BEILINSON’S THEOREM ON MODULAR CURVES.

Norbert Schappacher and Anthony J. Scholl

Contents

0. Introduction.

1. The Theorem.

2. Transformation of L-values.

3. Eisenstein series and modular units.

4. Whittaker functions and L-factors.

5. Evaluation of the regulator integral.

6. Non-vanishing of the regulator integral.

7. Integrality.

References.

0. Introduction.

The purpose of this chapter is to give a detailed account of the known evidence ([Be1], §5) for Beilinson’s conjecture concerning the values at s = 2 of L-functions of modular forms of weight two. We do not discuss here the results concerning the L-values at other integers s ≥ 2 (for which see [Be2]), nor do we treat the case of the product of two modular curves (also to be found in [Be1]).

On the whole we follow the method of Beilinson, apart from two differences.

Firstly, we give a statement of the main theorem in terms of modular curves, rather than motives of modular forms. In particular, we have tried to work throughout with modular curves of finite level, rather than passing to the inverse limit. Secondly, our proof of the “integrality” statement (Theorem 1.1.2(iii) here; Theorem 5.1.1 in [Be1]) is rather different from that proposed by Beilinson; this seems necessary since the “integral” refinement of the Manin-Drinfeld theorem (see [Be1], §5.5) does not hold in general. For further remarks on this see 1.1.3(iii) and 7.4 below.

In the course of preparing the talks of which this is an expanded account, Ramakrishnan’s expository preprint [Ra] was helpful in a number of places. We are also grateful to many people for helpful discussions during the conference, and in particular to G. Harder and R. Weissauer.

1. The theorem.

1.0. We first review the formalism of modular curves and their cohomology. For details, see [DR], [La]. 1.0.0. For any integer n ≥ 3, there exists a moduli scheme M_{n} for elliptic curves E with level n structure (Z/nZ)^{2} \underrightarrow{∼} E[n]. We have the following description of its complex points:–

M_{n}(C) = GL_{2}(Z)\H^{±} × GL_{2}(Z/nZ)

where H^{±} = C − R. A point on the right represented by (τ, γ) ∈ H^{±} × GL_{2}(Z/nZ) corresponds to E = C/Z + Zτ with γ · \binom{1/n}{τ/n} (mod Z + Zτ) as basis for E[n]. Writing G for the algebraic group GL_{2} over Q, G_{f} for its points in the finite adèle ring A_{f} of Q, and K_{n} for the compact open subgroup

K_{n} = ker{p_{n} : GẐ → G_{Z/nZ}}

one has

M_{n}(C) = G_{Z}\H^{±} × GẐ/K_{n} = G_{Q}\H^{±} × G_{f}/K_{n}.

1.0.1. More generally, for any compact open subgroup K of G_{f}, there is a modular curve M_{K} defined over Q with

M_{K}(C) = G_{Q}\H^{±} × G_{f}/K.

There is a compactification

M_{K} ⟶ \overline{M}_{K}

where \overline{M}_{K} is a smooth projective (not necessarily geometrically connected) curve over Q, and M_{K} is the complement of a finite set M_{K}^{∞} of cusps. We denote by \overline{M}_{K/Z} a regular model of \overline{M}_{K} over Z (see §7 below).

Examples of \overline{M}_{K} include the familiar modular curves

X_{1}(n) when K = K_{1}(n) =^{def} p_{n}^{−1} ( \begin{matrix}∗ & ∗ \\ 0 & 1 \end{matrix} )

X_{0}(n) when K = K_{0}(n) =^{def} p_{n}^{−1} ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} )

and \overline{M}_{n} itself, usually denoted X(n). Usually we will assume that K ⊆ GẐ (the general case may be reduced to this by conjugation). 1.0.2. If K′ ⊆ K are open compact subgroups of G_{f}, there is a natural map θ_{K′/K} : \overline{M}_{K′} → \overline{M}_{K}. The projective limit M = \underleftarrow{lim}_{K} M_{K} is the moduli scheme for elliptic curves E with ‘universal level structure’ Ẑ^{2} \underrightarrow{∼} \underleftarrow{lim}_{n} E[n].

1.1. We now turn to Beilinson’s conjecture concerning the leading coefficient of L(H^{1}(\overline{M}_{K}), s) at s = 0. 1.1.0. In Beilinson’s formulation, the image of the regulator map

r_{D} = r_{0,1} : H_{A}^{2}(\overline{M}_{K/Z}, Q(2)) → H_{D}^{2}(\overline{M}_{K/R}, R(2))

is conjectured to be a Q-structure of

\begin{aligned}H_{D}^{2}(\overline{M}_{K}, R(2)) & = H_{B}^{1}(\overline{M}_{K}(C), R(1))^{−} \\ & = H_{B}^{1}(\overline{M}_{K/R}, R(1)) \end{aligned}

and the determinant (“regulator”) of a linear map taking this Q-structure onto H_{B}^{1}(\overline{M}_{K/R}, Q(1)) is conjectured to be, up to a factor in Q^{∗}, the leading coefficient of L(H^{1}(\overline{M}_{K}), s) at s = 0. (We have denoted by \overline{M}_{K/R} the analytic space over R associated to \overline{M}_{K} —see chapter VIII, §1.) We recall that by definition, H_{A}^{2}(\overline{M}_{K/Z}, Q(2)) is the image of the homomorphism

K_{2}^{(2)}(\overline{M}_{K/Z}) ⟶ K_{2}^{(2)}(\overline{M}_{K})

and that it does not depend on the choice of regular model.

Beilinson’s theorem exhibits, for each K, a subspace of H_{A}^{2}(\overline{M}_{K/Z}, Q(2)) whose image under r_{D} is a Q-structure of H_{B}^{1}(\overline{M}_{K/R}, R(1)) with the desired regulator.

1.1.1. This subspace is defined in terms of O^{∗}(M_{K}), the so-called modular units of level K. Write {O^{∗}(M_{K}), O^{∗}(M_{K})} for the Q-subspace of H_{A}^{2}(M_{K}, Q(2)) generated by all symbols {u, v} with u, v ∈ O^{∗}(M_{K})⊗Q = H_{A}^{1}(M_{K}, Q(1)). Recall from chapter VIII, 5.1–2 that the map

H_{A}^{2}(\overline{M}_{K}, Q(2)) ⟶ H_{A}^{2}(M_{K}, Q(2))

is injective. Define

Q_{K} = H_{A}^{2}(\overline{M}_{K}, Q(2)) ∩ \{O^{∗}(M_{K}), O^{∗}(M_{K})\}

and

P_{K} = _{K′⊆K}^{⋃} θ_{K′/K∗}(Q_{K′}) ⊂ H_{A}^{2}(\overline{M}_{K}, Q(2))

the union being taken over all open subgroups K′ ⊆ K (cf. 1.0.2 above).

1.1.2. Theorem.

(i) r_{D}(P_{K}) is a Q-structure of

H_{D}^{2}(\overline{M}_{K/R}, R(2)) = H_{B}^{1}(\overline{M}_{K/R}, R(1)).

(ii) Let g be the genus of \overline{M}_{K}. Then

det r_{D}(P_{K}) = L^{(g)}(H^{1}(\overline{M}_{K}), 0) · det H_{B}^{1}(\overline{M}_{K/R}, Q(1)).

(iii) P_{K} ⊆ H_{A}^{2}(\overline{M}_{K/Z}, Q(2)).

1.1.3. Remarks. (i) Theorem 1.1.2(i) fails in general if P_{K} is replaced by Q_{K}. For example, when \overline{M}_{K} = X_{0}(p) for a prime p there are only two cusps, so that O^{∗}(M_{K}) = Q^{∗} · u^{Z} for some modular unit u; and in this case it is easy to see that r_{D}(Q_{K}) = 0, using the relation {u, u} = 0.

(ii) In some cases it is known, however, that Q_{K} is sufficient. For example, this holds when \overline{M}_{K} = X_{0}(27), the elliptic Fermat curve, as can be shown using methods similar to those of chapter VIII—see the concluding example in [Ro]. Also, for the elliptic curve E = \overline{M}_{K} = X_{0}(20) (which does not have complex multiplication) one has E(Q) = M_{K}^{∞}, so that the calculations of Bloch and Grayson [BG] for the curve “20B” show that again r_{D}(Q_{K}) ≠ 0.

(iii) The proof of part (iii) of the theorem, which is independent of the other parts, will be given in §7 below. In the talks presented at the conference we were only able to prove a weaker result, namely that

r_{D}(P_{K}) = r_{D} (P_{K} ∩ H_{A}^{2}(\overline{M}_{K/Z}, Q(2))) .

It seems to be generally believed that r_{D} is injective on H_{A}^{2}(\overline{M}_{K}, Q(2)), and not just on the integral subspace (see for example [BG], where r_{D} is denoted M). We regard part (iii) of the theorem as evidence for this conjecture, since it is not hard to see that if (iii) did not hold, then the rest of the theorem, together with the weaker statement mentioned above, would imply that the conjecture were false.

1.2. We next decompose the regulator map according to the automorphic representations of G_{f}, and reduce assertions (i) and (ii) of 1.1.2 to statements involving automorphic forms of weight 2. 1.2.0. Define Ω^{1}(\overline{M}) = \underrightarrow{lim}_{K} Ω^{1}(\overline{M}_{K}). The following facts are well known (see for example [La]).

The natural action of G_{f} on Ω^{1}(\overline{M}) gives rise to a decomposition

Ω^{1}(\overline{M}) ⊗ \overline{Q} = _{π}^{⨁} V_{π}

where π : G_{f} → GL(V_{π}) are irreducible admissible \overline{Q}-representations of G_{f}, pairwise non-isomorphic.

