On a Canonical Class of Green Currents

for the Unit Sections of Abelian Schemes

For Christophe Soulé, on the occasion of his sixtieth birthday

V. Maillot and D. Rössler

Received: December 12, 2014

Communicated by Takeshi Saito

Abstract. We show that on any abelian scheme over a complex quasi-projective smooth variety, there is a Green current for the zerosection, which is axiomatically determined up to ∂ and ¯∂-exact differential forms. On an elliptic curve, this current specialises to a Siegel function. We prove generalisations of classical properties of Siegel functions, like distribution relations and reciprocity laws. Furthermore, as an application of a refined version of the arithmetic Riemann-Roch theorem, we show that the above current, when restricted to a torsion section, is the realisation in analytic Deligne cohomology of an element of the (Quillen) K_{1} group of the base, the corresponding denominator being given by the denominator of a Bernoulli number. This generalises the second Kronecker limit formula and the denominator 12 computed by Kubert, Lang and Robert in the case of Siegel units. Finally, we prove an analog in Arakelov theory of a Chern class formula of Bloch and Beauville, where the canonical current plays a key role.

2010 Mathematics Subject Classification: 14G40, 14K15, 11G16, 11G55, 1 Introduction

In this article, we show that on any abelian scheme over a complex quasiprojective smooth variety, there is a Green current 𝔤 for the the cycle given by the zero-section of the abelian scheme, which is uniquely determined, up to ∂ and ¯∂-exact forms, by three axioms — see Theorem 1.1. We proceed to show that this Green current is naturally compatible with isogenies (ie it satisfies distribution relations), up to ∂ and ¯∂-exact forms (see Theorem 1.2.1), and that it intervenes in an Arakelov-theoretic generalization of a formula of Bloch and Beauville (see [2, p. 249] for the latter), which is proven here without resorting to the Fourier-Mukai transform. See Theorem 1.2.2. Furthermore, we show that if the basis of the abelian scheme is a point (ie if the abelian scheme is an abelian variety), then 𝔤 is a harmonic Green current (see Theorem 1.2.3). The current 𝔤 is also shown to be compatible with products (see Theorem 1.2.5). Finally, we show that the restriction of 𝔤 to the complement of the zero-section has a spectral interpretation. Up to a sign, it is given there by the degree (g − 1) part of the analytic torsion form of the Poincaré bundle of the abelian scheme. See Theorem 1.3.1 for this. In point 2 of the same theorem, we show that the restriction of the higher analytic torsion form to torsion sections, which never meet the zero-section, lies in the rational image of the Beilinson regulator from K_{1} to analytic Deligne cohomology and we give a multiplicative upper bound for the denominators involved. To prove Theorem 1.3.1, we make heavy use of the arithmetic Riemann-Roch theorem in higher degrees proven in [22] and to compute the denominators described in Theorem 1.3.2, we apply the Adams-Riemann-Roch theorem in Arakelov geometry proven in [42]. If one specializes to elliptic schemes (i.e. abelian schemes of relative dimension 1) the results proven in Theorems 1.1, 1.2 and 1.3 one recovers many results contained in the classical theory of elliptic units. In particular, on elliptic schemes the current 𝔤 is described by a Siegel function and the spectral interpretation of 𝔤 specializes to the second Kronecker limit formula. The reciprocity law for elliptic units (ie the analytic description of the action of the Galois group on the elliptic units) is also easily obtained and (variants) of the results of Kubert-Lang and Robert on the fields of definition of elliptic units are recovered as a special case of the above denominator computations. Details about elliptic schemes are given in section 5 where references to the classical literature are also given. The reader will notice that even in the case of elliptic schemes, our methods of proof are quite different from the classical ones. The current 𝔤 can also be used to describe the realisation in analytic Deligne cohomology of the degree 0 part of the polylogarithm on abelian schemes introduced by J. Wildeshaus in [45] (see also [25]). The fact that this should be the case was a conjecture of G. Kings. His conjecture is proven in [26]. Here is a detailed description of the results. Let (R, Σ) be an arithmetic ring. By definition, this means that R is an excellent regular ring, which comes with a finite conjugation-invariant set Σ of embeddings into ℂ (see [19, 3.1.2]). For example R might be ℤ with its unique embedding into ℂ, or ℂ with the identity and complex conjugation as embeddings. Recall that an arithmetic variety over R is a scheme, which is flat and of finite type over R. In this text, all arithmetic varieties over R will also be assumed to be regular, as well as quasi-projective over R. For any arithmetic variety X over R, we write as usual

X(ℂ) := ∐_{σ∈Σ}(X ×_{R,σ} ℂ)(ℂ) = ∐_{σ∈Σ} X(ℂ)_{σ}.

Let D^{p,p}(X_{ℝ}) (resp. A^{p,p}(X_{ℝ})) be the ℝ-vector space of currents (resp. differential forms) γ on X(ℂ) such that

• γ is a real current (resp. differential form) of type (p, p);

• F_{∞}^{∗}γ = (−1)^{p}γ,

where F_{∞} : X(ℂ) → X(ℂ) is the real analytic involution given by complex conjugation. We then define

D̃^{p,p}(X_{ℝ}) := D^{p,p}(X_{ℝ})/(im ∂ + im ¯∂)

(resp.

Ã^{p,p}(X_{ℝ}) := A^{p,p}(X_{ℝ})/(im ∂ + im ¯∂) ).

All these notations are standard in Arakelov geometry. See [44] for a compendium. It is shown in [19, Th. 1.2.2 (ii)] that the natural map Ã^{p,p}(X_{ℝ}) → D̃^{p,p}(X_{ℝ}) is an injection. If Z a closed complex submanifold of X(ℂ), we shall write more generally D_{Z}^{p,p}(X_{ℝ}) for the ℝ-vector space of currents γ on X(ℂ) such that

• γ is a real current of type (p, p);

• F_{∞}^{∗}γ = (−1)^{p}γ;

• the wave-front set of γ is included in the real conormal bundle of Z in X(ℂ).

Similarly, we then define the ℝ-vector spaces

D̃_{Z}^{p,p}(X_{ℝ}) := D_{Z}^{p,p}(X_{ℝ})/D_{Z,0}^{p,p}(X_{ℝ})

where D_{Z,0}^{p,p}(X_{ℝ}) is the set of currents γ ∈ D_{Z}^{p,p}(X_{ℝ}) such that: there exists a complex current α of type (p − 1, p) and a complex current β of type (p, p − 1) such that γ := ∂α + ¯∂β and such that the wave-front sets of α and β are included in the real conormal bundle of Z in X(ℂ). See [24] for the definition (and theory) of the wave-front set.

It is a consequence of [11, Cor. 4.7] that the natural morphism D̃_{Z}^{p,p}(X_{ℝ}) → D̃^{p,p}(X_{ℝ}) is an injection.^{1} Thus the real vector space D̃_{Z}^{p,p}(X_{ℝ}) can be identified with a subspace of the real vector space D̃^{p,p}(X_{ℝ}). Furthermore, it is a consequence of [11, Th. 4.3] that for any R-morphism f : Y → X of arithmetic varieties, there is a natural morphism of ℝ-vector spaces

f^{∗} : D̃_{Z}^{p,p}(X_{ℝ}) → D̃_{f(ℂ)^{∗}(Z)}^{p,p}(Y_{ℝ}),

provided f(ℂ) is transverse to Z. This morphism extends the morphism Ã^{p,p}(X_{ℝ}) → Ã^{p,p}(Y_{ℝ}), which is obtained by pulling back differential forms. Fix now S an arithmetic variety over R. Let π : A → S be an abelian scheme over S of relative dimension g. We shall write as usual A^{∨} → S for the dual abelian scheme. Write ϵ (resp. ϵ^{∨}) for the zero-section of A → S (resp. A^{∨} → S) and also S_{0} (resp. S_{0}^{∨}) for the image of ϵ (resp. ϵ^{∨}). We denote by the symbol P the Poincaré bundle on A ×_{S} A^{∨}. We equip the Poincaré bundle P with the unique metric h_{P} such that the canonical rigidification of P along the zerosection A^{∨} → A×_{S} A^{∨} is an isometry and such that the curvature form of h_{P} is translation invariant along the fibres of the map A(ℂ) ×_{S(ℂ)} A^{∨}(ℂ) → A^{∨}(ℂ). We write \overline{P} := (P, h_{P}) for the resulting hermitian line bundle. Write \overline{P}^{0} be the restriction of P to A ×_{S} (A^{∨}\S_{0}^{∨}). The aim of this text is now to prove the following three theorems.

Theorem 1.1. There is a unique class of currents 𝔤_{A} ∈ D̃^{g−1,g−1}(A_{ℝ}^{∨}) with the following three properties:

(a) Any element of 𝔤_{A} is a Green current for S_{0}^{∨}(ℂ).

(b) The identity (S_{0}^{∨}, 𝔤_{A}) = (−1)^{g}p_{2,∗}(ĉh(\overline{P}))^{(g)} holds in ĈH^{g}(A^{∨})_{ℚ}.

(c) The identity 𝔤_{A} = [n]_{∗}𝔤_{A} holds for all n ⩾ 2.

