CM PERIODS AND THE ANTICYCLOTOMIC MAIN CONJECTURE

HARUZO HIDA

We describe a close relation between an integral version of Shimura’s period conjecture (applied to CM forms) and the anticyclotomic main conjecture.

1. Minimal automorphic forms

In order to define optimally integral automorphic period for GL(2), the use of new forms is not often effective. We describe the definition of automorphic forms of minimal level in order to state the integral period conjecture.

We always assume our automorphic representation of GL(2) is holomorphic at archimedean places and principal at henselian places. We call such an automorphic representation globally principal.

Let F be a totally real field with integer ring O, and F_{𝔩} be the completion at a prime ideal 𝔩. We take an admissible irreducible representation π = π(ε_{1}, ε_{2}) of GL_{2}(F_{𝔩}) in the principal series; so, ε_{j} : F_{𝔩}^{×} → ℂ are quasi characters, and ε_{+} = ε_{1}ε_{2} is the central character of π. Write 𝔩^{e} for the conductor of π; so, it is the product of the conductors of ε_{j}. The new vector v ∈ π(ε_{1}, ε_{2}) is characterized by the property that π(u)v = ε_{+}(d)v for u = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(O_{𝔩}) with c ∈ 𝔩^{e}. We write U_{0}(𝔩^{e})_{𝔩} for the group of such matrices in GL_{2}(O_{𝔩}) whose reduction modulo 𝔩^{e} is upper triangular. Since π(ε_{1}, ε_{2}) = ε_{1} ⊗ π(1, ε^{−}) for ε^{−} = ε_{1}^{−1}ε_{2}, for the new vector v^{−} ∈ π(1, ε^{−}), we have the minimal vector w = ε_{1} ⊗ v^{−} ∈ π, which is characterized by

π(u)w = ε_{1}(ad − bc)ε^{−}(d)w for u = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ U_{0}(𝔩^{e_{0}})_{𝔩}

for the conductor 𝔩^{e_{0}} of ε^{−}. Obviously e_{0} ≤ e, and u ↦ ε_{1}(ad − bc)ε^{−}(d) is a character of U_{0}(𝔩^{e_{0}}) which coincides with the classical Neben character ε^{−} on SL_{2}(F_{𝔩}). Since π(ε_{1}, ε_{2}) ≅ π(ε_{2}, ε_{1}), the minimal vector depends on the choice of order of ε_{j}, but the minimal level 𝔩^{e_{0}} is uniquely determined by π.

The minimal vectors can be defined globally. Let B be a quaternion algebra over F (including M_{2}(F)). We fix a maximal order O_{B} of B and assume

Ô_{B} = O_{B} ⊗_{ℤ} ℤ̂ ≅ M_{2}(Ô),

where ℤ̂ = ∏_{p} ℤ_{p} and Ô = O ⊗_{ℤ} ℤ̂ = ∏_{𝔩} O_{𝔩}. Let G = G_{B} = Res_{F/ℚ}B^{×}. To have an globally principal automorphic representation on G, we need to assume the above unramifiedness condition. We fix a regular weight κ = (κ_{1}, κ_{2}) ∈ ℤ[I]^{2} for I = Hom_{field}(F, \overline{ℚ}) (regularity ⇔ κ_{1} > κ_{2}). We assume that κ_{1} + κ_{2} = [κ] ∑_{σ} σ for an integer [κ]. Fix

a prime p ≥ 5 unramified in F/ℚ

and two embeddings i_{p} : \overline{ℚ} ↪ ℂ_{p} = ˆ \overline{ℚ}_{p} and i_{∞} : \overline{ℚ} ↪ ℂ. Then we may regard (κ_{1,σ}, κ_{2,σ}) as the archimedean or p–adic Hodge weight of a rank two motive. We write S_{κ}^{B}(ℂ) for the space of holomorphic cusp forms of “Neben type” ε on G(𝔸) whose Galois representation and rank two motive has Hodge type κ_{σ} at the p–adic or complex place corresponding to σ ∈ I. Here the Neben-typus ε is made of the following three data: ε_{j} : Ô^{×} → ℂ^{×} (j = 1, 2) is continuous characters and ε_{+} : F_{𝔸}^{×} → ℂ^{×} is a Hecke character with ε_{+}|Ô^{×} = ε_{1}ε_{2} and ε_{+}(x_{∞}) = N(x_{∞})^{1−[κ]} for x_{∞} ≫ 0. The level 𝔑 is the conductor of ε^{−} = ε_{1}^{−1}ε_{2}. A Hecke eigenform f ∈ S_{κ}^{B}(ℂ) is a minimal vector in the globally principal automorphic representation π generated by f, and f ∈ S_{κ}^{B}(ℂ) satisfies