Let H(G_{f}, K) denote the Hecke algebra of compactly supported functions on G_{f} with values in \overline{Q} which are biinvariant under K. Write H_{K} for the image of H(G_{f}, K) in the ring of correspondences on \overline{M}_{K}. Then H_{K} acts faithfully on Ω^{1}(\overline{M}_{K}) ⊗ \overline{Q}, and

Ω^{1}(\overline{M}_{K}) ⊗ \overline{Q} = _{π}^{⨁} V_{π}^{K}

where those of the spaces V_{π}^{K} which are nonzero are pairwise non-isomorphic H_{K}- modules. We define

m(π, K) = dim\overline{Q} V_{π}^{K} < ∞

and we write e_{π}^{K} ∈ H_{K} for the projector

Ω^{1}(\overline{M}_{K}) ⊗ \overline{Q} ⟶ V_{π}^{K}.

1.2.1. For each σ : \overline{Q} ⟶ C and each π as above, we obtain a complex representation π^{σ} of G_{f}, and therefore corresponding L-functions and ϵ-factors L(π^{σ}, s), ϵ(π^{σ}, s).

Following Beilinson we normalise the L-functions so that the functional equation is given by the substitution s ↦ 2 − s (this represents a shift of \frac{1}{2} from the conventions of [JL]).

In verifying Beilinson’s conjecture it is convenient not to specify a preferred embedding of \overline{Q} in C, and accordingly one defines L(π, s) to be the function taking values in C^{Hom(\overline{Q},C)} whose σ-component is L(π^{σ}, s). Since π can be defined over an algebraic number field, L(π, s) actually takes values in the subring

\overline{Q} ⊗ C ⊂ C^{Hom(\overline{Q},C)}.

We similarly define the ϵ-factors ϵ(π, s) as \overline{Q} ⊗ C-valued functions of s.

1.2.2. Theorem.

L(H^{1}(\overline{M}_{K}), s) = _{π}^{∏} L(π, s)^{m(π,K)}.

1.2.3. Remarks. (i) Up to a finite number of Euler factors, this result was obtained by Eichler and Shimura. By the work of Igusa, Langlands, Deligne and Carayol, it is now completely proved; see [Ca].

(ii) Note that although the individual factors on the right are \overline{Q} ⊗ C-valued functions, their product takes values in C ⊂ \overline{Q} ⊗ C.

(iii) We should point out that in the decomposition of the l-adic cohomology H_{ét}^{1}(\overline{M}_{K} ⊗ \overline{Q}, Q_{l}) under the action of H_{K}, the π-isotypical component corresponds to the factors L(π̌, s), where π̌ is the representation contragredient to π, rather than L(π, s). But since m(π, K) = m(π̌, K) this gives the theorem as stated. However, when we decompose the regulator map r_{D} under the action of G_{f} in 1.2.6 below, the π-component will actually contribute a factor of L′(π, 0) (not L′(π̌, 0)) to the value of the regulator. This can be accounted for by the general form of Beilinson’s conjecture, in which it is the L-function of the dual of the motive which occurs (see [Be1], 3.4(b)). 1.2.4. We next recall the isomorphism (given by Poincaré duality)

H_{B}^{1}(\overline{M}_{K/R}, R(1)) \underrightarrow{∼} Hom(Ω^{1}(\overline{M}_{K}), R).

Applying the projector eK ; π̌, we obtain a diagram

eK ; π̌H_{B}^{1}(\overline{M}_{K/R}, \overline{Q}(1)) ⟶ eK ; π̌H_{B}^{1}(\overline{M}_{K/R}, \overline{Q} ⊗ R(1))
↓≀
Hom\overline{Q}(V_{π}^{K}, \overline{Q})  ⟶  Hom\overline{Q}(V_{π}^{K}, \overline{Q} ⊗ R)

in which the two groups on the left define \overline{Q}-structures on the free \overline{Q}⊗R-modules on the right. Since V_{π}^{K} is an irreducible H_{K}-module, and the right-hand isomorphism is an H_{K}-isomorphism, we must have

eK ; π̌H_{B}^{1}(\overline{M}_{K/R}, \overline{Q}(1)) = c^{+}(π) · Hom\overline{Q}(V_{π}^{K}, \overline{Q})

for some c^{+}(π) ∈ (\overline{Q} ⊗ R)^{∗}/\overline{Q}^{∗}, independent of K (provided V_{π}^{K} ≠ (0)). This constant is none other than Deligne’s period for the motive attached to π whose Lfunction is L(π, s + 1) ([DP], 2.6, 7.4 and Chapter II of this volume)—see also 2.2.1 below. 1.2.5. We observe that H_{K} acts on H_{A}^{2}(\overline{M}_{K}, Q(2)) ⊗ \overline{Q}, leaving stable the subspace P_{K} ⊗ \overline{Q}. This is immediate from the definition of P_{K}. Theorem 1.1.2(i) and (ii) are now a consequence of the next theorem.

1.2.6. Theorem. For every π and every K,

eK ; π̌r_{D}(P_{K} ⊗ \overline{Q}) = L′(π̌, 0)c^{+}(π) · Hom\overline{Q}(V_{π}^{K}, \overline{Q})

⊂ Hom\overline{Q}(V_{π}^{K}, \overline{Q} ⊗ R) = eK ; π̌H_{B}^{1}(\overline{M}_{K/R}, \overline{Q} ⊗ R(1)).

1.2.7. We will actually prove a slightly different statement. Denote by < , >_{K} the \overline{Q}-linear pairing given by Poincaré duality (1.2.4):

H_{B}^{1}(\overline{M}_{K/R}, \overline{Q} ⊗ R(1)) × Ω^{1}(\overline{M}_{K}) ⊗ \overline{Q} ⟶^{< , >_{K}} \overline{Q} ⊗ R.

1.2.8. Theorem. Let π, V_{π} be as in 1.2.0.

(i) For every open compact subgroup K ⊂ G_{f}, for every ω ∈ V_{π}^{K}, and for every ξ ∈ Q_{K} ⊗ \overline{Q}, we have

<r_{D}(ξ), ω >_{K} ∈ c^{+}(π)L′(π̌, 0) · \overline{Q} ⊂ \overline{Q} ⊗ R.

(ii) There exist K, ω and ξ as in (i) such that

<r_{D}(ξ), ω >_{K} ≠ 0.

1.2.9. Let us first show that 1.2.8 implies the preceding theorem 1.2.6 and therefore also 1.1.2(i) and (ii).

First, note the adjointness property, for K′ ⊆ K, and θ = θ_{K′/K} as in 1.0.2, 1.1.1: if ξ′ ∈ H_{A}^{2}(\overline{M}_{K′}, Q(2)) ⊗ \overline{Q} and ω ∈ V_{π}^{K}, then

<r_{D}(ξ′), θ^{∗}ω >_{K′} =<r_{D}(θ_{∗}ξ′), ω >_{K} .

From this and the identity θ_{∗}θ^{∗}P_{K} = P_{K}, we see that 1.2.8(i)—for all K—implies

eK ; π̌r_{D}(P_{K} ⊗ \overline{Q}) ⊆ L′(π̌, 0)c^{+}(π) · Hom\overline{Q}(V_{π}^{K}, \overline{Q}).

Assuming this to be true (for all K) consider the commutative diagram

eK′ ; π̌r_{D}(P_{K′} ⊗ \overline{Q})  ⟶^{θ_{∗}}  eK ; π̌r_{D}(P_{K} ⊗ \overline{Q})
↓i_{K′}  ↓i_{K}
L′(π̌, 0)c^{+}(π) · Hom\overline{Q}(V_{π}^{K′}, \overline{Q}) ⟶^{θ^{∗}} L′(π̌, 0)c^{+}(π) · Hom\overline{Q}(V_{π}^{K}, \overline{Q})

To prove that i_{K} is an isomorphism, it suffices to prove that i_{K′} is an isomorphism, for some K′; by the irreducibility of V_{π}^{K′} this follows from eK′ ; π̌r_{D}(P_{K′} ⊗\overline{Q}) ≠ 0, which in turn is a consequence of 1.2.8(ii).

1.3. The final reformulation of Theorem 1.1.2 to be done in this section makes explicit the regulator map r_{D}. 1.3.0. Recall from Chapter VIII, 1.6 the projection

pr_{D} : H_{B}^{1}(M_{K/R}, R(1)) ⟶ H_{B}^{1}(\overline{M}_{K/R}, R(1)).

The commutative diagram

H_{A}^{2}(\overline{M}_{K}, Q(2)) ⟶^{r_{D}} H_{B}^{1}(\overline{M}_{K/R}, R(1))
↓  ↓ ↑pr^{D}
H_{A}^{2}(M_{K}, Q(2)) ⟶^{r_{D}} H_{B}^{1}(M_{K/R}, R(1))

suggests that we compute, for modular units u, v ∈ O^{∗}(M_{K}) ⊗_{Z} \overline{Q}, the element

pr_{D}(r_{D}(\{u, v\})) ∈ H_{B}^{1}(\overline{M}_{K/R}, R(1)).

This is done by the following formula, valid for all ω ∈ Ω^{1}(\overline{M}_{K} ⊗ \overline{Q}), which is none other than a \overline{Q}-linear extension of VIII, (1.10) (compare also Chapter III, in particular 1.8–1.12):

< pr_{D}(r_{D}(\{u, v\}), ω >_{K}= \frac{1}{2πi} ∫_{M_{K}(C)} log |u| \overline{d log v} ∧ ω.