Here the morphism p_{2} is the second projection A ×_{S} A^{∨} → A^{∨} and [n] : A^{∨} → A^{∨} is the multiplication-by-n morphism. The symbol ĉh(·) refers to the arithmetic Chern character and ĈH^{•}(·) is the arithmetic Chow group. See [19, 1.2] for the notion of Green current. Supplement. The proof of Theorem 1.1 given below shows that if S is assumed proper over Spec R, then the condition (b) can be replaced by the following weaker condition : (b)’ The identity of currents dd^{c}𝔤 + δ_{S_{0}^{∨}(ℂ)} = (−1)^{g}p_{2,∗}(ch(\overline{P}))^{(g)} holds. Here dd^{c} := \frac{i}{2π} ∂ ¯∂ and δ_{S_{0}^{∨}(ℂ)} is the Dirac current associated to S_{0}^{∨}(ℂ) in A^{∨}(ℂ). Furthermore, ch(\overline{P}) is the Chern character form of the hermitian bundle \overline{P}. See [20, Intro.] for this. Remarks. (1) The condition (b) apparently makes the current 𝔤_{A} dependent on the arithmetic structure of A. We shall show in 1.2.4 below that this is not the case. In particular, in defining 𝔤_{A}, we could have assumed that R = ℂ. The settting of arithmetic varieties is used in Theorem 1.1 because it is the natural one for property (b). Formula (15) gives a purely analytic expression for 𝔤_{A}. (2) It is tempting to try to refine Theorem 1.1 by using in property (b) the arithmetic Chow groups defined by J.-I. Burgos (in [10]) rather than the arithmetic Chow groups of Gillet-Soulé. One would then obtain a class of forms with certain logarithmic singularities, rather than a class of currents. Such a refinement does not seem to be easily attainable though, because of the lack of covariant functoriality of the spaces of forms mentioned in the last sentence. The next theorem gives some properties of the class of currents 𝔤_{A}. Let L be a rigidified line bundle on A. Endow L with the unique hermitian metric h_{L}, which is compatible with the rigidification and whose curvature form is translation-invariant on the fibres of A(ℂ) → S(ℂ). Let \overline{L} := (L, h_{L}) be the resulting hermitian line bundle. Let ϕ_{L} : A → A^{∨} be the polarisation morphism induced by L.

1many thanks to J.-I. Burgos for bringing this to our attention

Documenta Mathematica 20 (2015) 631–668

Theorem 1.2. 1. Let ι : A → B be an isogeny of abelian schemes over S. Then the identity ι_{∗}^{∨}(𝔤_{B}) = 𝔤_{A} holds.

2. Suppose that L is ample relatively to S and symmetric. Then the equalities

(S_{0}^{∨}, 𝔤_{A}) = (−1)^{g}p_{2,∗}(ĉh(\overline{P})) = \frac{1}{g!\sqrt{deg(ϕ_{L})}}ϕ_{L,∗}(ĉ_{1}(\overline{L})^{g})

are verified in ĈH^{∗}(A^{∨})_{ℚ}.

3. If S → Spec R is the identity on Spec R then any element of 𝔤_{A} is a harmonic Green current for S_{0}^{∨}(ℂ), where A^{∨}(ℂ) is endowed with a conjugation invariant Kähler metric, whose Kähler form is translation invariant.

4. The class 𝔤_{A} is invariant under any change of arithmetic rings (R, Σ) → (R′, Σ′).

5. Let B → S be another abelian scheme and let π_{A^{∨}} : A^{∨} ×_{S} B^{∨} → A^{∨} (resp. π_{B^{∨}} : A^{∨} ×_{S} B^{∨} → B^{∨}) be the natural projections. Then

𝔤_{A×_{S}B} = π_{A^{∨}}^{∗}(𝔤_{A}) ∗ π_{B^{∨}}^{∗}(𝔤_{B})  (1)

6. The class of currents 𝔤_{A} lies in D̃_{S_{0}^{∨}(ℂ)}^{g−1,g−1}(A_{ℝ}^{∨}).

7. Let T be a an arithmetic variety over R and let T → S be a morphism of schemes over R. Let A_{T} be the abelian scheme obtained by base-change and let BC : A_{T} → A be the corresponding morphism. Then BC(ℂ) is tranverse to S_{0}^{∨}(ℂ) and BC^{∗}𝔤_{A} = 𝔤_{A_{T}}.

Here ι^{∨} : B^{∨} → A^{∨} is the isogeny, which is dual to ι. For the notion of harmonic Green current, see [8] and [29]. The pairing ∗ appearing in the equation (1) is the ∗-product of Green currents. See [19, par. 2.2.11, p. 122] for the definition. Recall that an S-isogeny between the abelian schemes A and B is a flat and finite S-morphism A → B, which is compatible with the group-scheme structures. The symbol ĉ_{1}(·) refers the first arithmetic Chern class; see [20, Intro.] for this notion. Theorem 1.2.1 generalizes to higher degrees the distribution relations of Siegel units. See section 5 below for details. If the morphism S → Spec R is the identity on Spec R and R is the ring of integers of a number field, then it is shown in [29, Prop. 11.1 (ii)] that Theorem 1.2.3 implies Theorem 1.2.2. Still in the situation where S → Spec R is the identity on Spec R, another construction of a Green current for S_{0}^{∨}(ℂ) is described in A. Berthomieu’s thesis [3]. The current constructed by Berthomieu is likely to be harmonic (it is not proven in [3], but according to the author [private communication] it can easily be shown). The current constructed in [3] satisfies the identity in Theorem 1.3.1 by construction. The last theorem relates the current 𝔤_{A} to the Bismut-Köhler analytic torsion form of the Poincaré bundle (see [6, Def. 3.8, p. 668] for the definition). Let λ be a (1, 1)-form on A(ℂ) defining a Kähler fibration structure on the fibration A(ℂ) → S(ℂ) (see [6, par. 1] for this notion). With the form λ, one can canonically associate a hermitian metric on the relative cotangent bundle Ω_{A/S} and we shall write \overline{Ω}_{A/S} for the resulting hermitian vector bundle. We suppose that λ is translation invariant on the fibres of the map A(ℂ) → S(ℂ) as well as conjugation invariant. We shall write

T (λ, \overline{P}^{0}) ∈ Ã((A^{∨}\S_{0}^{∨})_{ℝ}) := ⨁_{p⩾0} Ã^{p,p}((A^{∨}\S_{0}^{∨})_{ℝ})

for the Bismut-Köhler higher analytic torsion form of \overline{P}^{0} along the fibration

A(ℂ) ×_{S(ℂ)} (A^{∨}(ℂ)\S_{0}^{∨}(ℂ)) ⟶ A^{∨}(ℂ)\S_{0}^{∨}(ℂ).

For any regular arithmetic variety X over R, the (Beilinson) regulator map gives rise to a morphism of groups

reg_{an} : K_{1}(X) ⟶ ⨁_{p⩾0} H_{D,an}^{2p−1}(X_{ℝ}, ℝ(p)).

To define the space H_{D,an}^{2p−1}(X_{ℝ}, ℝ(p)) and the map reg_{an}, let us first write H_{D,an}^{∗}(X, ℝ(·)) for the analytic real Deligne cohomology of X(ℂ). By definition,

H_{D,an}^{q}(X, ℝ(p)) := ℍ^{q}(X(ℂ), ℝ(p)_{D,an})

where ℝ(p)_{D,an} is the complex of sheaves of ℝ-vector spaces

0 → ℝ(p) → O_{X(ℂ)} →^{d} Ω_{X(ℂ)}^{1} → ⋯ → Ω_{X(ℂ)}^{p−1} → 0

on X(ℂ) (for the ordinary topology). Here ℝ(p) := (2iπ)^{p} ℝ ⊆ ℂ. We now define

H_{D,an}^{2p−1}(X_{ℝ}, ℝ(p)) := {γ ∈ H_{D,an}^{2p−1}(X, ℝ(p)) | F_{∞}^{∗}γ = (−1)^{p}γ}.

By construction, the regulator map K_{1}(X) → ⊕_{p⩾0}H_{D,an}^{2p−1}(X, ℝ(p)) (see [12] for a direct construction of the regulator and further references) factors through ⊕_{p⩾0}H_{D,an}^{2p−1}(X_{ℝ}, ℝ(p)) and thus gives rise to a map K_{1}(X) → ⊕_{p⩾0}H_{D,an}^{2p−1}(X_{ℝ}, ℝ(p)). This is the definition of the map reg_{an}. It is shown in [12, par 6.1] that there is a natural inclusion H_{D,an}^{2p−1}(X_{ℝ}, ℝ(p)) ↪ Ã^{p−1,p−1}(X_{ℝ}). For the next theorem, define

N_{2g} := 2 · denominator [(−1)^{g+1}B_{2g}/(2g)],

where B_{2g} is the 2g-th Bernoulli number. Recall that the Bernoulli numbers are defined by the identity of power series:

∑_{t⩾1} B_{t} \frac{u^{t}}{t!} = \frac{u}{exp(u) − 1}.

