0 if and only if it holds for f over F′ with μ(F′_{∞}, f) = 0 .

For an example of Theorem 4.2, consider the eigenform

Δ = q ∏_{n≥1}(1 − q^{n})^{24}

of weight 12 and level 1. We take p = 11. It is well known that Δ is congruent modulo 11 to the newform associated to the elliptic curve X_{0}(11). The 11-adic main conjecture is known for X_{0}(11) over Q; it has trivial μ-invariant and λinvariant equal to 1 (see, for instance, [1, Example 5.3.1]. We should be clear here that the non-triviality of λ in this case corresponds to a trivial zero of the p-adic L-function; we are using the Greenberg Selmer group which does account for the trivial zero.) It follows from [1] that the 11-adic main conjecture also holds for Δ over Q, again with trivial μ-invariant and λ-invariant equal to 1. Theorem 4.2 thus allows us to conclude that the main conjecture holds for Δ over any abelian 11-extension of Q. For a specific example, consider F = Q(ζ_{23})^{+}; it is a cyclic 11-extension of Q. We can easily use Theorem 4.1 to compute its λ-invariant: using that τ(23) = 18643272 one finds that h_{23}(Δ) = 0, so that λ(Q(ζ_{23})^{+}, Δ) = 11. For a more interesting example, take F to be the unique subfield of Q(ζ_{1123}) which is cyclic of order 11 over Q. In this case we have

τ(1123) ≡ 2 (mod 11)

so that we have h_{1123}(Δ) = 20. Thus, in this case, Theorem 4.1 shows that λ(F, Δ) = 31.

4.3. The supersingular case. As mentioned in the introduction, the underlying principle of this paper is that the existence of a formula relating the λ-invariants of congruent Galois representations should imply a Kida-type formula for these invariants. We illustrate this now in the case of modular forms of weight two that are supersingular at p. Let f be an eigenform of weight 2 and level N with Fourier coefficients in K some finite extension of Q_{p}. Assume further than p ∤ N and that a_{p}(f) is not a p-adic unit. In [8], Perrin-Riou associates to f a pair of algebraic and analytic μ-invariants over Q_{∞} which we denote by μ_{±}^{⋆}(Q_{∞}, f). (Here ⋆ denotes either “alg” or “an” for algebraic and analytic respectively.) Moreover, when μ_{+}^{⋆}(Q_{∞}, f) = μ_{−}^{⋆}(Q_{∞}, f) or when a_{p}(f) = 0, she also defines corresponding λ-invariants λ_{±}^{⋆}(Q_{∞}, f). When a_{p}(f) = 0 these invariants coincide with the Iwasawa invariants of [6] and [9]. We also note that in [8] only the case of elliptic curves is treated, but the methods used there generalize to weight two modular forms. We extend the definition of these invariants to the cyclotomic Z_{p}-extension of an unramified abelian extension F of Q. We define

μ_{±}^{⋆}(F_{∞}, f) = ∑_{ψ∈Gal(F/Q)^{∨}} μ_{±}^{⋆}(Q_{∞}, f_{ψ}) and λ_{±}^{⋆}(F_{∞}, f) = ∑_{ψ∈Gal(F/Q)^{∨}} λ_{±}^{⋆}(Q_{∞}, f_{ψ})

for ⋆ ∈ {alg, an}. The following transition formula follows from the congruence results of [2].

Theorem 4.3. Let f be as above and consider a p-extension of number fields F′/F with F′/Q unramified at p. If μ_{±}^{⋆}(F_{∞}, f) = 0, then μ_{±}^{⋆}(F′_{∞}, f) = 0. Moreover, if this is the case, then

λ_{±}^{⋆}(F′_{∞}, f) = [F′_{∞} : F_{∞}] · λ_{±}^{⋆}(F_{∞}, f) + ∑_{w′∈R(F′_{∞}/F_{∞})}m(F′_{∞,w′}/F_{∞,w}, V_{f}).

In particular, if the main conjecture is true for f over F (with μ_{±}^{⋆}(F_{∞}, f) = 0), then the main conjecture is true for f over F′ (with μ_{±}^{⋆}(F′_{∞}, f) = 0).

Proof. The proof of this theorem proceeds along the lines of the proof of Theorem 3.1 replacing the appeals to the results of [1, 11] to the results of [2]. The main result of [2] is a formula relating the λ_{±}^{⋆}-invariants of congruent supersingular weight two modular forms. This formula has the same shape as the formulas that appear in [1] and [11] which allows for the proof to proceed nearly verbatim. □

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Robert Pollack Department of Mathematics Boston University, Boston, MA USA rpollack@math.bu.edu Tom Weston Dept. of Mathematics, University of Massachusetts, Amherst, MA USA weston@math.umass.edu