Note that this is an identity in \overline{Q} ⊗ C … 1.3.1. Let us look again at the commutative diagram in 1.3.0. In view of the Manin-Drinfeld theorem (3.4.0 below) we can apply Bloch’s lemma (Lemma 5.2 of Chapter VIII). Thus given u, v ∈ O^{∗}(M_{K}) there is a finite extension F of Q and an element α ∈ ϕ_{∗}{F^{∗}, O^{∗}(M_{K} ⊗_{Q} F)} such that ξ = α + {u, v} belongs to the image of H_{A}^{2}(\overline{M}_{K}, Q(2)) in H_{A}^{2}(M_{K}, Q(2)). (Here ϕ is the basechange morphism M_{K} ⊗ F ⟶ M_{K}.) From the diagram above we have pr_{D} (r_{D} ({u, v})) = r_{D}(ξ). Granted this, the following theorem clearly implies 1.2.8 and therefore also parts (i), (ii) of the main theorem 1.1.2.

1.3.2. Theorem. Let π, V_{π} be as in 1.2.0.

(i) For all K, all u, v ∈ O^{∗}(M_{K}) ⊗_{Z} \overline{Q}, we have

∫_{M_{K}(C)} log |u| \overline{d log v} ∧ ω ∈ 2πi c^{+}(π)L′(π̌, 0) · \overline{Q} ⊂ \overline{Q} ⊗ C.

(ii) For some K, some ω ∈ V_{π}^{K}, and some u, v ∈ O^{∗}(M_{K}) ⊗_{Z} \overline{Q},

∫_{M_{K}(C)} log |u| \overline{d log v} ∧ ω ≠ 0.

2. Transformation of L-values.

2.0. In this section, we are going to rewrite the product

(2.0.0)  2πi c^{+}(π)L′(π̌, 0) ∈ (\overline{Q} ⊗ C)^{∗}/\overline{Q}^{∗}

which occurs in 1.3.2. 2.0.1. In this section, every expression is to be regarded as an element of (\overline{Q} ⊗ C)^{∗}/\overline{Q}^{∗}. We always regard C as embedded in \overline{Q} ⊗ C via the second inclusion. Thus if a, b are complex numbers, then in writing a = b we signify that a and b are equal up to a nonzero rational factor.

2.1. The functional equation for the L-function of π implies that

(2.1.0)  L′(π̌, 0) = π^{−2}ϵ(π, 2)^{−1}L(π, 2).

2.1.1. Now, the ϵ-factor of a motive is equal (in the sense of 2.0.1) to that of its determinant motive; see [DP], 5.5. But the determinant representation of the ladic realisation of the motive associated to π is the central character ω_{π} of π, and ω−1 ; π̌ = ω_{π}—cf. [DP], 7.1, 7.4 where the “contragredient” normalisation is used. ω_{π} is an even Dirichlet character of Q since π has weight 2. 2.1.2. If χ is any even \overline{Q}-valued Dirichlet character of Q, its ϵ-factor ϵ(χ), evaluated at any integer s, is equal to the standard Gauss sum of χ; see [DP], 6.4. It is characterised by the following properties—see [Scha], II §3, in particular 3.4; compare also [Bl].

(i) ϵ(χ) ∈ (\overline{Q} ⊗ C)^{∗}.

(ii) If Aut(C/Q) acts on \overline{Q} ⊗ C via the second factor, then for all τ ∈ Aut(C/Q),

one has

ϵ(χ)^{τ} = χ(τ) · ϵ(χ),

where χ is defined on Aut(C/Q) via the reciprocal of the cyclotomic character (“geometric Frobenius”).

This characterisation applies in particular to ϵ(ω_{π}) = ϵ(π, 2), by 2.1.1; as a general consequence, note that

2.1.3. If χ, χ′ are two even Dirichlet characters of Q, then

ϵ(χχ′) = ϵ(χ) · ϵ(χ′).

2.2. We now transform the product (2.0.0) into an expression involving a certain auxiliary Dirichlet character χ, which will vary over all even Dirichlet characters of Q. The following fact is essential for the proof of 1.3.2(ii).

2.2.0. Lemma. Given π, there exists an even Dirichlet character χ ≠ 1, ω_{π}^{−1} of Q, such that the corresponding value at s = 1 of the twisted L-function does not vanish:

L(π ⊗ χ, 1) ∈ (\overline{Q} ⊗ C)^{∗}.

The standard proof of this lemma uses the fact that the first homology of the modular curves is generated by modular symbols—see [Shi], Theorem 2 and remark, pp. 212–214 for a result which covers all we need here.

2.2.1. Proposition. Let χ be as in 2.2.0. Then

L(π ⊗ χ, 1) = c^{+}(π) · ϵ(χ),

where c^{+}(π) is as in 1.2.4 above.

In fact, Deligne’s conjecture is true for motives of modular forms, by [DP], §7. Thus one gets

L(π ⊗ χ, 1) = c^{+}(π ⊗ χ).

The remaining identity c^{+}(π ⊗ χ) = ϵ(χ)c^{+}(π) is essentially a consequence of the fact that the de Rham realisation of the motive of χ is spanned by an element ϵ(χ) satisfying (i) and (ii) of 2.1.2—see [DP], 6.3; [Scha], II,§3; [Bl].

2.2.2. In transforming the product (2.0.0), via 2.1.0 and 2.2.1, we shall pick up in the denominator the term

ϵ(π, 2) · ϵ(χ) = ϵ(ω_{π}) · ϵ(χ) = ϵ(ω_{π}χ),

by 2.1.1 and 2.1.3. But Deligne’s conjecture is true (due to Siegel) for Artin motives; see [DP], 6.7. Thus, since s = 2 is critical for the L-function of an even Dirichlet character—cf. [DP], 5.1.8—we see that

(2.2.3)  π^{2}ϵ(ω_{π}χ) = L(ω_{π}χ, 2)

where the L-function on the right is the Dirichlet L-function attached to the character ω_{π}χ. 2.2.4. Remark. We have been using freely the notion of motive and certain special cases of Deligne’s conjecture in this section. For the reader who may feel uneasy about this, we want to point out that all the motives and their properties that were needed in this section can be obtained in the category of motives for absolute Hodge cycles—see [DMOS] and [Scha], in particular Chapter V, §1. But they also exist in the strongest sense required by Grothendieck—cut out by projectors in the category of algebraic correspondences modulo rational equivalence.

Collecting now everything that has been established in this section, we obtain the following result.

2.3. Theorem. For all χ satisfying L(π⊗χ, 1) ∈ (\overline{Q}⊗C)^{∗}, we have in (\overline{Q}⊗C)^{∗}/\overline{Q}^{∗},

c^{+}(π) · L′(π̌, 0) = \frac{L(π, 2) · L(π ⊗ χ, 1)}{L(ω_{π}χ, 2)}.

3. Eisenstein series and modular units.

3.0. This and the following section prepare the evaluation, in §5, of the regulator integral occurring in 1.3.2(i). 3.0.0. Let us begin with some notations for subgroups of G = GL_{2}:

B = ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} ); N = ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ); D = ( \begin{matrix}∗ & 0 \\ 0 & 1 \end{matrix} ) .

If H is any of G, B, N, D and R is a Z-algebra, we write H_{R} for the group of R-valued points of H; H_{f} for H_{A_{f}}, where A_{f} is the ring of finite adèles of Q; and

H_{Q}^{+} = {g ∈ H_{Q} : det g > 0} .

We also write H_{p} for H_{Q_{p}}.

If b = ( \begin{matrix}b_{1} & ∗ \\ 0 & b_{2} \end{matrix} ) ∈ B_{f}, we write the modulus as

|b|_{f} = |\frac{b_{1}}{b_{2}} |_{f},

where the right hand side is finite idèle modulus. Thus, for b ∈ B_{Q}^{+}, one has |b| = b_{2}/b_{1}. 3.0.1. The cusps of \overline{M}_{K}(C) can be written as

M_{K}^{∞}(C) = ±NẐ\GẐ/K.

For k ∈ GẐ, denote the corresponding cusp by [k]. The width w(k, K) of [k] is the least w > 0 such that

( \begin{matrix}1 & w \\ 0 & 1 \end{matrix} ) ∈ ±kKk^{−1} ∩ B_{Q}^{+}.

A uniformising parameter on \overline{M}_{K}(C) in a neighbourhood of [k] is then

(z, k) ↦ exp(2πiz/w(k, K)),

for Im z > 0. 3.0.2. If

t = _{p}^{∏} t_{p} : Aut(C/Q) ⟶ Ẑ^{∗} = _{p}^{∏} Z_{p}^{∗}

denotes the cyclotomic character, giving the action of Aut(C/Q) on exp(2πi · Q), then Aut(C/Q) acts on the cusps by the rule

[k]^{τ} = [( \begin{matrix}t(τ) & 0 \\ 0 & 1 \end{matrix} ) k] .

3.1.0. Let ϕ : ±NẐ\GẐ → \overline{Q} be a locally constant function. Define

deg ϕ : Ẑ^{∗} ⟶ \overline{Q} ⊂ \overline{Q} ⊗ C

by

(deg ϕ)(a) = ∫_{SL_{2}(Ẑ)} ϕ (( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) g) dg,

relative to the Haar measure of total mass 1 on SL_{2}(Ẑ). If ϕ is right invariant under the open subgroup K ⊆ GẐ, then it just gives a divisor on \overline{M}_{K}(C), with coefficients in \overline{Q}, supported on M_{K}^{∞}(C); the connected components of \overline{M}_{K}(C) are indexed by Ẑ^{∗}/ det(K), and deg ϕ measures the degree on each connected component.