Theorem 1.3. 1. The class of differential forms Td(ϵ^{∗}\overline{Ω}_{A/S}) · T (λ, \overline{P}^{0}) lies in Ã^{g−1,g−1}((A^{∨}\S_{0}^{∨})_{ℝ}) and the equality

𝔤_{A}|_{A^{∨}(ℂ)\S_{0}^{∨}(ℂ)} = (−1)^{g+1} Td(ϵ^{∗}\overline{Ω}_{A/S}) · T (λ, \overline{P}^{0})

holds. In particular T (λ, \overline{P}^{0})^{(g−1)} does not depend on λ.

2. Suppose that λ is the first Chern form of a relatively ample rigidified line bundle, endowed with its canonical metric. Let σ ∈ A^{∨}(S) be an element of finite order n, such that σ^{∗}S_{0}^{∨} = ∅. Then

g · n · N_{2g} · σ^{∗}T (λ, \overline{P}^{0}) ∈ image(reg_{an}(K_{1}(S))).

A the end of section 4.2 (see the end of the proof of Lemma 4.5), we give a statement, which is slightly stronger than Theorem 1.3.2 (but more difficult to formulate). Theorem 1.3.1 can be viewed as a generalization to higher degrees of the second Kronecker limit formula (see [31, chap. 20, par. 5, p. 276] for the latter). Theorem 1.3.2 generalizes to higher degrees part of a classical statement on elliptic units and their fields of definition. See section 5 below. Remark. It would be interesting to have an analogue of Theorem 1.3.2, where reg_{an} is replaced by the analytic cycle class cyc_{an} (see (3) below). If S ≃ Spec R and R is the ring of integers in a number field, then reg_{an} and cyc_{an} can be identified but this is not true in general. In particular, this suggests that the Bernoulli number 12 = N_{2}/2, which appears in the denominators of elliptic units (see the last section) should be understood as coming from the natural integral structure of the group K_{1}(·)_{ℚ} and not from the natural integral structure of the corresponding motivic cohomology group ⨁_{p} H_{M}^{2p−1}(·, ℤ(p))_{ℚ}. Some of the results of this article were announced in [37]. We shall provide many bibliographical references to ease the reading but the reader of this article is nevertheless assumed to have some familiarity with the language of Arakelov theory, as expounded for instance in [44]. Acknowledgments. We thank J.-I. Burgos for patiently listening to our explanations on the contents of this article and for his (very) useful comments over a number of years. We also thank C. Soulé, G. Kings, as well as S. Bloch and A. Beilinson for their interest. J. Kramer made several interesting remarks on the contents of this article and his input was very useful. Many thanks also to K. Köhler for his explanations on the higher analytic torsion forms of abelian schemes. Finally, we are grateful to J. Wildeshaus for his feedback and for answering many questions on abelian polylogarithms. Notations. Here are the main notational conventions. Some of them have already been introduced above. Recall that we wrote π : A → S for the structure morphism of the abelian scheme A over S. We also write π^{∨} : A^{∨} → S for the structure morphism of the abelian scheme A^{∨} over S. Write μ = μ_{A} : A×_{S}A → A for the addition morphism and p_{1} : A×A^{∨} → A, p_{2} : A×A^{∨} → A^{∨} for the obvious projections. We shall also also write \underline{p}_{1}, \underline{p}_{2} : A × A → A and \underline{p}_{1}^{∨}, \underline{p}_{2}^{∨} : A^{∨} × A^{∨} → A^{∨} for more obvious projections. Recall that we wrote ϵ (resp. ϵ^{∨}) for the zero-section of A → S (resp. A^{∨} → S) and also that we wrote S_{0} (resp. S_{0}^{∨}) for the image of ϵ (resp. ϵ^{∨}). Write ω_{A} := det(Ω_{A/S}) for the determinant of the sheaf of differentials of A over S. We let dd^{c} := \frac{i}{2π} ∂ ¯∂.

2 Proof of Theorem 1.1

If M is a smooth complex quasi-projective variety, we shall write H_{D}^{∗}(M, ℝ(·)) for the Deligne-Beilinson cohomology of M. We recall its definition. Let M̄ be a smooth complex projective variety, which contains M as an open subscheme. We call M̄ a compactfication of M. Suppose furthermore that M̄\M is the underlying set of a reduced divisor with normal crossings D. From now on, we view M and M̄ as complex analytic spaces and we work in the category of complex analytic spaces. Let j : M ↪ M̄ be the given open embedding. There is a natural subcomplex Ω• ; M̄(log D) of j_{∗}Ω_{M}^{•}, called the complex of holomorphic differential forms on M with logarithmic singularities along D. The objects of Ω• ; M̄(log D) are locally free sheaves. We redirect the reader to [9, chap. 10] for the definition and further bibliographical references. Write F^{p}Ω• ; M̄(log D) for the subcomplex

Ωp ; M̄(log D) ⟶ Ωp+1 ; M̄(log D) ⟶ ⋯

of Ω• ; M̄(log D). Write f_{p}^{0} : F^{p}Ω• ; M̄(log D) → j_{∗}Ω_{M}^{•} for the inclusion morphism. Abusing notation, we shall identify Rj_{∗}ℝ(p) with the complex, which is the image by j_{∗} of the canonical flasque resolution of ℝ(p). Similarly, we shall write Rj_{∗}Ω_{M}^{•} for the simple complex associated to the image by j_{∗} of the canonical flasque resolution of Ω_{M}^{•} ( the latter being a double complex). Write f_{p} : F^{p}Ω• ; M̄(log D) → Rj_{∗}Ω_{M}^{•} for the morphism obtained by composing f_{p}^{0} with the canonical morphism j_{∗}Ω_{M}^{•} → Rj_{∗}Ω_{M}^{•}. There is a natural morphism of complexes ℝ(p) → Ω_{M}^{•} (where ℝ(p) is viewed as a complex with one object sitting in degree 0) and by the functoriality of the flasque resolution, we obtain a morphism r_{p} : Rj_{∗}ℝ(p) → Rj_{∗}Ω_{M}^{•}. We now define the complex

ℝ(p)_{D} := simple(Rj_{∗}ℝ(p) ⊕ F^{p}Ω• ; M̄(log D) ⟶^{u_{p}} Rj_{∗}Ω_{M}^{•})

where u_{p} := r_{p}−f_{p}. By definition, Deligne-Beilinson cohomology is now defined by the formula

H_{D}^{q}(M, ℝ(p)) := ℍ^{q}(M, ℝ(p)_{D}).

Notice that by construction, H_{D}^{q}(M, ℝ(p)) = H_{D,an}^{q}(M, ℝ(p)) if M is compact (so that D is empty). More generally, there is a natural "forgetful" morphism of ℝ-vector spaces H_{D}^{q}(M, ℝ(p)) → H_{D,an}^{q}(M, ℝ(p)) (what is forgotten is the logarithmic structure) ; see [10, before Prop. 1.3] for this. It can be proven that Deligne-Beilinson cohomology does not depend on the choice of the compactification M̄. By its very definition, we have a canonical long exact sequence of ℝ-vector spaces

⋯ → H^{q−1}(M, ℂ) → H_{D}^{q}(M, ℝ(p)) → H^{q}(M, ℝ(p))⊕F^{p}H^{q}(M, ℂ) → ⋯ (2)

where F^{p}H^{q}(M, ℂ) is the p-th term of the Hodge filtration of the mixed Hodge structure on H^{q}(M, ℂ). Furthermore the ℝ-vector space H_{D}^{q}(M, ℝ(p)) has a natural structure of contravariant functor from the category of smooth quasiprojective varieties over ℂ to the category of ℝ-vector spaces (see [10, Prop. 1.3] for this). If we equip the singular cohomology spaces H^{q}(M, ℂ) and H_{D}^{q}(M, ℝ(p)) with their natural contravariant structure, then the sequence (2) becomes an exact sequence of functors. If X is an arithmetic variety, then we define

H_{D}^{q}(X_{ℝ}, ℝ(p)) := {γ ∈ H_{D}^{q}(X(ℂ), ℝ(p)) | F_{∞}^{∗}γ = (−1)^{p}γ}.

Before beginning with the proof of Theorem 1.1, we shall prove the following key lemma.

Lemma 2.1. Let n ⩾ 2. The eigenvalues of the ℝ-endomorphism [n]^{∗} of the Deligne-Beilinson cohomology ℝ-vector space H_{D}^{2p−1}(A^{∨}(ℂ), ℝ(p)) lie in the set {1, n, n^{2}, …, n^{2p−1}}.