A one-hour talk at Luminy, June 18 2002.

f(αxuz) = ε_{+}(z)ε(u)f(x) for z ∈ F_{𝔸}^{×}, u ∈ U_{0}(𝔑) = ∏_{𝔩} U_{0}(𝔩^{e(𝔩)})_{𝔩}

(for 𝔑 = ∏_{𝔩} 𝔩^{e(𝔩)}), where ε ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ε_{1}(ad − bc)ε^{−}(d_{𝔑}) for ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ U_{0}(𝔑). By the

Jacquet-Langlands correspondence, the same π^{(∞)} appears for all B, and we fix one such π^{(∞)}.

We take a discrete valuation ring W associated to i_{p} in \overline{ℚ} big enough so that it contains all Hecke eigenvalues in the space S_{κ}^{B}(ℂ). We write W for the p–adic completion of W. Let X_{0}^{B}(𝔑)(ℂ) be the quotient G(ℚ)\(G(𝔸^{(∞)}) × X)/F_{𝔸}^{×}U_{0}(𝔑) for the symmetric domain X of G(ℝ). Assume p ∤ 𝔑. As long as p is unramified in F/ℚ, we find the canonical model X_{0}^{B}(𝔑) over W and a vector bundle \underline{ω}^{κ}(ε)_{/W} such that H^{0}(X_{0}^{B}(𝔑)_{/ℂ}, \underline{ω}^{κ}(ε)) = S_{κ}^{B}(ℂ). We define the W–integral subspace

S_{κ}^{B}(W) = H^{0}(X_{0}^{B}(𝔑)_{/W}, \underline{ω}^{κ}(ε)).

Choosing a generator [y_{p}]^{κ_{2}} ∈ W of y_{p}^{κ_{2}}W, we check that the integralized Hecke operator 𝕋(y) = [y_{p}]^{−κ_{2}}T (y) for y ∈ Ô ∩ (F_{𝔸}^{(∞)})^{×} preserves S_{κ}^{B}(W). By our definition, S_{κ}^{B}(ℂ) is a semi-simple Hecke module. When we say f_{B} a minimal form, we mean a generator f_{B} over W of the eigenspace in S_{κ}^{B}(W) of π^{(∞)}. Fixing a Haar measure on G(𝔸) in a standard way (so that volume of U_{0}(𝔑) is equal to 1), we have an invariant measure in X_{0}^{B}(𝔑) and hence have the Petersson inner product (·, ·)_{𝔑} on S_{κ}^{B}(ℂ). It is easy to verify

the class (f_{B}, f_{B})_{𝔑} in ℂ^{×}/W^{×} is independent of the minimal form f_{B} in π.

2. Period conjecture

Since κ is regular, the space S_{κ}^{B}(A) can be related to the Betti (cuspidal) cohomology group H^{B}(A) = H_{cusp}^{r}(X_{0}^{B}(𝔑), L^{B}(κ, ε; A)) (r = dim X_{0}^{B}(𝔑)) by the Eichler-Shimura isomorphism δ for a locally constant sheaf L^{B}(κ, ε; ℂ) over X_{0}^{B}(𝔑), whose fiber is a polynomial representation of G(ℝ): ⨂_{σ} (det^{κ_{2,σ}} ·Sym(ρ_{σ}^{⊗k_{σ}−2})) (k = κ_{1} − κ_{2} + I) for a p–integral inclusion ρ_{σ} : O_{B} ↪ M_{2}(W) times inducing σ on O.