If z ∈ H^{±}, write

(3.1.1)  I(z) = \{ \begin{matrix}Im(z) & if z ∈ H, i.e., Im(z) > 0 \\ 0 & if −z ∈ H \end{matrix}

By the Iwasawa decomposition, we may write any g ∈ G_{f} in the form g = b · k, with b ∈ B_{Q}^{+}, and k ∈ GẐ. Then define

ϕ̂_{s} : G_{f} ⟶ \overline{Q} ⊗ C

by

(3.1.2)  ϕ̂_{s}(g) = |b|_{f}^{s}ϕ(k)

for s ∈ C. This makes the function on H^{±} × G_{f}

(3.1.3)  (z, g) ↦ ϕ̂_{s}(g)I(z)^{s}

well-defined, and invariant under left translation by B_{Q}^{+}. The real analytic Eisenstein series associated with ϕ is then defined, for Re s > 1, by the absolutely convergent series

(3.1.4)  E_{ϕ}(z, g; s) = −2π _{γ∈B_{Q}^{+}\G_{Q}}^{∑} ϕ̂_{s}(γg)I(γz)^{s}.

This expression is to be regarded as taking values in \overline{Q} ⊗ C.

The following facts are consequences of Selberg’s theory—see [GJ], or [Ku] for a more classical account. (3.1.5) E_{ϕ} has a meromorphic continuation to the s-plane, with at worst a simple

pole at s = 1. The residue of E_{ϕ} at s = 1 is a locally constant function on H^{±} × G_{f}.

(3.1.6) As a function on H^{±} × G_{f}, E_{ϕ} is left G_{Q}-invariant, and right K-invariant if ϕ is. It satisfies the differential equation △E_{ϕ} = s(s−1)E_{ϕ}, for the Laplacian △ on H^{±}.

(3.1.7) If deg ϕ = 0, then E_{ϕ} has no pole at s = 1. In this case, E_{ϕ}(z, g; 1) is harmonic on H^{±}, and the difference

E_{ϕ}(z, k; 1) − (−2πy)ϕ(k)

is bounded as y = Im z tends to +∞. From 3.1.7, it follows that, if deg ϕ = 0 and ϕ is right K-invariant, then the

differential form on H^{±} × G_{f}

(3.1.8)  η_{ϕ} = 2∂_{z}E_{ϕ}(z, g; 1)

may be viewed as a holomorphic \overline{Q} ⊗ C-valued 1-form on M_{K}(C), with at worst simple poles at the cusps, and its residue at the cusp [k] ∈ M_{K}^{∞}(C) is ϕ(k)/w(k, K). 3.1.9. Define

Eis^{K} = {η_{ϕ} | ϕ is K-invariant and deg ϕ = 0} ,

and

Eis = _{K}^{⋃} Eis^{K}.

Notice that a priori the space Eis is a \overline{Q}-subspace of the space Ω^{1}(M) ⊗ \overline{Q} ⊗ C, where Ω^{1}(M) = \underrightarrow{lim}_{K} Ω^{1}(M_{K}); it is however actually contained in Ω^{1}(M) ⊗ \overline{Q}, as a consequence of the Manin-Drinfeld theorem—see Theorem 3.4.0 below.

3.2.0. For any ϕ (not necessarily of degree 0) the function

ϕ̂ = ϕ̂_{1} : G_{f} ⟶ \overline{Q} ⊂ \overline{Q} ⊗ C

is a locally constant function satisfying

ϕ̂(bng) = |b|_{f} ϕ̂(g)

for all b ∈ B_{Q}^{+}, n ∈ NẐ, g ∈ G_{f}.

Under the action of diagonal matrices, the space of all such functions ϕ̂ decomposes as

(3.2.1)  _{χ_{1},χ_{2}}^{⨁} Ind(χ_{1}, χ_{2}),

where (χ_{1}, χ_{2}) runs over all pairs of Dirichlet characters

χ_{i} : A_{Q}^{∗}/Q^{∗} ⟶ \overline{Q}^{∗}

whose product χ_{1}χ_{2} is even, and where Ind(χ_{1}, χ_{2}) is the space of all locally constant functions ϕ̂ : G_{f} → \overline{Q} satisfying

ϕ̂(bg) = χ_{1,f}(b_{1})χ_{2,f}(b_{2})|b|_{f} ϕ̂(g),

for all b = ( \begin{matrix}b_{1} & ∗ \\ 0 & b_{2} \end{matrix} ) ∈ B_{f}. In the terminology of [JL], we have

Ind(χ_{1}, χ_{2}) = B(μ, ν) = ⨂ ; p′ B(μ_{p}, ν_{p}),

where μ, ν : A_{f}^{∗} → \overline{Q}^{∗} are the characters

\begin{aligned}μ(x) & = χ_{1,f}(x)|x|f^{\frac{1}{2}} \\ ν(x) & = χ_{2,f}(x)|x|_{f}^{− \frac{1}{2}} . \end{aligned}

3.2.2. The assignment η_{ϕ} ↦ ϕ̂ defines a G_{f}-equivariant inclusion

Eis ⟶ _{χ_{1},χ_{2}}^{⨁} Ind(χ_{1}, χ_{2}).

We write

Eis(χ_{1}, χ_{2}) = Eis ∩ Ind(χ_{1}, χ_{2}).

3.2.3. Proposition. Eis = ⨁ Eis(χ_{1}, χ_{2}). Furthermore, Eis(χ_{1}, χ_{2}) = Ind(χ_{1}, χ_{2}) unless χ_{1} = χ_{2} = χ, in which case Eis(χ, χ) is the unique G_{f}-invariant subspace of Ind(χ, χ) of codimension one.

Proof. The map η_{ϕ} ↦ ϕ̂ identifies Eis(χ_{1}, χ_{2}) with

{ϕ̂ ∈ Ind(χ_{1}, χ_{2}) | deg (ϕ̂|_{GẐ}) = 0} .

If ϕ̂ ∈ Ind(χ_{1}, χ_{2}) and a, b ∈ Ẑ^{∗}, then

(deg (ϕ̂|_{GẐ})) (a) = ∫_{SL_{2}(Ẑ)} ϕ̂ (( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) g) dg

= ∫_{SL_{2}(Ẑ)} ϕ̂ (( \begin{matrix}ab & 0 \\ 0 & b^{−1} \end{matrix} ) g) dg

= χ_{1,f}(a)\frac{χ_{1,f}}{χ_{2,f}} (b) · (deg (ϕ̂|_{GẐ})) (1).

Thus, deg (ϕ̂|_{GẐ}) is determined by its value at 1, and is zero if χ_{1} ≠ χ_{2}. Hence, in this case we have indeed that Eis(χ_{1}, χ_{2}) = Ind(χ_{1}, χ_{2}).

If, however, χ_{1} = χ_{2}, then μν^{−1} = |·|_{f}, and so B(μ_{p}, ν_{p}) has a unique nonzero invariant subspace B_{s}(μ_{p}, ν_{p}), the so-called special representation (see [JL], Theorem 3.3). It has codimension 1, and is the subspace of functions ϕ̂_{p} on G_{p} which belong to B(μ_{p}, ν_{p}) and satisfy

∫_{SL_{2}(Z_{p})} ϕ̂_{p}(g) dg = 0.

Eis(χ_{1}, χ_{2}) is then identified with the subspace of ⨂ ; p′ B(μ_{p}, ν_{p}) spanned by tensors ⊗_{p}ϕ̂_{p} such that ϕ̂_{p} belongs to B_{s}(μ_{p}, ν_{p}) for at least one p, and this is the unique invariant subspace of B(μ, ν) of codimension one.

3.3. Remarks. (i) Even if deg ϕ ≠ 0, we can still define a G_{Q}-invariant 1-form η_{ϕ} on H^{±} × G_{f}; for the residue of E_{ϕ} at s = 1 is locally constant on H^{±} × G_{f}, and we can write

η_{ϕ} = lim_{s→1} 2∂_{z}E_{ϕ}.

This will be an “almost holomorphic” form, as introduced by Hecke ([He], pp. 411– 413).

(ii) The representations Eis(χ_{1}, χ_{2}) are highly reducible. However, if χ_{1} ≠ χ_{2}, then for infinitely many primes p, the p-components χ_{i,p} will be unequal, whence the local factors B(μ_{p}, ν_{p}) will be irreducible. In particular, for such primes p, there can be no quotient of Eis(χ_{1}, χ_{2}) whose restriction to G_{p} is abelian. We will come back to this in 7.4 below.

(iii) If K is an open subgroup of GẐ, then for every p such that G_{Z_{p}} ⊆ K, the Hecke operator T_{p} ∈ H(G_{f}, K) (cf. 1.2.0 above) is defined as \frac{1}{p} times the character-

istic function of the double coset K ( \begin{matrix}1 & 0 \\ 0 & ϖ \end{matrix} ) K, with ϖ ∈ A_{f}^{∗} having component p at the place p, and 1 elsewhere. Then T_{p} acts as multiplication by pχ_{1}(p) + χ_{2}(p) on Ind(χ_{1}, χ_{2})^{K}.

3.4. We next recall the well-known theorem of Manin and Drinfeld:

3.4.0. Theorem (Manin-Drinfeld). Let C ⊂ Pic^{0}(\overline{M}_{K} ⊗ C) be the subgroup of classes of divisors supported on the cusps of \overline{M}_{K}(C). Then C is finite.