Proof. The existence of the exact sequence of functors (2) shows that we have the following exact sequence of ℝ-vector spaces

H^{2p−2}(A^{∨}(ℂ), ℂ) → H_{D}^{2p−1}(A^{∨}(ℂ), ℝ(p)) →

→ H^{2p−1}(A^{∨}(ℂ), ℝ(p)) ⊕ F^{p}H^{2p−1}(A^{∨}(ℂ), ℂ)

a and that the differentials in this sequence are compatible with the natural contravariant action of [n]. Hence it is sufficient to prove the conclusion of the lemma for the ℝ-vector spaces H^{2p−2}(A^{∨}(ℂ), ℂ), H^{2p−1}(A^{∨}(ℂ), ℝ(p)) and H^{2p−1}(A^{∨}(ℂ), ℂ). These spaces are more easily tractable and can be approximated by the Leray spectral sequence

E_{2}^{rs} = H^{r}(S(ℂ), R^{s}π^{∨}(ℂ)_{∗}(K)) ⇒ H^{r+s}(A^{∨}(ℂ), K)

where K is ℝ or ℂ. Now notice that since [n] is an S-morphism, this spectral sequence carries a natural contravariant action of [n], which is compatible with the aforementionned action of [n] on its abutment. Consider the index 2p − 1. We know that [n]^{∗} acts on R^{s}π^{∨}(ℂ)_{∗}(K) by multiplication by n^{s}. This may be deduced from known results on abelian varieties using the proper base-change theorem. Hence [n]^{∗} acts on H^{r}(S(ℂ), R^{s}π^{∨}(ℂ)_{∗}(K)) by multiplication by n^{s} as well. Now the existence of the spectral sequence shows that H^{2p−1}(A^{∨}(ℂ), K) has a natural filtration, which consists of subquotients of the spaces H^{r}(S(ℂ), R^{s}π^{∨}(ℂ)_{∗}(K)), where r + s = 2p − 1. Since s ⩽ 2p − 1, this proves the assertion for the index 2p − 1. The index 2p − 2 can be treated in an analogous fashion. □ Proof of uniqueness. Let 𝔤_{A} and 𝔤_{A}^{0} be elements of D̃^{g−1,g−1}(A_{ℝ}^{∨}) satisfying (a), (b) and (c). Let κ_{A} := 𝔤_{A}^{0} − 𝔤_{A} ∈ D̃^{g−1,g−1}(A^{∨}) be the error term. Recall the fundamental exact sequence

CH^{g,g−1}(A^{∨}) ⟶^{cyc_{an}} Ã^{g−1,g−1}(A_{ℝ}^{∨}) →^{a} ĈH^{g}(A^{∨}) → CH^{g}(A^{∨}) → 0 (3)

(see [19, th. 3.3.5] for this). Here CH^{g,g−1}(·) is Gillet-Soulé’s version of one of Bloch’s higher Chow groups. The group ĈH^{g}(A^{∨}) is the g-th arithmetic Chow group and CH^{g}(A^{∨}) is the g-th ordinary Chow group. By construction, there are maps

CH^{g,g−1}(A^{∨}) ⟶^{cyc} H_{D}^{2g−1}(A_{ℝ}^{∨}, ℝ(g))⟶^{forgetful}H_{D,an}^{2g−1}(A_{ℝ}^{∨}, ℝ(g))→Ã^{g−1,g−1}(A_{ℝ}^{∨})

whose composition is cyc_{an}. Here cyc is the cycle class map into Deligne-Beilinson cohomology; the second map from the left is the forgetful map and the third one is the natural inclusion mentioned before Theorem 1.3. Now let n ⩾ 2. Let

V := image(H_{D}^{2g−1}(A_{ℝ}^{∨}, ℝ(g)) ⟶^{forgetful} H_{D,an}^{2g−1}(A_{ℝ}^{∨}, ℝ(g)))  (4)

be the image of the forgetful map from H_{D}^{2g−1}(A_{ℝ}^{∨}, ℝ(g)) to H_{D,an}^{2g−1}(A_{ℝ}^{∨}, ℝ(g)). By (b), we know that κ_{A} ∈ V. Furthermore V is invariant under [n]^{∗}. In fact, by Lemma 2.1, [n]^{∗} restrict to an injective morphism V → V, which is thus an isomorphism, since V is finite dimensional. Now the projection formula shows that the equation [n]_{∗}[n]^{∗} = n^{2g} is valid in H_{D,an}^{2g−1}(A_{ℝ}^{∨}, ℝ(g)) and we conclude that that V is also invariant under [n]_{∗}. The same equation [n]_{∗}[n]^{∗} = n^{2g} now shows that the eigenvalues of [n]_{∗} on V lie in the set {n^{2g}, n^{2g−1}, …, n}. In particular, [n]_{∗} has no non-vanishing fixed point in V. Since [n]_{∗}κ_{A} = κ_{A} by (c), this proves that κ_{A} = 0. Proof of existence. As very often, the proof of existence is inspired by the proof of uniqueness. Let 𝔤 ∈ D̃^{g−1,g−1}(A_{ℝ}^{∨}) be a class of Green currents for S_{0}^{∨} satisfying (b). To see that there is such a 𝔤, pick any Green current 𝔤′ for S_{0}^{∨}(ℂ), such that F_{∞}^{∗}𝔤′ = (−1)^{g−1}𝔤′. This exists by [19, th.1.3.5]. Now a basic property of the Fourier-Mukai transformation for abelian schemes (see [32, Lemme 1.2.5]) implies that (−1)^{g}p_{2,∗}(ch(P))^{(g)} = S_{0}^{∨} in CH^{g}(A^{∨})_{ℚ}. Hence, looking at the sequence (3), we see that there exists α ∈ Ã^{g−1,g−1}(A_{ℝ}^{∨}) such that (a⊗ℚ)(α) = (S_{0}^{∨}, 𝔤′) − (−1)^{g}p_{2,∗}(ĉh(\overline{P}))^{(g)}. If we define 𝔤 := 𝔤′ − α, we obtain the required class of Green currents. Now fix n ⩾ 2 and let c := 𝔤 − [n]_{∗}𝔤. We shall prove below (see (6)) that

[n]_{∗}p_{2∗}(ĉh(\overline{P}))^{(g)} = p_{2∗}(ĉh(\overline{P}))^{(g)}.

This implies that c lies in the space V defined in (4) above. Now recall that we proved that [n]_{∗} sends V on V and that 1 is not a root of the characteristic polynomial of [n]_{∗} as an endomorphism of V. Hence the linear equation in x

x − [n]_{∗}x = c

has a unique solution in V. Call this solution c_{0}. By construction the current 𝔤_{0} := 𝔤 + c_{0} satisfies the equation 𝔤_{0} − [n]_{∗}𝔤_{0} = 0. Now let m ⩾ 2 be another natural number. We have seen above that 𝔤_{0} − [m]_{∗}𝔤_{0} lies in V. On the other hand

[n]_{∗}(𝔤_{0} − [m]_{∗}𝔤_{0}) = [n]_{∗}𝔤_{0} − [m]_{∗}[n]_{∗}𝔤_{0} = 𝔤_{0} − [m]_{∗}𝔤_{0}

hence 𝔤_{0}−[m]_{∗}𝔤_{0} is a fixed point of [n]_{∗} in V. This implies that 𝔤_{0}−[m]_{∗}𝔤_{0} = 0. This proves that 𝔤_{0} satisfies (a), (b) and (c).

3 Proof of Theorem 1.2

3.1 Proof of 1.2.1

By the definition of the dual isogeny, there is a diagram

\begin{aligned}A ×_{S} B^{∨} & \overline{ι × Id>} & B ×_{S} B^{∨} & \overline{>} & B^{∨} \\ A ×_{S} A^{∨} & \begin{aligned}Id × ι^{∨} \\ & ∨ \end{aligned} & \overline{>} & A^{∨} & \begin{aligned}ι^{∨} \\ & ∨ \end{aligned} \end{aligned}

such that

(Id × ι^{∨})^{∗}P_{A} ≃ (ι × Id)^{∗}P_{B}

Documenta Mathematica 20 (2015) 631–668

and such that the outer square is cartesian. Here P_{A} := P and P_{B} is the Poincaré bundle of B over S. First notice that the arithmetic Riemann-Roch theorem [22] implies that

ĉh((ι × Id)_{∗}(\overline{O}_{A×_{S}B^{∨}})) = deg ι

in ĈH^{•}(B ×_{S} B^{∨})_{ℚ}. Now we compute

ι^{∨,∗}p_{2∗}^{A}(ĉh(\overline{P}_{A})) = p_{2∗}^{B}(ĉh(\overline{P}_{B})ĉh((ι × Id)_{∗}(\overline{O}_{A×_{S}B^{∨}})))
= (deg ι) · p_{2∗}^{B}(ĉh(\overline{P}_{B})).  (5)

Here we used the projection formula for arithmetic Chow theory (see [19, Th. 4.4.3, 7.]) and the fact that the push-forward map in arithmetic Chow theory commutes with base-change. We may now compute

ι_{∗}^{∨}ι^{∨,∗}p_{2∗}^{A}(ĉh(\overline{P}_{A})) = (deg ι) · p_{2∗}^{A}(ĉh(\overline{P}_{A})) = (deg ι) · ι_{∗}^{∨}p_{2∗}^{B}(ĉh(\overline{P}_{B}))).