Writing I(B) ⊂ I for the set of archimedean places splitting B. Then ℌ^{I(B)} (ℌ: upper half complex plane) gives the universal covering of X_{0}^{B}(𝔑). We let {±1}^{I} act on ℌ^{I(B)} (ignoring the component outside I(B)) so that −1 at σ acts by z_{σ} ↦ −\overline{z}_{σ}. We consider the eigenspace H^{B}(ℂ)[s] with eigenvalue s_{σ} = ±1. We have

δ_{s} : S_{κ}^{B}(ℂ) ≅ H^{B}(ℂ)[s].

We choose a generator ϕ_{f_{B},s} of the π^{(∞)}–eigenspace of H^{B}(W)[s] over W. Then we define the complex period Ω_{B}(f_{B}, s) by

δ_{s}(f_{B}) = (2π)^{κ_{1}(B)−κ_{2}(B)}Ω_{B}(f_{B}, s)ϕ_{f_{B},s}.

Here κ_{j}(B) is the projection of κ_{j} to ℤ[I(B)].

Conjecture 2.1. Suppose a_{p} ∈ W^{×} if f_{B}|𝕋(p) = a_{p}f_{B} (the ordinarity condition).

(1) We have a factorization Ω_{B}(f_{B}, s(B)) = ∏_{σ∈I(B)} P (π^{(∞)}, s_{σ}σ) for a factor P (π^{(∞)}, ±σ) ∈ ℂ^{×}/W^{×} independent of B with s = ∑_{σ} ±σ ∈ {±1}^{I}.

(2) For a finite totally real extension E/F with the base-change π_{E}^{(∞)} of π^{(∞)}, we have P (π_{E}^{(∞)}, ±σ) = P (π^{(∞)}, ±σ|_{F}) for σ : E ↪ \overline{ℚ} (independence of F).

Original Shimura’s conjecture is modulo \overline{ℚ}^{×} instead of “modulo W^{×}” and contains more specification than the two conditions above (see [Sh1] Conjecture 5.12 and also [Sh2], [Sh3]). The identity modulo \overline{ℚ}^{×} has been proven under some mild assumptions by Yoshida (see [Y]).

We can verify that Ω_{B}(f_{B}, s(B)) = Ω_{B}(f_{B}, −s(B)) mod W^{×}, when f has complex multiplication. This implies P (π^{(∞)}, σ) = P (π^{(∞)}, −σ) mod W^{×}.

3. CM periods and Iwasawa theory

We would like to relate Conjecture 2.1 with the anticyclotomic main conjecture of the Iwasawa theory. We now assume that π is an automorphic induction of a Hecke character λ of a p–ordinary CM type Σ of a CM field M/F. Here a CM type is ordinary if we have an abelian variety over W of CM type (M, Σ) with ordinary good reduction modulo 𝔪_{W}. Identifying Σ with I by the restriction map, λ is of type Σ if λ(x_{∞}) = x_{∞}^{−κ_{1}−κ_{2}c} for a regular weight κ (and complex conjugation c). Defining λ^{−}(x) = λ(x^{c−1}), π(f) is globally principal if and only if the conductor ℭ(λ^{−}) is a product of primes split in M/F (we call such λ^{−} split), which we assume.

Let H be the W–subalgebra of End_{W}(S_{κ}^{B}(W)) generated over W by Hecke operators 𝕋(y) for all y. This H is independent of B (by Jacquet-Langlands-Shimizu correspondence). Let T be the local ring of H of π^{(∞)} and define

H[s(B), T] := T (H^{B}(W)[s(B)]) and S[B, T] := T (S_{κ}^{B}(W)) .

We assume solution of Eichler’s basis problem modulo p for a given s and B, that is,

(E1) H[s(B), T] ≅ T as H–modules; (E2) S[B, T] ≅ T as T –modules.