Proof. Choose a prime p ≥ 7 such that G_{Z_{p}} ⊆ K. C is a quotient of

H̃^{0}(M_{K}^{∞}(C), Z) = {ϕ : ±NẐ\GẐ/K → Z | deg ϕ = 0} ,

and, by 3.3(iii), the eigenvalues of T_{p} on C ⊗ \overline{Q} have absolute value ≥ p − 1. On the other hand, the characteristic polynomial X_{p}(T) of T_{p} on Ω^{1}(\overline{M}_{K}) has rational coefficients, and all its roots have absolute value ≤ 2\sqrt{p}. For some positive integer N, the correspondence NX_{p}(T_{p}) annihilates Pic^{0}(\overline{M}_{K}), and since p − 1 > 2\sqrt{p} for p ≥ 7, we get that C ⊗ Q = 0. 3.4.1. In §7 below we shall use a similar argument in the course of proving 1.1.2(iii).

3.5.0. Let u ∈ O^{∗}(M_{K} ⊗ C), for some open compact subgroup K ⊆ GẐ. If ord_{[k]} u denotes the multiplicity of u at the cusp [k] ∈ M_{K}^{∞}(C), then the function

div(u) : GẐ ⟶ \overline{Q}

k ↦ \frac{ord_{[k]} u}{w(k, K)}

depends only on u, not on the choice of K. It is left ±NẐ-invariant, right K-invariant, and has degree zero (see 3.1.0). Conversely, by the Manin-Drinfeld theorem any ϕ with this property is of the form div(u) for some u ∈ O^{∗}(M_{K} ⊗ C) ⊗_{Z} \overline{Q}.

3.5.1. Proposition. If ϕ = div(u) : ±NẐ\GẐ/K → Q, then

η_{ϕ} = d log u;

the function

E_{ϕ}(z, g; 1) − log |u|

is locally constant.

Proof. Since ϕ is Q-valued, E_{ϕ}(z, g; s) takes values in C ⊂ \overline{Q} ⊗ C, and E_{ϕ}(z, g; 1) is real-valued. By 3.1.7, the difference E_{ϕ}(z, g; 1) − log |u| is harmonic and bounded on M_{K}(C), whence locally constant. The first claim follows immediately.

3.5.2. Corollary. Let O^{∗}(M ⊗ C) = \underrightarrow{lim} O^{∗}(M_{K} ⊗ C). Then

d log O^{∗}(M ⊗ C) ⊗ \overline{Q} = Eis ⊂ Ω^{1}(M) ⊗ \overline{Q}.

We can therefore write the integral occurring in 1.3.2 as

(3.5.3)  ∫_{M_{K}(C)} (E_{ϕ}(z, g; 1) + c) η̄_{ξ} ∧ ω,

where ξ = div v, ϕ = div u and c is locally constant on M_{K}(C). Since η̄_{ξ} ∧ ω = 2 d(E_{ξ}(z, g; 1)ω), Stokes’ theorem shows that we may assume that c = 0.

3.5.4. Proposition. d log O^{∗}(M) ⊗_{Z} \overline{Q} = _{χ even}^{⨁} Eis(1, χ).

Proof. Let Q^{c} denote the algebraic closure of Q in C (to be distinguished from \overline{Q}). Then

O^{∗}(M) = O^{∗}(M ⊗ C)^{Aut(C/Q)} = O^{∗}(M ⊗ Q^{c})^{Gal(Q^{c}/Q)}

whence by Hilbert 90

d log O^{∗}(M) = (d log O^{∗}(M ⊗ Q^{c}))^{Gal(Q^{c}/Q)} .

If τ ∈ Gal(Q^{c}/Q) and u ∈ O^{∗}(M ⊗ Q^{c}), then by 3.0.2,

(div u^{τ}) (k) = (div u) (( \begin{matrix}t(τ) & 0 \\ 0 & 1 \end{matrix} ) k)

for all k ∈ GẐ. Thus if u ∈ O^{∗}(M) and ϕ = div u, then ϕ̂ ∈ _{χ}^{⨁} Ind(1, χ).

4. Whittaker functions and L-factors.

4.0. In this section we continue to prepare for the integral evaluation of §5, and also include some facts needed for later sections. Most of the results required from representation theory can be found in [JL], [Ge] or [Go]. 4.0.0. For every prime p, let ψ_{p} : Q_{p} → C^{∗} be the additive character such that

ψ_{p}(p^{−r}) = exp(−2πip^{−r}).

Then ψ = ∏_{p} ψ_{p}: A_{f} → C^{∗} has the property that (x_{∞}, x_{f}) ↦ exp(2πix_{∞}) · ψ(x_{f}) is a non-trivial character of Q\A. 4.0.1. If π_{C} is any irreducible admissible complex representation of G_{f}, all of whose local components are infinite-dimensional, then it has a unique realisation W(π_{C}) in the space of Whittaker functions

W = \{f : G_{f} → C | f (( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) g) = ψ(x)f(g) for all x ∈ A_{f}, g ∈ G_{f} \} .

If π_{C} occurs in Ω^{1}(\overline{M} ⊗ C), the space of (complex) cusp forms of weight 2 (cf. 1.2.0 above), then Fourier expansion gives an explicit mapping of the space of π_{C} into W; if ω = 2πif(g, z) dz belongs to the space of π_{C} (here z ∈ H^{±}, g ∈ G_{f}), then the mapping is:

(4.0.2)  ω ↦ W_{ω}(g) = ∫_{Q\A} f(z + x_{∞}, ( \begin{matrix}1 & x_{f} \\ 0 & 1 \end{matrix} ) g) e^{−2πi(z+x_{∞})}\overline{ψ(x_{f})} dx

where x = (x_{∞}, x_{f}) ∈ R × A_{f} = A. The inverse mapping is given by

(4.0.3)  ω = ∑_{{a∈Q \atop a>0}} W_{ω} (( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) g) d(e^{2πiaz}).

4.1. Now suppose that π is an irreducible factor of Ω^{1}(\overline{M}) ⊗ \overline{Q}. Then, as in 1.2.0, for every embedding σ : \overline{Q} ⟶ C, we obtain by extension of scalars a complex representation π^{σ} of G_{f}. The resulting Whittaker models W(π^{σ}) fit together to give a “rational” Whittaker model, in a sense to be explained now. 4.1.0. Recall the definition of the cyclotomic character t : Aut(C/Q) → Ẑ^{∗} from 3.0.2, and let Aut(C/Q) act on \overline{Q}⊗C via the second factor. In terms of the inclusion \overline{Q} ⊗ C ⊂ C^{Hom(\overline{Q},C)}, for τ ∈ Aut(C/Q) and (x_{σ})_{σ} ∈ \overline{Q} ⊗ C we have

(x_{σ})^{τ} = (x_{τ^{−1}σ}^{τ})_{σ}.

(The action of Aut(C/Q) is a left action.)

4.1.1. Theorem. There is a unique realisation of π in a \overline{Q}-subspace W^{rat}(π) of the \overline{Q} ⊗ C-module

W^{rat} = \{f : G_{f} → \overline{Q} ⊗ C | f (( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) g) = ψ(x)f(g) for all x ∈ A_{f} and g ∈ G_{f} f(g)^{τ} = f (( \begin{matrix}t(τ) & 0 \\ 0 & 1 \end{matrix} ) g) for all τ ∈ Aut(C/Q)\} .

The mapping (4.0.2) gives rises to a (canonical) isomorphism V_{π} \underrightarrow{∼} W^{rat}(π).

This theorem is implicit in [Ha]. The main point is that the curves \overline{M}_{K} are defined over Q, and the action of τ ∈ Aut(C/Q) on the set of cusps

M_{K}^{∞}(C) = ±NẐ\GẐ/K

is left multiplication by ( \begin{matrix}t(τ) & 0 \\ 0 & 1 \end{matrix} ). This enables one to keep track on the action of Aut(C/Q) on the Fourier expansions of automorphic forms at the various cusps of \overline{M}_{K} (and the theorem is no more than a description of this action). 4.1.2. The model W(π_{C}) of 4.0.1 is a restricted tensor product ⨂ ; p′ W(π_{C,p}) of Whittaker models of the local representations π_{C,p} of G_{p}. The rational Whittaker model also admits such a factorisation; if π = ⊗_{p}′ π_{p} is as above, then

W^{rat}(π) = ⨂ ; p′ W^{rat}(π_{p})

where W^{rat}(π_{p}) is the unique realisation of π_{p} in

W_{p}^{rat} = \{f : G_{p} → \overline{Q} ⊗ C | f (( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) g) = ψ_{p}(x)f(g) for all x ∈ Q_{p} and g ∈ G_{p} f(g)^{τ} = f (( \begin{matrix}t_{p}(τ) & 0 \\ 0 & 1 \end{matrix} ) g) for all τ ∈ Aut(C/Q)\} .

4.2. The restriction of functions in the Whittaker model to the subgroup ( \begin{matrix}∗ & 0 \\ 0 & 1 \end{matrix} ) of G is injective; the resulting space of functions is the Kirillov model. Restricting functions in W^{rat}(π), resp. W^{rat}(π_{p}), gives realisations K^{rat}(π), K^{rat}(π_{p}) of the representations π, π_{p} in spaces of \overline{Q} ⊗ C-valued functions on A_{f}^{∗} and Q_{p}^{∗}, respectively. In particular, we have

K^{rat}(π_{p}) ⊂ K_{p}^{rat} = \{f : Q_{p}^{∗} → \overline{Q} ⊗ C | f locally constant, and f(x)^{τ} = f(t_{p}(τ)x) for all x ∈ Q_{p}^{∗} and τ ∈ Aut(C/Q)\} .