In other words, we have

p_{2∗}^{A}(ĉh(\overline{P}_{A})) = ι_{∗}^{∨}p_{2∗}^{B}(ĉh(\overline{P}_{B}))).  (6)

Furthermore, since ι^{∨} restricts to an isomorphism between the zero-sections, the class of currents ι_{∗}^{∨}(𝔤_{B}) consists of Green currents for S_{0,A}^{∨}. All this shows that 𝔤_{A} − ι_{∗}^{∨}(𝔤_{B}) lies inside the space V defined in 4. Recall that V is the image of the forgetful map H_{D}^{2g−1}(A_{ℝ}^{∨}, ℝ(g)) ⟶^{forgetful} H_{D,an}^{2g−1}(A_{ℝ}^{∨}, ℝ(g)). To conclude, notice that for any n ⩾ 2, we have

[n]_{∗}(𝔤_{A} − ι_{∗}^{∨}(𝔤_{B})) = 𝔤_{A} − ι_{∗}^{∨}(𝔤_{B})

since [n] commutes with ι^{∨}. It was shown just before the proof of existence in the proof of Theorem 1.1 that [n]_{∗} leaves V invariant and has no non-vanishing fixed points in V. Thus 𝔤_{A} − ι_{∗}^{∨}(𝔤_{B}) = 0.

3.2 Proof of 1.2.2

We shall prove the equivalent identities

\frac{(−1)^{g}}{g!\sqrt{deg(ϕ_{L})}}ϕ_{L,∗}(ĉ_{1}(\overline{L})^{g}) = p_{2,∗}(ĉh(\overline{P}))^{(g)}  (7)

and

p_{2,∗}(ĉh(\overline{P}))^{(k)} = 0  (8)

if k ≠ g. For the equality (8), notice that in view of (5) and the fact that [n]^{∨} = [n], we have

[n]^{∗}(p_{2,∗}(ĉh(\overline{P}))) = n^{2g} · p_{2,∗}(ĉh(\overline{P}))  (9)

for any n ⩾ 2. On the other hand, since (Id × [n])^{∗}P̄ = P̄^{⊗n}, we have also

[n]^{∗}(p_{2,∗}(ĉh(\overline{P}))) = ∑_{k⩾0} n^{k+g} · p_{2,∗}(ĉh(\overline{P}))^{(k)}  (10)

and comparing equations (9) and (10) as polynomials in n proves equation (8). We now proceed to the proof of equation (7). Notice that the line bundle μ^{∗}L⊗\underline{p}_{1}^{∗}L^{∨}⊗\underline{p}_{2}^{∗}L^{∨} on A×_{S} A carries a natural rigidification on the zero section

A ⟶^{(Id,ϵ)} A ×_{S} A and that the same line bundle is algebraically equivalent to 0 on each geometric fibre of the morphism \underline{p}_{2} : A ×_{S} A → A. Hence there is a unique morphism ϕ_{L} : A → A^{∨}, the polarisation morphism induced by L, such that there is an isomorphism of rigidified line bundles

(Id × ϕ_{L})^{∗}P ≃ μ^{∗}L ⊗ \underline{p}_{1}^{∗}L^{∨} ⊗ \underline{p}_{2}^{∗}L^{∨}.  (11)

Furthermore, if we endow the line bundles on both sides of (11) with their natural metrics, this isomorphism becomes an isometry, because both line bundles carry metrics that are compatible with the rigidification and the curvature forms of both sides are translation invariant (in fact 0) on the fibres of the map \underline{p}_{2}(ℂ). We shall now give a more concrete expression for

\underline{p}_{2∗}(ĉh(μ^{∗}\overline{L})ĉh(\underline{p}_{1}^{∗}\overline{L}^{∨})ĉh(\underline{p}_{2}^{∗}\overline{L}^{∨})).

We first make the calculation

\underline{p}_{2∗}(ĉh(μ^{∗}\overline{L})ĉh(\underline{p}_{1}^{∗}\overline{L}^{∨})ĉh(\underline{p}_{1}^{∗}\overline{L})) = \underline{p}_{2∗}(ĉh(μ^{∗}\overline{L}))
= \underline{p}_{2∗}(α^{∗}\underline{p}_{1}^{∗}ĉh(\overline{L})) = \underline{p}_{2∗}(\underline{p}_{1}^{∗}ĉh(\overline{L}))

where α : A ×_{S} A → A ×_{S} A is the \underline{p}_{2}-automorphism α := (μ, \underline{p}_{2}). Now notice that for any n ⩾ 2,

\underline{p}_{2∗}(([n] × Id)_{∗}([n] × Id)^{∗}\underline{p}_{1}^{∗}ĉh(\overline{L})) = n^{2g}\underline{p}_{2∗}(\underline{p}_{1}^{∗}ĉh(\overline{L}))
= \underline{p}_{2∗}(([n] × Id)^{∗}\underline{p}_{1}^{∗}ĉh(\overline{L})) = \underline{p}_{2∗}(∑_{l⩾1} n^{2l}(\underline{p}_{1}ĉh(\overline{L}))^{(l)}).

Here we used the isometric isomorphism [n]^{∗}\overline{L} ≃ \overline{L}^{⊗n^{2}} (recall that L is symmetric). We deduce that

\underline{p}_{2∗}(\underline{p}_{1}^{∗}ĉh(\overline{L})) = \underline{p}_{2∗}(\underline{p}_{1}^{∗}ĉh(\overline{L})^{(g)}) = \sqrt{deg(ϕ_{L})}

Thus, using the projection formula, we see that

\underline{p}_{2∗}(ĉh(\underline{p}_{1}^{∗}\overline{L})ĉh(μ^{∗}\overline{L})ĉh(\underline{p}_{1}^{∗}\overline{L}^{∨})\underline{p}_{2}^{∗}ĉh(\overline{L}^{∨})) = \sqrt{deg(ϕ_{L})} ĉh(\overline{L}^{∨})

which implies that

p_{2∗}(ĉh(p_{1}^{∗}\overline{L})ĉh(\overline{P})) = \frac{1}{\sqrt{deg(ϕ_{L})}}ϕ_{L,∗}ĉh(\overline{L}^{∨}).

Now notice that in the proof of equation (7), we may assume without restriction of generality that L is relatively generated by its sections, which is to say that the natural morphism π^{∗}π_{∗}L → L is surjective. Indeed, for any n ⩾ 2, we have

\frac{1}{g!\sqrt{deg(ϕ_{L^{⊗n}})}}ϕ_{L^{⊗n},∗}(ĉ_{1}(\overline{L}^{⊗n})^{g}) = \frac{1}{g!\sqrt{deg(ϕ_{L})}}ϕ_{L,∗}([n]_{∗}ĉ_{1}(\overline{L})^{g})

and for any k ⩾ 0,

[n]_{∗}ĉ_{1}(\overline{L})^{k} = n^{−2k}[n]_{∗}[n]^{∗}ĉ_{1}(\overline{L})^{k} = n^{2g−2k}ĉ_{1}(\overline{L})^{k}.  (12)

Hence

\frac{1}{g!\sqrt{deg(ϕ_{L^{⊗n}})}}ϕ_{L^{⊗n},∗}(ĉ_{1}(\overline{L}^{⊗n})^{g}) = \frac{1}{g!\sqrt{deg(ϕ_{L})}}ϕ_{L,∗}(ĉ_{1}(\overline{L})^{g}).

We may thus harmlessly replace \overline{L} by \overline{L}^{⊗n}, where n is some large positive integer. In particular, we may assume (and we do) that the morphism π^{∗}π_{∗}L → L is surjective, since L is relatively ample. Now let E := π^{∗}π_{∗}L ⊗ L^{∨} and let

P_{0}^{•} : ⋯ → Λ^{r}(E) → Λ^{r−1}(E) → ⋯ → E → O → 0

be the associated Koszul resolution. Let

P_{1}^{•} : 0 → P → p_{1}^{∗}E^{∨} ⊗ P → ⋯ → p_{1}^{∗}Λ^{r−1}(E)^{∨} ⊗ P → p_{1}^{∗}Λ^{r}(E)^{∨} ⊗ P → ⋯

be the complex P ⊗ p_{1}^{∗}(P_{0}^{•})^{∨}. All the bundles appearing in the complex P_{1}^{•} have natural hermitian metrics and we let ηP̄_{1} be the corresponding Bott-Chern class. Notice the equalities

ηP̄_{1} = ĉh(Λ_{−1}(\overline{E}^{∨}))ĉh(\overline{P}) = c^{top}(\overline{E})T̂d^{−1}(\overline{E})ĉh(\overline{P})

in ĈH^{•}(A×_{S} A^{∨}) (see [5, last paragraph]). Here Λ_{−1}(\overline{E}^{∨}) is the formal ℤ-linear combination ∑_{r⩾0}(−1)^{r}Λ^{r}(\overline{E}^{∨}). Since rk(E) may be assumed arbitrarily large (since we may replace L by some of its tensor powers), we see that we may

On a Canonical Class of Green Currents … 645

assume that ηP̄_{1} = 0 in ĈH^{•}(A ×_{S} A^{∨})_{ℚ}. Thus we may compute

p_{2∗}(ĉh(\overline{P})) = p_{2∗}(ĉh[−p_{1}^{∗}Λ_{−1}(\overline{E}^{∨}) + \overline{O}]ĉh(\overline{P}))

= − ∑_{r=1}^{rk(E)}(−1)^{r}p_{2∗}[ĉh(Λ^{r}(π^{∗}π_{∗}(\overline{L})^{∨}))ĉh(p_{1}^{∗}\overline{L}^{⊗r})ĉh(\overline{P})]