Under the following two conditions (tw1-2), the assumption (E1) is valid when r ≤ 1 (as proven by K. Fujiwara for r ≤ 1 by the method of Taylor-Wiles), and the solution of the anticyclotomic main conjecture actually implies (E1) when r = [F : ℚ]. As for (E2), it is valid (without conditions on r) under (tw1-2) (again applying Fujiwara’s techniques to coherent cohomology). Here are the two conditions (tw1-2):

(tw1) λ^{−} mod 𝔪_{W} is non-trivial on M_{𝔭}^{×} for all prime factors 𝔭|p; (tw2) λ^{−} ∘ N_{M[\sqrt{p^{∗}}]/M} mod 𝔪_{W} is non-trivial for p^{∗} = (−1)^{(p−1)/2}p.

Since f_{B} is minimal, we can split T ⊗_{ℤ} ℚ = (W ⊗_{ℤ} ℚ) ⊕ X as an algebra direct sum. Writing X for the image of T in X, we find W ⊗_{𝕋} X ≅ W/η(λ)W and

(3.1) η(λ) = \frac{(2π)^{2(κ_{1}(B)−κ_{2}(B))}(f_{B}, f_{B})_{𝔑}}{Ω_{B}(f_{B}, s(B))^{2}} mod W^{×} independent of B.

We write λ̂ : Gal(\overline{M}/M) → \overline{ℚ}_{p}^{×} for the p–adic Galois character associated to λ. By Wiles-Taylor and Fujiwara, under (tw1-2), we have

(3.2) |η(λ)|_{p}^{−[W:ℤ_{p}]} = |Sel(Ad(Ind_{M}^{F} λ̂))| if [W : ℤ_{p}] < ∞.

The Selmer group Sel(Ad(Ind_{M}^{F} λ̂)) of the adjoint square Galois representation Ad(Ind_{M}^{F} λ̂) is a subgroup of Galois cohomology H^{1}(F, Ad(Ind_{M}^{F} λ̂) ⊗_{ℤ_{p}} ℚ_{p}/ℤ_{p}) defined by a standard procedure (for example, see [MFG] Sections 5.1-2). Since Ad(Ind_{M}^{F} λ̂) = ^{(\underline{M/F} )} ⊕ Ind_{M}^{F} λ̂^{−}, we find

(3.3) Sel(Ad(Ind_{M}^{F} λ)) ≅ Hom(Cl_{M}/Cl_{F} , W ⊗_{ℤ} (ℚ/ℤ)) ⊕ Sel(λ̂^{−}),

where Cl_{F} (resp. Cl_{M}) is the class group of F (resp. M).

Conjecture 3.1 (Main conjecture). Let φ ↦ L_{p}^{Σ}(φ) ∈ ℂ_{p} be the p–adic Hecke L– function of type Σ defined on continuous Galois characters φ : Gal(\overline{M}/M) → W^{×}. For each split Hecke character λ of M of CM type Σ, we have

η(λ) = (h(M)/h(F)) · L_{p}^{Σ}(λ̂^{−}) mod W^{×}.

This statement is equivalent to the standard formulation of the anticyclotomic main conjecture (as in [HT]), although it is highly non-trivial to prove the equivalence. We have stated the conjecture in this form, since it is then easier to see its relevance to the period conjecture.

4. CM Periods

Let R be the integer ring of M. Then we may normalize the abelian variety X_{Σ}(R) of CM type Σ so that H_{1}(X_{Σ}(R)(ℂ), ℤ) = Rγ. This abelian variety extends to an abelian scheme over W (for sufficiently large W) by the theory of complex multiplication (of Shimura-Taniyama-Weil) and a theorem of Serre-Tate on good reduction. If ℭ(λ^{−}) is prime to p, up to units in W,

\frac{(2π)^{2(κ_{1}−κ_{2})}(f, f)}{Ω(Σ)^{2(κ_{1}−κ_{2})}} = (h(M)/h(F))L_{p}^{−}(λ̂^{−}) for f = f_{M_{2}(F)}, (4.1)

where Ω(Σ) ∈ (ℂ^{×}/W^{×})^{Σ} is the Néron period of X_{Σ}(R): Ω(σ, Σ) = ∫_{γ} ω_{σ} for the σ– component ω_{σ} of a differential ω generating H^{0}(X_{Σ}(R), Ω_{X_{Σ}(R)/W}) over O ⊗_{ℤ} W ≅ W^{Σ}. This combined with (3.1) shows

\frac{(2π)^{2(κ_{1}−κ_{2})}(f, f)}{Ω(σ, Σ)^{2(κ_{1}−κ_{2})}} =^{main conj.} η(λ) = \frac{(2π)^{2(κ_{1}−κ_{2})}(f, f)}{Ω(f, s)^{2}}.