From [JL], Proposition 2.9(i), one obtains:

4.2.0. Lemma. The space K^{rat}(π_{p}) contains all functions in K_{p}^{rat} of compact support.

4.3. Using the Kirillov model one defines local L-factors. For all p, associated to π_{p} there is a \overline{Q} ⊗ C-valued function L(π_{p}, s), which is in fact the reciprocal of a polynomial in p^{−s} with coefficients in \overline{Q} ⊂ \overline{Q} ⊗ C. The fact that these coefficients are in \overline{Q} expresses the fact that if π_{E} is a representation of G_{p} defined over a subfield E ⊂ C which is Galois over Q, then the L-factor L(π_{E}, s)—in the usual sense—is a polynomial in p^{−s} with coefficients in E, and if α ∈ Gal(E/Q), the L-factor of π^{α} is obtained by applying α to the coefficients of L(π_{E}, s). 4.3.0. In case π_{p} is spherical (that is, has a nonzero G_{Z_{p}}-invariant vector) we have

L(π_{p}, s) = ∫_{Q_{p}^{∗}} |a|_{p}^{s−1}f(a) d^{∗}a,

where f is the unique spherical function in K^{rat}(π_{p}) such that f(1) = 1. 4.3.1. The global L-function is defined as the Euler product (for Re s > 3/2)

L(π, s) = _{p}^{∏} L(π_{p}, s);

it is a \overline{Q} ⊗ C-valued function, whose σ-component is the usual L-function associated to π^{σ}—cf. 1.2.1 above.

4.4. One also defines Whittaker models for the induced representations Ind(χ_{1}, χ_{2}) (see [Go], §I.9)—note that these are in general reducible, and are not automorphic representations in the usual sense of the word. The associated L-function is the product of the Dirichlet L-series

L(χ_{1}, s) · L(χ_{2}, s + 1).

The composite mapping

Eis(χ_{1}, χ_{2}) ⟶ Ind(χ_{1}, χ_{2}) \underrightarrow{∼} W^{rat}(Ind(χ_{1}, χ_{2}))

can be described by the Fourier expansion map (4.0.2); however in the inversion formula (4.0.3) a constant term must be added.

4.5. We now describe what we need from the theory of L-functions for GL_{2} ×GL_{2}— see [J]. 4.5.0. Suppose that π is an irreducible constituent of Ω^{1}(\overline{M}) ⊗ \overline{Q}, and that π′ is one of the representations Ind(χ_{1}, χ_{2}). For f ∈ K^{rat}(π_{p}) and g ∈ K^{rat}(π′_{p}), write

(4.5.1)  I(f, g; s) = ∫_{Q_{p}^{∗}} |a|_{p}^{s−1}f(a)\overline{g(a)} d^{∗}a

(where complex conjugation acts on \overline{Q} ⊗ C via the second factor).

4.5.2. Proposition. (i) I(f, g; s) ∈ \overline{Q}(p^{−s}) ⊂ \overline{Q} ⊗ C(p^{−s}).

(ii) If f and g are spherical functions with f(1) = g(1) = 1, then

I(f, g; s) = \frac{L(π_{p} ⊗ \overline{χ}_{1,p}, s + 1) · L(π_{p} ⊗ \overline{χ}_{2,p}, s)}{L(\overline{χ}_{1,p}\overline{χ}_{2,p}ω_{π,p}, 2s)},

where ω_{π} is the central character of π.

The proof can be extracted from §§14, 15 of [J], in particular Proposition 14.4 and its proof, and Lemma 15.9.4. The only point which this reference does not make explicit is that I has coefficients in \overline{Q}. The key observation here is that conjugating the coefficients by τ ∈ Aut(C/Q) amounts to replacing f(a)\overline{g(a)} with (f(a)\overline{g(a)})^{τ} in the integral, hence with f (t_{p}(τ)a) \overline{g (t_{p}(τ)a)}. This is the same as substituting t_{p}(τ)a for a everywhere in the integral—which however leaves it unchanged. 4.5.3. Now suppose that f ∈ W^{rat}(π), g ∈ W^{rat}(π′) where π and π′ are as in 4.5.0, and let ϕ be any function as in 3.1.0 with the additional property that, for every τ ∈ Aut(C/Q) and k ∈ GẐ,

ϕ (( \begin{matrix}t(τ) & 0 \\ 0 & 1 \end{matrix} ) k) = ϕ(k) ∈ \overline{Q} ⊂ \overline{Q} ⊗ C.

Consider the integral

J(f, g, ϕ; s) = ∫_{A_{f}^{∗}×GẐ} |a|_{f}^{s−1}ϕ(k)f (( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) k) \overline{g (( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) k)} d^{∗}a dk

where dk denotes the Haar measure on GẐ with total mass one. By restricting to sufficiently small open subgroups of GẐ (leaving f, g, ϕ invariant under right translation) and using 4.5.2 and its proof one sees that J can be meromorphically continued to the s-plane (cf. [J], pp. 124ff.), and that, for some A = A(f, g, ϕ) ∈ \overline{Q},

J(f, g, ϕ; 1) = A · \frac{L(π ⊗ \overline{χ}_{1}, 2)L(π ⊗ \overline{χ}_{2}, 1)}{L(\overline{χ}_{1}\overline{χ}_{2}ω_{π}, 2)}.

4.5.4. We need one further local fact; by Lemma 4.2.0, we see that if S is any nonempty compact open subset of Q_{p}^{∗}, there exist f ∈ K^{rat}(π_{p}) and g ∈ K^{rat}(π′_{p}) such that

supp(f) ∪ supp(g) ⊆ S and I(f, g; 1) = 1.

5. Evaluation of the regulator integral.

5.0. We now begin the calculation of the integral of 1.3.2,

∫_{M_{K}(C)} log |u|\overline{d log v} ∧ ω

where π, V_{π} is fixed as in 1.2.0, K is an open compact subgroup of G_{f}, ω ∈ V_{π}^{K}, and u, v ∈ O^{∗}(M_{K}). 5.0.0. Write ϕ = div(u) and ξ = div(v). By virtue of (3.5.3) this integral equals

(5.0.1)  ∫_{M_{K}(C)} E_{ϕ}(z, g; 1)η̄_{ξ} ∧ ω.

Note that η_{ξ} is a \overline{Q}-linear combination of elements of Eis(1, χ), where χ runs over all even Dirichlet characters, by 3.5.4.

5.1. The main tool in the evaluation of (5.0.1) is Rankin’s trick. Let (π, V_{π}) be an irreducible constituent of Ω^{1}(\overline{M}) ⊗ \overline{Q}, and let (π′, V_{π′}) be a representation of G_{f} which is either also an irreducible constituent of Ω^{1}(\overline{M}) ⊗ \overline{Q}, or else one of the representations Ind(χ_{1}, χ_{2}). In the former case write V_{π′}^{0} = V_{π′}, and in the latter set V_{π′}^{0} = Eis(χ_{1}, χ_{2}) ⊆ V_{π′}. If ω ∈ V_{π}^{K} and η ∈ (V_{π′}^{0})^{K}, the associated Whittaker functions W_{ω}, W_{η} are defined (by 4.0 and 4.4 above).

5.1.0. Proposition (“Rankin’s trick”). For any ϕ : ±NẐ\GẐ/K → \overline{Q}, and any s with Re s > 1,

\begin{matrix}∫_{M_{K}(C)} E_{ϕ}(z, g; s) η̄∧ & ω = πi\frac{Γ(s + 1)}{(4π)^{s−1}} [GẐ : ±K] \\ × ∫_{A_{f}^{∗}×GẐ} ϕ(k)W_{ω} (( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) k) \overline{W_{η} (( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) k)} |a|_{f}^{s−1} d^{∗}a dk \end{matrix}

where dk is the Haar measure on GẐ with total mass 1.

Proof. The integral is

\begin{aligned}_{G_{Q}\H^{±}×G_{f} /K}^{∫} −2π & _{γ∈B_{Q}^{+}\G_{Q}}^{∑} ϕ̂_{s}(γg)I(γz)^{s} η̄ ∧ ω \\ & =_{B_{Q}^{+}\H^{±}×G_{f} /K}^{∫} −2πϕ̂_{s}(g)I(z)^{s} η̄ ∧ ω \end{aligned}

(since η̄ ∧ ω is G_{Q}-invariant). Now if Im(z) > 0 then

\begin{matrix}η̄ ∧ ω = & \overline{(_{b>0}^{∑} W_{η} (( \begin{matrix}b & 0 \\ 0 & 1 \end{matrix} ) g) d (e^{2πibz}) + C)} \\ ∧ _{a>0}^{∑} W_{ω} (( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) g) d (e^{2πiaz}) \end{matrix}

where C denotes the “constant term” in the Fourier expansion of η. The only terms contributing to the integral are those for which a = b, and so it may be rewritten as

_{B_{Q}^{+}\H×G_{f} /K}^{∫} −2πϕ̂_{s}(g)y^{s} _{a>0}^{∑} \overline{W_{η} (( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) g)}W_{ω} (( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) g) \overline{d (e^{2πiaz})} ∧ d (e^{2πiaz})

= 16π^{3}i_{Z_{Q}N_{Q}\H×G_{f} /K}^{∫} ϕ̂_{s}(g)y^{s}\overline{W_{η}(g)}W_{ω}(g)e^{−4πy} dx ∧ dy

= πi\frac{Γ(s + 1)}{(4π)^{s−1}} [GẐ : ±K]_{Z_{Q}N_{f} \G_{f}}^{∫} ϕ̂_{s}(g)\overline{W_{η}(g)}W_{ω}(g) dg

where dg is the measure on Z_{Q}N_{f}\G_{f} derived from the Haar measure on G_{f} for which GẐ has measure 1. Observing that

Z_{Q}N_{f}\G_{f} = ±NẐ\D_{f}GẐ

and that, in terms of the decomposition

G_{f} = N_{f}A_{f}GẐ, g = ( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) ( \begin{matrix}a_{1} & 0 \\ 0 & a_{2} \end{matrix} ) k

the Haar measure is dg = |\frac{a_{1}}{a_{2}} |_{f}^{−1}dx d^{∗}a_{1} d^{∗}a_{2} dk, we obtain the required formula.