= − ∑_{r=1}^{rk(E)}(−1)^{r}p_{2∗}[ĉh(p_{1}^{∗}\overline{L}^{⊗r})ĉh(\overline{P})]ĉh(Λ^{r}(π^{∗}π_{∗}(\overline{L}))^{∨})

= − ∑_{r=1}^{rk(E)}(−1)^{r} \frac{1}{\sqrt{deg(ϕ_{L^{⊗r}})}} ϕ_{L^{⊗r},∗}(ĉh(\overline{L}^{∨,⊗r}))ĉh(Λ^{r}(π^{∗}π_{∗}(\overline{L}))^{∨})

= − \frac{1}{\sqrt{deg(ϕ_{L})}} ϕ_{L,∗}[ ∑_{r=1}^{rk(E)}(−1)^{r}r^{−g}([r]_{∗}(ĉh(\overline{L}^{∨,⊗r}))ĉh(Λ^{r}(π^{∗}π_{∗}(\overline{L}))^{∨}))]

= − \frac{1}{\sqrt{deg(ϕ_{L})}} ϕ_{L,∗}[ ∑_{r=1}^{rk(E)}(−1)^{r}r^{−g}([∑_{s⩾0} r^{2g−2s}ĉh(\overline{L}^{∨,⊗r})^{(s)}]ĉh(Λ^{r}(π^{∗}π_{∗}(\overline{L}))^{∨}))]

= − \frac{1}{\sqrt{deg(ϕ_{L})}} ϕ_{L,∗}[ ∑_{r=1}^{rk(E)} ∑_{s⩾0}(−1)^{r}r^{g−s}ĉh(\overline{L}^{∨})^{(s)}ĉh(Λ^{r}(π^{∗}π_{∗}(\overline{L}))^{∨})].

Now notice that the expression [n]_{∗}p_{2∗}(ĉh(\overline{P})) = p_{2∗}((Id × [n])_{∗}ĉh(\overline{P}))

= p_{2∗}((Id × [n])_{∗}(Id × [n])^{∗} ∑_{k⩾0} n^{−k}ĉh(\overline{P})^{(k)}) = p_{2∗}(∑_{k⩾0} n^{2g−k}ĉh(\overline{P})^{(k)})

is a Laurent polynomial in n ⩾ 2. The equation (12) shows that the expression

[n]_{∗}( − \frac{1}{\sqrt{deg(ϕ_{L})}}ϕ_{L,∗}[ ∑_{r=1}^{rk(E)} ∑_{s⩾0}(−1)^{r}r^{g−s}ĉh(\overline{L}^{∨})^{(s)}ĉh(Λ^{r}(π^{∗}π_{∗}(\overline{L}))^{∨})])

is also a Laurent polynomial in n ⩾ 2. We may thus identify the coefficients of these polynomials. We obtain the following : if g + k is even, then p_{2∗}(ĉh(\overline{P}))^{(k)} =

−\frac{1}{\sqrt{deg(ϕ_{L})}}ϕ_{L,∗}[ĉh(\overline{L}^{∨})^{((g+k)/2)}[ ∑_{r=1}^{rk(E)}(−1)^{r}r^{g−(g+k)/2}ĉh(Λ^{r}(π^{∗}π_{∗}(\overline{L}))^{∨})]]

and

p_{2∗}(ĉh(\overline{P}))^{(k)} = 0

if g + k is odd. Note that we have already proven the stronger fact that p_{2∗}(ĉh(\overline{P}))^{(k)} = 0 if k ≠ g. Thus

p_{2∗}(ĉh(\overline{P}))^{(g)} = −\frac{1}{\sqrt{deg(ϕ_{L})}}ϕ_{L,∗}[ĉh(\overline{L}^{∨})^{g}[ ∑_{r=1}^{rk(E)}(−1)^{r}ĉh(Λ^{r}(π^{∗}π_{∗}(\overline{L}))^{∨})]].

Documenta Mathematica 20 (2015) 631–668

Using furthermore that the left-hand side expression in the last equality is of pure degree g in ĈH^{•}(A^{∨})_{ℚ}, we deduce that

p_{2∗}(ĉh(\overline{P}))^{(g)} = −\frac{1}{\sqrt{deg(ϕ_{L})}}ϕ_{L,∗}[ĉh(\overline{L}^{∨})^{g}[ ∑_{r=1}^{rk(E)}(−1)^{r}\binom{rk(E)}{r}]].

Now notice that by the binomial formula ∑_{r=1}^{rk(E)}(−1)^{r}\binom{rk(E)}{r} = (1−1)^{rk(E)}−1 = −1. This proves equation (7).

3.3 Proof of 1.2.3

Let Z be an analytic cycle of pure codimension c on A(ℂ). In view of the assumption on S, A(ℂ) is a finite disjoint union of abelian varieties and so we may (and do) choose a translation invariant Kähler form on A(ℂ). A current 𝔤 on A(ℂ) of type (c−1, c−1) is said to be a harmonic Green current for Z (with respect to the Kähler form), if it satisfies the following properties : 𝔤 is a Green current for Z, the differential form dd^{c}𝔤+δ_{Z} is harmonic and ∫_{A(ℂ)} 𝔤∧κ = 0 for any harmonic form κ on A(ℂ). Notice now that a differential form on A(ℂ) is harmonic if and only if it is translation invariant (see for instance [15, p. 648]). Hence the concept of harmonic Green current is independent of the choice of the translation invariant Kähler form. The property (a) in Theorem 1.1 shows that 𝔤_{A} is a Green current for S_{0}^{∨}(ℂ) and the property 2 in Theorem 1.2 shows that dd^{c}𝔤 + δ_{S_{0}^{∨}(ℂ)} is harmonic. Let now κ be a harmonic form of type (1, 1) on A(ℂ). We know that κ is d-closed and that [n]^{∗}κ = n^{2} · κ for any n ⩾ 2. We may thus compute

∫_{A(ℂ)} 𝔤_{A}∧κ = ∫_{A(ℂ)}[n]_{∗}(𝔤_{A}∧κ) = n^{−2} ∫_{A(ℂ)}[n]_{∗}(𝔤_{A}∧[n]^{∗}κ) = n^{−2} ∫_{A(ℂ)} 𝔤_{A}∧κ

and hence ∫_{A(ℂ)} 𝔤_{A} ∧ κ = 0. Thus 𝔤_{A} is harmonic.

3.4 Proof of 1.2.4

In the next section, we shall give an expression for 𝔤_{A}, which depends only on A_{ℂ} (see the formula (15)). This implies the assertion.

3.5 Proof of 1.2.5

The proof of 1.2.5 is postponed to the end of the proof of Theorem 1.3.1. See the paragraph before subsection 4.2.

3.6 Proof of 1.2.6

This is a direct consequence of Theorem 1.1.1 and [11, Cor. 4.7 (i)] (thanks to J.-I. Burgos for providing this proof).

3.7 Proof of 1.2.7

We leave the proof that BC(ℂ) is transverse to S_{0}^{∨}(ℂ) to the reader. The equation BC^{∗}𝔤_{A} = 𝔤_{A_{T}} follows from formula (15) for 𝔤_{A}, which will be proved in the next section, together with [11, Th. 9.11 (ii)] and the fact that the higher analytic torsion forms of Bismut-Köhler are compatible with base-change.

4 Proof of Theorem 1.3

4.1 Proof of 1.3.1

This is the most difficult point to prove. We shall construct a class of currents 𝔤_{A}^{0} which naturally restricts to the degree (g −1, g −1) part of the analytic torsion and we shall prove that 𝔤_{A}^{0} satisfies the axioms defining 𝔤_{A}. The arithmetic Riemann-Roch theorem in higher degrees plays a crucial role here.

4.1.1 Definition of 𝔤_{A}^{0}

Let

V : 0 → O_{A} → V_{0} → ⋯ → V_{r} → V_{r+1} → ⋯

be a resolution of O_{A} by π_{∗}-acyclic vector bundles. Dualising, we get a resolution

V^{∨} : ⋯ → V_{r+1}^{∨} → V_{r}^{∨} → ⋯ → V_{0}^{∨} → O_{A} → 0

of O_{A} on the left. The first hypercohomology spectral sequence of the complex V^{∨} ⊗ P for the functor p_{2,∗} provides an exact sequence

H : ⋯ → R^{g}p_{2∗}(V_{r}^{∨} ⊗ P) → R^{g}p_{2∗}(V_{r−1}^{∨} ⊗ P) → … → R^{g}p_{2∗}(V_{0}^{∨} ⊗ P) → ϵ^{∨}_{∗}(ω_{A/S}^{∨}) → 0.