One might expect that the period relation would hold by putting P (π^{(∞)}, ±σ) = (Ω(σ, Σ)π)^{κ_{1,σ}−κ_{2,σ}}. But this is an easy trap. Shimura proved a further factorization

Ω(σ, Σ) = π ∏_{τ∈Σ} p_{M}(σ, τ) in ℂ^{×}/\overline{ℚ}^{×}

for a constant p_{M}(σ, τ) ∈ ℂ^{×}/\overline{ℚ}^{×} well defined for all σ, τ ∈ Σ ⊔ Σc = I_{M}. We extend this map to a bilinear form p_{M} : ℤ[I_{M}] × ℤ[I_{M}] → ℂ^{×}/\overline{ℚ}^{×}. Then p_{M} satisfies Shimura’s period relation ([ACM] 32.5) given as follows: (P1) π · p_{M}(σ, Σ) = Ω(σ, Σ) = ∫_{γ} ω_{σ}; (P2) p_{M}(ξc, η) = p_{M}(ξ, ηc) = p_{M}(ξ, η)^{−1}; (P3) p_{M}(ξ, Res_{M}^{L}(η)) = p_{L}(Inf_{M}^{L}(ξ), η) and p_{M}(Res_{M}^{L}(η), ξ) = p_{L}(η, Inf_{M}^{L}(ξ)) for

ξ ∈ ℤ[I_{M}] and η ∈ ℤ[I_{L}] for each CM extension L/M; (P4) p_{σ^{−1}(M)}(ξσ, ησ) = p_{M}(ξ, η). Shimura proved a statement ([ACM] 32.4) equivalent to (P1-4) in 1978, and a (more accessible) version of the proof can be found in [ACM] Chapter VII. This also follows from the proof of the absolute Hodge conjecture for abelian varieties later given by Deligne (see [DMOS] Chapter I).

If (P1-4) hold in ℂ^{×}/W^{×}, we may put P (π^{(∞)}, σ) = π^{(κ_{1}−κ_{2})_{σ}}p_{M}(κ_{1} − κ_{2}, σ), and we may easily verify the period conjecture under (P1-4) in ℂ^{×}/W^{×}, because

Ω(Σ)^{κ_{1}−κ_{2}} = π^{κ_{1}−κ_{2}} ∏_{σ∈Σ} p_{M}(κ_{1} − κ_{2}, σ). (4.2)

Thus, assuming (E1-2), if the p–integral period relation holds for P (λ_{F} , ±σ; A) = π^{κ_{1,σ}−κ_{2,σ}}p_{M}(κ_{1} − κ_{2}, σ), the main conjecture holds. Conversely, the main conjecture implies the integral factorization at least for B = M_{2}(F).

References

[ACM] G. Shimura, Abelian Varieties with Complex Multiplication and Modular Functions, Princeton University Press, 1998

[DMOS] P. Deligna, J. S. Milne, A. Ogus and K.-Y. Shih, Hodge Cycles, Motives, and Shimura Varieties, Lecture notes in Math. 900, Springer, 1982

[MFG] H. Hida, Modular Forms and Galois Cohomology, Cambridge University Press, 2000 [HT] H. Hida and J. Tilouine, On the anticyclotomic main conjecture for CM fields, Inven-

tiones Math. 117 (1994), 89–147

[Sh1] G. Shimura, Algebraic relations between critical values of zeta functions and inner products, Amer. J. Math. 104 (1983), 569–607

[Sh2] G. Shimura, On the critical values of certain Dirichlet series and the periods of automorphic forms, Inventiones Math. 94 (1988), 245–305

[Sh3] G. Shimura, On the fundamental periods of automorphic forms of arithmetic type, Inventiones Math. 102 (1990), 399–428

[Y] H. Yoshida, On a conjecture of Shimura concerning periods of Hilbert modular forms, Amer. J. Math. 117 (1995), 1019–1038