5.1.1. Remark. When deg(ϕ) = 0, the above yields

(5.1.2) \begin{aligned}_{∫M_{K}(C)} & E_{ϕ}(z, g; 1)η̄ ∧ ω = πi [GẐ : ±K] \\ & × lim_{s→1} ∫_{A_{f}^{∗}×GẐ} ϕ(k)W_{ω} (( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) k) \overline{W_{η} (( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) k)}|a|_{f}^{s−1} d^{∗}a dk. \end{aligned}

For the nonvanishing argument in §6 it is useful to note that the limit of the integrals in 5.1.0 exists for every ϕ (even though in general E_{ϕ} has a pole at s = 1). Indeed, since the residue of E_{ϕ} at s = 1 is locally constant, the same reasoning as in (3.5.3) above shows that it does not contribute to the integral.

5.2. It remains to collect up the loose ends of the argument. As observed in 5.0 above, the integral (5.0.1) is a \overline{Q}-linear combination of integrals of the form (5.1.2), with η ∈ (V_{π′}^{0})^{K} where π′ = Ind(1, χ) for even characters χ. Furthermore ϕ = div(u) satisfies the Galois invariance property required in 4.5.3. This paragraph tells us that the integral (5.0.1) lies in the \overline{Q}-subspace of \overline{Q} ⊗ C generated by the elements

2πi\frac{L(π, 2) · L(π ⊗ \overline{χ}, 1)}{L(\overline{χ}ω_{π}, 2)} ∈ \overline{Q} ⊗ C,

as χ varies over all even Dirichlet characters of Q. By 2.3, these elements are \overline{Q}- multiples of

2πi · c^{+}(π) · L′(π̌, 0).

This concludes the proof of part (i) of Theorem 1.3.2.

6. Nonvanishing of the regulator integral.

6.0. In this section we complete the proof of part (ii) of Theorem 1.3.2. Fix π as in 1.2.0, and let χ ≠ 1, \overline{ω}_{π} be an even Dirichlet character of Q such that L(π ⊗\overline{χ}, 1) ≠ 0 (see 2.2.0). In view of 3.5.1 and 3.5.4, in order to establish 1.3.2(ii) it suffices to prove the following.

6.0.0 Proposition. There exist ϕ : ±NẐDẐ\GẐ ⟶ Q of degree zero (3.1.0), ω ∈ V_{π}, and η ∈ Eis(1, χ), such that, for some K ⊆ GẐ fixing ϕ, ω and η under right translation, the integral

(6.0.1)  ∫_{M_{K}(C)} E_{ϕ}(z, g; 1) η̄ ∧ ω

is nonzero.

6.1.0. Let us first prove 6.0.0 without the requirement that deg(ϕ) be zero. Choose ω and η in such a way that their Whittaker functions factorise:

\begin{aligned}W_{ω} & = ⊗_{p}′ W_{ω,p} \\ W_{η} & = ⊗_{p}′ W_{η,p}, \end{aligned}

and such that for a suitable finite set S of primes, one has:

• for p ∉ S, W_{ω,p} and W_{η,p} are right GẐ-invariant; and

• for p ∈ S, we have (cf. 4.5.4 above)

I (W_{ω,p} (( \begin{matrix}· & 0 \\ 0 & 1 \end{matrix} )) , W_{η,p} (( \begin{matrix}· & 0 \\ 0 & 1 \end{matrix} )); 1) = 1.

For p ∈ S, denote by K_{p} an open subgroup of G_{Z_{p}} such that W_{ω,p} and W_{η,p} are right K_{p}-invariant. Choose ϕ to be the characteristic function of

NẐ(_{p∉S}^{∏} G_{Z_{p}} × _{p∈S}^{∏} K_{p} · D_{Z_{p}}).

Then in view of the choice of χ (from 6.0 above), it follows from Propositions 5.1.0 and 4.5.2 that the integral (6.0.1) does not vanish, as claimed. 6.1.1. Finally, given any ϕ (not necessarily of degree zero), ω and η such that the integral (6.0.1) is nonzero, denote by ψ the function on GẐ such that ψ̂ (in the notation of 3.2.0) is the (1, 1)-component of ϕ̂ in the decomposition 3.2.1. Then deg(ϕ) = deg(ψ), and the integral (6.0.1), with ϕ replaced by ψ, vanishes, since the product of the central characters of the three representations involved is 1·\overline{χ}·ω_{π} ≠ 1. Thus replacing ϕ by ϕ − ψ leaves (6.0.1) unchanged, while deg(ϕ − ψ) = 0.

7. Integrality.

7.0. In this section we prove Theorem 1.1.2(iii). The main ingredient is the analysis of the reduction mod p of modular units using supersingular points. 7.0.0. Throughout the section, we fix a prime p and an integer m ≥ 3 which is prime to p. Define

G^{(p)} = \begin{matrix}∏ & ′ \\ l≠p \end{matrix} G_{l}.

We denote by H(m, p) the Hecke algebra of compactly-supported functions on G^{(p)} with values in \overline{Q} which are biinvariant under G^{(p)} ∩ K_{m}. 7.0.1. We need to recall the structure of the reduction mod p of certain modular curves. If M_{K/Z} is a model for M_{K} over Spec Z, we will denote M_{K/Z} ⊗_{Z} F_{p} by M_{K/F_{p}}. In the cases

K = K_{m} ∩ K_{0}(p) or K(mp^{k}),

we take M_{K/Z} to be the regular model described in [DR], V.1.18, V.4 and [KM], Chapter 13. 7.0.2. There is a natural map M_{m/Z} ⟶ Spec Z[μ μ μ_{m}] induced by the Weil pairing, whose fibres in characteristic p are smooth and geometrically connected. We write Σ_{m,p} for the set of supersingular points of M_{m/Z} ⊗_{Z} \overline{F}_{p}, and S_{m,p} for the set of primes of Q(μ μ μ_{m}) lying over p (which is thus the same as the set of connected components of M_{m/Z} ⊗ F_{p}). If A is any abelian group, let A[Σ_{m,p}], A[S_{m,p}] be the groups of A-valued functions on Σ_{m,p}, S_{m,p}. There is a natural inclusion A[S_{m,p}] ⊂ A[Σ_{m,p}], and also a surjection γ : A[Σ_{m,p}] ⟶ A[S_{m,p}] given by summing along the fibres; write A[Σ_{m,p}]^{0} =^{def} ker γ. 7.0.3. Consider the modular curve \overline{M}_{K} for K = K_{0}(p) ∩ K_{m}. There is a short exact sequence

(7.0.4)  0 ⟶ T ⟶ Pic^{0}(\overline{M}_{K/F_{p}}) ⟶ Pic^{0}(\overline{M}_{m/F_{p}})^{2} ⟶ 0

in which T is a torus over F_{p} whose character group is Z[Σ_{m,p}]^{0} (cf. [DR] V.1.18 and I.3.7).

7.1.0. If (π, V_{π}) = \begin{matrix}⨂ & ′ \\ l \end{matrix}(π_{l}, V_{π,l}) is an irreducible representation of G_{f} occurring in Ω^{1}(\overline{M}) ⊗ \overline{Q}, let

V_{π}^{(p)} = \begin{matrix}⨂ & ′ \\ l≠p \end{matrix} V_{π,l};

it is an irreducible admissible representation of G^{(p)}.

We denote by sp(1) the (special) representation of G_{p} on the space of locally constant functions on P^{1}(Q_{p}) modulo constants.

7.1.1. Theorem. There is an isomorphism of H(m, p)-modules

\overline{Q}[Σ_{m,p}]/\overline{Q}[S_{m,p}] \underrightarrow{∼} _{π}^{⨁}(V_{π}^{(p)})^{K_{m}},

where the sum is taken over all irreducible (π, V_{π}) ⊂ Ω^{1}(\overline{M}) ⊗ \overline{Q} for which π_{p} is isomorphic to sp(1).

This result is well known, and in principle is due to Eichler. The result in the case m = 1 has been used by Mestre and Oesterlé to find Weil curves with prime conductor (see [M]). Since there does no seem to be an easily accessible proof of the general case in the literature, we sketch one here. First note that the right-hand space is isomorphic, as an H(m, p)-module, to the sum of spaces V_{π}^{K_{m}∩K_{0}(p)}, taken over all (π, V_{π}) for which p divides the conductor of π. This in turn is precisely the space of all holomorphic cusp forms of weight two on K(m) ∩ K_{0}(p) which are “p-new”. Now the reduction mod p of the corresponding part of the Jacobian of \overline{M}_{K_{m}∩K_{0}(p)} is the torus T. Since End T \underrightarrow{∼} End(Hom(Z[Σ_{m,p}]^{0}, Z)) by (7.0.4), the result follows.