Now endow the vector bundles V_{r} with conjugation-invariant hermitian metrics. The line bundle \overline{ω}_{A/S} is endowed with its L^{2}-metric. This metric does not depend on the choice of λ. This follows from the explicit formula for the L^{2}-metric on Hodge cohomology given in [38, Lemma 2.7]. The arithmetic Riemann-Roch [22] in higher degrees applied to \overline{P} and p_{2} is the identity

(−1)^{g}ĉh(∑_{r⩾0}(−1)^{r}R^{g}p_{2∗}(\overline{V} _{r}^{∨} ⊗ \overline{P})) − ∑_{r⩾0}(−1)^{r}T (λ, \overline{V} _{r}^{∨} ⊗ \overline{P})+

+ ∫_{p_{2}} Td(\overline{T}p_{2}) ch(\overline{P})ηV̄^{∨}

= p_{2,∗}(T̂d(\overline{T}p_{2})ĉh(\overline{P})) − ∫_{p_{2}} ch(P)R(Tp_{2}) Td(Tp_{2})

in ĈH^{•}(A^{∨})_{ℚ}. Here ηV̄^{∨} is the Bott-Chern secondary class of \overline{V}^{∨}, where O_{A} has index 0. We have identified λ with p_{1}^{∗}λ.

Notice first that

[∫_{p_{2}} ch(P)R(Tp_{2}) Td(Tp_{2})]^{(g−1)}
= [∫_{p_{2}} ch(P)R(Tπ) Td(Tπ)]^{(g−1)}
= [π^{∨,∗}(ϵ^{∗}(R(Tπ) Td(Tπ))) ∫_{p_{2}} ch(P)]^{(g−1)} = 0

where we used (7). Write T (\overline{H}) for the homogenous secondary class in the sense of Bismut-Burgos-Litcanu (see [11, sec. 6]) of the resolution H. By its very definition, −T (\overline{H})^{(g−1)} is a class of Green currents for S_{0}^{∨}(ℂ) and it is shown in [11, Th. 10.28] that

[ĉh(∑_{r⩾0}(−1)^{r}R^{g}p_{2∗}(\overline{V} _{r}^{∨} ⊗ \overline{P}))]^{(g)} = (S_{0}^{∨}, −T (\overline{H})^{(g−1)})  (13)

in ĈH^{g}(A^{∨})_{ℚ} and equation (7) shows that

[p_{2,∗}(T̂d(\overline{T}p_{2})ĉh(\overline{P}))]^{(g)} = [p_{2,∗}(T̂d(\overline{T}π)ĉh(\overline{P}))]^{(g)}

= [p_{2,∗}(ĉh(\overline{P}))π^{∨,∗}(ϵ^{∗}(T̂d(\overline{T}π)))]^{(g)} = p_{2,∗}(ĉh(\overline{P}))^{(g)}

hence we are led to the equality

p_{2,∗}(ĉh(\overline{P}))^{(g)} = (−1)^{g}(S_{0}^{∨}, −T (\overline{H})^{(g−1)}) − ∑_{r⩾0}(−1)^{r}T (λ, \overline{V} _{r}^{∨} ⊗ \overline{P})^{(g−1)}

+ [∫_{p_{2}} Td(\overline{T}p_{2}) ch(\overline{P})ηV̄^{∨}]^{(g−1)}. (14)

This motivates the definition:

\begin{aligned}𝔤_{A}^{0} & := & −T (\overline{H})^{(g−1)} + (−1)^{g+1} ∑_{r⩾0}(−1)^{r}T (λ, \overline{V} _{r}^{∨} ⊗ \overline{P})^{(g−1)} \\ & & +(−1)^{g}[∫_{p_{2}} Td(\overline{T}p_{2}) ch(\overline{P})ηV̄^{∨}]^{(g−1)} & (15) \end{aligned}

Lemma 4.1. The class of currents 𝔤_{A}^{0} does not depend on the resolution V, nor on the metrics on the bundles V_{r}, nor on the translation invariant Kähler form λ.

Proof. We first prove that the class of currents 𝔤_{A}^{0} does not depend on V and that it does not depend on the hermitian metrics or on the bundles V_{r}.

Suppose that there is a second resolution V′ dominating V :

0  0  0  0

V′ : 0 \overline{>} O_{A} ∨  \overline{>} V′_{0} ∨ \overline{>} ⋯ \overline{>} V′_{r} ∨ \overline{>} V′_{r+1} ∨  \overline{>} ⋯

V : 0 \overline{>} O_{A} \begin{matrix}Id \\ ∨ \end{matrix}  \overline{>} V_{0} ∨ \overline{>} ⋯ \overline{>} V_{r} ∨ \overline{>} V_{r+1} ∨  \overline{>} ⋯

Q :  0 ∨ \overline{>} Q_{0} ∨ \overline{>} ⋯ \overline{>} Q_{r} ∨ \overline{>} Q_{r+1} ∨  \overline{>} ⋯

0 ∨  0 ∨  0 ∨

By assumption the complex Q is exact and we assume that its objects are π_{∗}- acyclic. We endow everything with hermitian metrics. We shall write H′ for the exact sequence

H′ : ⋯ → R^{g}p_{2∗}(V_{r}^{′,∨} ⊗ P) → R^{g}p_{2∗}(V_{r−1}^{′,∨} ⊗ P) → …

hfill → R^{g}p_{2∗}(V_{0}^{′,∨} ⊗ P) → ϵ^{∨}_{∗}(ω_{A/S}^{∨}) → 0.

In order to emphasize the dependence of 𝔤_{A}^{0} on the resolution V together with the collection of hermitian metrics on the V_{r}, we shall write 𝔤0 ; \overline{V} := 𝔤_{A,\overline{V}}^{0} instead

of 𝔤_{A}^{0}. Recall that η\overline{V}^{∨} is the Bott-Chern secondary class of the sequence \overline{V}^{∨}, with \overline{O}_{A} sitting at the index 0. We shall accordingly write η\overline{V}_{′,∨} for the Bott-

Chern secondary class of the sequence \overline{V}^{′,∨}, with \overline{O}_{A} sitting at the index 0. By definition, we have

(−1)^{g}(𝔤0 ; \overline{V} − 𝔤_{\overline{V}′}^{0}) = (−1)^{g+1}T (\overline{H})^{(g−1)}

− ∑_{r⩾0}(−1)^{r}T (λ, \overline{V} _{r}^{∨} ⊗ \overline{P})^{(g−1)} + [∫_{p_{2}} Td(\overline{T}p_{2}) ch(\overline{P})ηV̄^{∨}]^{(g−1)}

− (−1)^{g+1}T (\overline{H}′)^{(g−1)} + ∑_{r⩾0}(−1)^{r}T (λ, \overline{V} _{r}^{′,∨} ⊗ \overline{P})^{(g−1)}

− [∫_{p_{2}} Td(\overline{T}p_{2}) ch(\overline{P})ηV̄^{′,∨}]^{(g−1)}.

Let now

C_{r} : 0 → Q_{r}^{∨} ⊗ P → V_{r}^{∨} ⊗ P → V_{r}^{′,∨} ⊗ P → 0

be the natural exact sequence. All the bundles appearing on C_{r} are endowed with natural hermitian metrics. By the symmetry formula [4, Th. 2.7, p. 271], we may compute

T (\overline{H}) − T (\overline{H}′) = c̃h(R^{g}π_{∗}(Q^{∨} ⊗ P)) − ∑_{r⩾0}(−1)^{r}c̃h(R^{g}π_{∗}(C_{r})).

On the other hand, the anomaly formula [6, Th. 3.10, p. 670] tells us that

∑_{r⩾0}(−1)^{r}T (λ, \overline{V} _{r}^{∨} ⊗ \overline{P}) − ∑_{r⩾0}(−1)^{r}T (λ, \overline{V} _{r}^{′,∨} ⊗ \overline{P})

= ∑_{r⩾0}(−1)^{r}T (λ, \overline{Q}_{r}^{∨} ⊗ \overline{P}) + (−1)^{g} ∑_{r⩾0}(−1)^{r}c̃h(R^{g}π_{∗}(C_{r}))

− ∫_{p_{2}} ∑_{r⩾0}(−1)^{r} Td(\overline{T}p_{2})c̃h(\overline{C}_{r})

and

∑_{r⩾0}(−1)^{r}T (λ, \overline{Q}_{r}^{∨} ⊗ \overline{P}) = ∫_{p_{2}} Td(\overline{T}p_{2})c̃h(\overline{Q}^{∨} ⊗ \overline{P}) − (−1)^{g}c̃h(R^{g}π_{∗}(Q^{∨} ⊗ P)).

Furthermore the symmetry formula [4, Th. 2.7, p. 271] again implies that

∫_{p_{2}} Td(\overline{T}p_{2})ηV̄^{′,∨} ch(\overline{P}) − ∫_{p_{2}} Td(\overline{T}p_{2})ηV̄^{∨} ch(\overline{P})

= − ∫_{p_{2}} Td(\overline{T}p_{2})c̃h(\overline{Q}^{∨} ⊗ \overline{P}) + ∫_{p_{2}} ∑_{r⩾0}(−1)^{r} Td(\overline{T}p_{2})c̃h(\overline{C}_{r}).