7.2.0. In what follows, the reductions mod p of modular curves and their components will always be assumed to be endowed with the reduced subscheme structure. 7.2.1. Consider M_{n/Z} for n = mp^{k}, k ≥ 1. We recall the structure of M_{n/F_{p}} and the covering M_{n/F_{p}} ⟶ M_{m/F_{p}}. 7.2.2. The irreducible components of M_{n/F_{p}} are indexed by S_{m,p} × P^{1}(Z/p^{k}Z). The action of G_{Z/nZ} = G_{Z/mZ} × G_{Z/p^{k}Z} on them is the product of the action of G_{Z/mZ} on S_{m,p} given by determinant followed by the Artin map, and the usual action of G_{Z/p^{k}Z} on P^{1}(Z/p^{k}Z) by linear fractional transformations. 7.2.3. If D is a G_{Z/mZ}-orbit of an irreducible component of M_{n/F_{p}}, then D is a smooth curve over F_{p} (but not in general irreducible). H(m, p) acts on D by correspondences. The covering D ⟶ M_{m/F_{p}} is finite and flat, everywhere of degree p^{k}ϕ(p^{k}), and is totally ramified over the supersingular points. Thus we can identify the supersingular points of D with Σ_{m,p}, and this identification is compatible with the action of H(m, p). By the previous paragraph, the irreducible components of D can be identified with S_{m,p}.

7.2.4. Let u ∈ O^{∗}(M_{n}), and let C be an irreducible component of M_{n/F_{p}}. Use the superscript ^{h} to denote the complement of the supersingular points. u gives rise to a unit (modulo roots of unity) on C^{h} as follows. Let ord_{C} denote the normalised valuation on the function field Q(M_{n}) at the generic point of C, and write e = ord_{C}(p) = p^{k}ϕ(p^{k}). Then if k(C) is the function field of C,

\begin{aligned}u_{C} & = ∂_{C}\{u, p\} \\ & = u^{e}/p^{ord_{C}(u)} \end{aligned}

may be viewed as belonging to k(C)^{∗}. Its only poles or zeros can be at singular points of the fibre M_{n/F_{p}}, so in fact u_{C} ∈ O^{∗}(C^{h}).

7.2.5. Theorem. u_{C} has the same order of pole or zero at each supersingular point of C.

Proof. It is convenient to consider, rather than C, the orbit D = ∐ C_{i} of C under G_{Z/mZ}; thus D^{h} = ∐ C_{i}^{h}. Then define u_{D} ∈ O^{∗}(D^{h}) to be the unit whose restriction to C_{i}^{h} is u_{C_{i}}. Consider the composite homomorphism:

O^{∗}(M_{n}) ⟶ O^{∗}(D^{h}) ⟶ Z[Σ_{m,p}]

where the first map is given by u ↦ u_{D}, and the second is the divisor map. The assertion of the theorem is that the image of this composite lies in Z[S_{m,p}]. But by composing with the quotient map and extending scalars we obtain a map

(O^{∗}(M_{n})/Q(μ μ μ_{n})^{∗}) ⊗_{Z} \overline{Q} ⟶ \overline{Q}[Σ_{m,p}]/\overline{Q}[S_{m,p}]

which is a homomorphism of H(m, p)-modules. The first space occurs in the space of Eisenstein series (by 3.5.2), and the second in the space of cusp forms (by Theorem 7.1.1)—so by the same principle as in the proof of the Manin-Drinfeld theorem (3.4.0 above) this map is zero.

7.3.0. Fix a positive integer n, which we assume is the product of two coprime integers, each ≥ 3. Recall that M_{n/Z} is the complement in \overline{M}_{n/Z} of the cuspidal subscheme M_{n/Z}^{∞}, itself a disjoint union of copies of Spec Z[μ μ μ_{n}]. Denote the set of connected components of M_{n/Z}^{∞} simply by “cusps”. We have boundary maps in absolute cohomology/homology (see [Be1], 2.2.3 and Chapter V)

∂_{∞} : H_{A}^{2}(M_{n}, Q(2)) ⟶ H_{A}^{1}(M_{n}^{∞}, Q(1)) = ∐ ; cuspsQ(μ μ μ_{n})^{∗} ⊗_{Z} Q

∂_{p} : H_{A}^{2}(M_{n}, Q(2)) ⟶ H_{A}^{′−1}(M_{n/F_{p}}, Q(0)) ⊂ H_{A}^{1}(M_{n/F_{p}}^{h}, Q(1))
= O^{∗}(M_{n/F_{p}}^{h}) ⊗_{Z} Q.

From the localisation sequence, we have (cf. 1.1.1 above)

\begin{matrix}Q_{n} =^{def} & \{O^{∗}(M_{n}), O^{∗}(M_{n})\} ∩ H_{A}^{2}(M_{n}, Q(2)) \\ = & \{O^{∗}(M_{n}), O^{∗}(M_{n})\} ∩ ker ∂_{∞}. \end{matrix}

7.3.1. Theorem. Q_{n} ⊆ H_{A}^{2}(\overline{M}_{n/Z}, Q(2)).

7.3.2. Remark. This is sufficient to prove Theorem 1.1.2(iii). Indeed, from the definition of P_{K} and the remarks in 1.2.9 above, if ξ ∈ P_{K} then for some n ≥ 3 with K_{n} ⊆ K, there exists α ∈ Q_{n} satisfying ξ = θ_{K_{n}/K∗}α. Now θ_{K_{n}/K∗} maps H_{A}^{•}(\overline{M}_{n/Z}, ∗) into H_{A}^{•}(\overline{M}_{K/Z}, ∗), since the graph of θ_{K_{n}/K} extends to a correspondence on the regular models over Z. Proof. We have to show that ∂_{p}(Q_{n}) = 0 for every p. If p |̸n then this follows at once from the localisation sequence, since in this case H_{A}^{′−1}(M_{n/F_{p}}, Q(0)) = O^{∗}(M_{n/F_{p}})⊗_{Z} Q. So let us restrict to the case n = mp^{k}, k ≥ 1. We have the following diagram with exact rows, in which the solid arrows commute up to sign:

[figure: 0; Q_{n}; {O^{∗}(M_{n}), O^{∗}(M_{n})}; ∂_{∞}; _{cusps}^{∐} Q(μ μ μ_{n})^{∗} ⊗_{Z} Q]

[figure: 0; O^{∗}(\overline{M}_{n/F_{p}}^{h}) ⊗_{Z} Q; α_{p}; O^{∗}(M_{n/F_{p}}^{h}) ⊗_{Z} Q; ∂_{p}; ϵ_{p}; _{cusps}^{∐} _{S_{m,p}}^{∐} Q; θ_{p}]

Here the maps ∂_{p}, ∂_{∞} are those in the localisation sequence. The map

ϵ_{p} : O^{∗}(M_{n/F_{p}}^{h}) ⊗_{Z} Q ⟶ _{M_{n/Fp}^{∞}}^{∐} Q

is the infinite part of the divisor map (noting that the points of M_{n/F_{p}}^{∞} are just (cusps) × S_{m,p}). Finally θ_{p} is the “content” map

Q(μ μ μ_{n})^{∗} ⟶^{(ord_{℘})_{℘}} _{℘∈S_{m,p}}^{∐} Q.

We therefore obtain a map α_{p} as indicated making the left-hand square commutative. It is enough to show that α_{p}(Q_{n}) = 0. But for any component C, we have in O^{∗}(C^{h})⊗_{Z} Q

∂_{C}\{f, g\}^{e} = \frac{f_{C}^{ord_{C} g}}{g_{C}^{ord_{C} f}} .

Applying 7.2.5 above we see that if ξ ∈ Q_{n} then α_{p}(ξ), when restricted to any irreducible component C̄^{h} of \overline{M}_{n/F_{p}}^{h}, has the same order at each supersingular point. But every such C̄^{h} is the complement, in the complete curve C̄, of the set of supersingular points, and the total degree of the divisor of α_{p}(ξ) on C̄ is of course zero. Therefore α_{p}(ξ) = 0 as required.

7.4. Remark. In [Be1, Be2], Beilinson suggests another proof of 7.3.1 above. This relies however on the assumption that

O^{∗}(M_{n}) ⊗_{Z} Q = Q(μ μ μ_{n})^{∗} · O^{∗}(M_{n/Z}) ⊗_{Z} Q

(which would imply an “integral” version of the Manin-Drinfeld theorem, see [Be2], 5.2.4). This does not hold in general; for example, if p divides n, the modular unit Δ(pz)/Δ(z) belongs to the space on the left, but not to that on the right. The difficulty arises because the Eisenstein representations Eis(χ_{1}, χ_{2}) are highly reducible, and can be intertwined with many different irreducible representations of G_{f}. The argument of 5.5 of [Be1] produces certain maps

Eis(χ_{1}, χ_{2}) ⟶ U_{p}

where U_{p} is a G_{f}-module on which, for every l ≠ p, G_{l} acts via a sum of abelian characters. Using remark 3.3(ii) above, one can then obtain the relation (compare 3.5.4 above)

d log O^{∗}(M_{Z}) ⊗_{Z} \overline{Q} ⊇ ⨁ ; {χ≠1 \atop χ even} Eis(1, χ).

This is enough to prove the weaker version of 1.1.2(iii) described in 1.1.3(iii). One can also obtain a precise description of d log O^{∗}(M_{Z}) ⊗_{Z} \overline{Q} analogous to 3.5.4, but we shall not go into this here.

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