Putting everything together, we see that

(−1)^{g}(𝔤0 ; \overline{V} − 𝔤_{\overline{V}′}^{0})

= [ − (−1)^{g}c̃h(R^{g}π_{∗}(Q^{∨} ⊗ P)) + (−1)^{g} ∑_{r⩾0}(−1)^{r}c̃h(R^{g}π_{∗}(C_{r}))

− (−1)^{g} ∑_{r⩾0}(−1)^{r}c̃h(R^{g}π_{∗}(C_{r})) + ∫_{p_{2}} ∑_{r⩾0}(−1)^{r} Td(\overline{T}p_{2})c̃h(\overline{C}_{r})

− ∫_{p_{2}} Td(\overline{T}p_{2})c̃h(\overline{Q}^{∨} ⊗ \overline{P}) + (−1)^{g}c̃h(R^{g}π_{∗}(Q^{∨} ⊗ P))

+ ∫_{p_{2}} Td(\overline{T}p_{2})c̃h(\overline{Q}^{∨} ⊗ \overline{P}) − ∫_{p_{2}} ∑_{r⩾0}(−1)^{r} Td(\overline{T}p_{2})c̃h(\overline{C}_{r})]^{(g−1)} = 0.

Homological algebra tells us that there always exists a resolution dominating simultaneously two other ones. Furthermore, we might assume that this resolution satisfies the above conditions of π_{∗}-acyclicity (see [30, chap XX, par. 3, proof of Th. 3.5, p. 773]). Hence we have proven that 𝔤0 ; \overline{V} does not depend on V and that it does not depend on the hermitian metrics on the bundles V_{r}.

We shall now prove that 𝔤_{A}^{0} does not depend on the choice of λ. So let λ′ be another Kähler fibration, which is translation invariant on the fibres. To emphasize the dependence of 𝔤_{A}^{0} on λ, let us write 𝔤_{A}^{0} := 𝔤^{0,λ}. Write \overline{H}^{λ} (resp.

\overline{H}^{λ′}) for the sequence H together with the hermitian metrics induced by λ (resp. λ′). We compute as before

(−1)^{g}(𝔤^{0,λ} − 𝔤^{0,λ′})

= (−1)^{g+1}T (\overline{H}^{λ})^{(g−1)} − ∑_{r⩾0}(−1)^{r}T (λ, \overline{V} _{r}^{∨} ⊗ \overline{P})^{(g−1)}

+ [∫_{p_{2}} Td(Tp_{2}, λ) ch(\overline{P})ηV̄^{∨}]^{(g−1)}

− (−1)^{g+1}T (\overline{H}^{λ′})^{(g−1)} + ∑_{r⩾0}(−1)^{r}T (λ′, \overline{V} _{r}^{∨} ⊗ \overline{P})^{(g−1)}

− [∫_{p_{2}} Td(Tp_{2}, λ′) ch(\overline{P})ηV̄^{∨}]^{(g−1)}.

We compute

T (\overline{H}^{λ′}) − T (\overline{H}^{λ}) = − ∑_{r⩾0}(−1)^{r}c̃h(R^{g}π_{∗}(V_{r}^{∨} ⊗ P), λ′, λ)

and by the anomaly formula [6, Th. 3.10, p. 670]

∑_{r⩾0}(−1)^{r}T (λ′, \overline{V} _{r}^{∨} ⊗ \overline{P}) − ∑_{r⩾0}(−1)^{r}T (λ, \overline{V} _{r}^{∨} ⊗ \overline{P})

= (−1)^{g} ∑_{r⩾0}(−1)^{r}c̃h(R^{g}π_{∗}(V_{r}^{∨} ⊗ P), λ′, λ)

− ∫_{p_{2}} ∑_{r⩾0}(−1)^{r}T̃d(λ′, λ) ch(\overline{V} _{r}^{∨} ⊗ \overline{P}).

Now notice that by the Leibniz formula (see [42, 6.2, (7)] for details)

∫_{p_{2}} ( Td(Tp_{2}, λ) − Td(Tp_{2}, λ′)) ch(\overline{P})ηV̄^{∨} = − ∫_{p_{2}} dd^{c}(T̃d(λ′, λ)) ch(\overline{P})ηV̄^{∨}

= − ∫_{p_{2}} T̃d(λ′, λ)dd^{c}(ch(\overline{P})ηV̄^{∨})

= ∑_{r⩾0}(−1)^{r} ∫_{p_{2}} T̃d(λ′, λ) ch(\overline{V} _{r}^{∨} ⊗ \overline{P}) − ∫_{p_{2}} T̃d(λ′, λ) ch(\overline{P}).

Now, by the projection formula, since λ and λ′ are translation invariant, we have

∫_{p_{2}} T̃d(λ′, λ) ch(\overline{P}) = ϵ^{∨,∗}(T̃d(λ′, λ)) ∫_{p_{2}} ch(\overline{P}) and by the formula (7) we have [ϵ^{∨,∗}(T̃d(λ′, λ)) ∫_{p_{2}} ch(\overline{P})]^{(g−1)} = 0. Assembling everything, we get that 𝔤^{0,λ} − 𝔤^{0,λ′} = 0. □

4.1.2 End of proof of 1.3.1

We keep the notation of the last subsection. By construction, the class of currents 𝔤_{A}^{0} has the property that

𝔤_{A}^{0}|_{A^{∨}(ℂ)\S_{0}^{∨}(ℂ)} = (−1)^{g+1}T (λ, \overline{P}^{0})^{(g−1)}  (16)

in Ã^{g−1,g−1}(A^{∨}\S_{0}^{∨}) and contemplating equation (13), we see that the elements of 𝔤_{A}^{0} are Green currents for S_{0}^{∨}. Looking at the equation (14) we see that 𝔤_{A}^{0} satisfies (a) and (b) in Theorem 1.1. Now we want to prove that

𝔤_{A}^{0} = [n]_{∗}𝔤_{A}^{0}  (17)

for all n ⩾ 2. Fix n ⩾ 2. We claim that to prove equation (17), we may assume that ker [n]_{A} is a constant diagonalisable subgroup-scheme of A. Indeed, both sides of equation (17) depend on A_{ℂ}^{∨} only. In proving equation (17), we thus may (and do) replace R by its fraction field Frac(R). We may also assume without restriction of generality that S is connected, hence integral (since S is regular by assumption). Now let S′ be the normalisation of S in the composite of the field extensions of the function field κ(S) of S, which are defined by the residue fields of the n-torsion points in the generic fibre A_{κ(S)} of A. Then b : S′ → S is finite (see [23, II, 6.3.10]) and étale (see [39, Cor. 20.8, p. 147]). Since [n]_{A} is étale, the group scheme of n-torsion points on A_{S′} is then a constant group scheme. If we again replace S′ by a finite étale cover, we may assume that Γ(S′, O_{S′}) contains the n-th roots of 1 and the group scheme of n-torsion points on A_{S′} then become diagonalisable (and constant). Now, by the projection formula the pull-back morphism b^{∗} : Ã^{g−1,g−1}(A_{ℝ}^{∨}) → Ã^{g−1,g−1}(A_{S′,ℝ}^{∨}) is injective. Furthermore, since an étale finite morphism is a local isomorphism in the category of complex manifolds, we have b^{∗}([n]_{∗}𝔤_{A}^{0}) = [n]_{∗}(b^{∗}𝔤_{A}^{0}). For the same reason, we also have b^{∗}𝔤_{A}^{0} = 𝔤_{A_{S′}}^{0} .

Lemma 4.2. There is an isometric isomorphism

[n]_{∗}O_{A} ≃ ⨁_{M∈A^{∨}[n](S)} M

where the left-hand side is endowed with its L^{2}-metric and the direct summands on the right-hand side are endowed with the metric induced by the Poincaré bundle.

Proof. Let us denote by F_{A} : D^{b}(A) → D^{b}(A^{∨}) the Fourier-Mukai transformation. If X is a scheme, the category D^{b}(X) is a full subcategory of the category D(X) derived from the category of sheaves in O_{X}-modules. Its objects are the complexes with a finite number of non-zero homology sheaves, all of which are coherent. The functor F_{A} is given by the formula

K^{•} ↦ Rp_{2,∗}(p_{1}^{∗}K^{•} ⊗ P).

It can be proven that in our situation, there is a natural isomorphism of functors

F_{A^{∨}} ∘ F_{A}(·) ≃ (([−1]^{∗}(·)) ⊗ ω_{A}^{∨})[−g]

(here we have identified A with (A^{∨})^{∨}) and

F_{A} ∘ [n]_{∗}(·) ≃ [n]^{∗} ∘ F_{A}(·).

See [32] for this. Now we compute

F_{A^{∨}} ∘ F_{A}([n]_{∗}(O_{A})) ≃ [n]_{∗}(O_{A}) ⊗ ω_{A}^{∨}[−g] ≃ F_{A^{∨}} ∘ [n]^{∗}(F_{A}(O_{A}))

≃ F_{A^{∨}}([n]^{∗}(ϵ_{∗}^{∨}(ϵ^{∨,∗}(ω_{A}^{∨}))[−g])) ≃ π_{A,∗}(P_{A}|_{A×_{S}ker [n]_{A^{∨}}}) ⊗ ω_{A}^{∨}[−g]

and we thus obtain a canonical isomorphism

[n]_{∗}O_{A} ≃ ⨁_{M∈A^{∨}[n](S)} M.  (18)

Now we make a different computation. Let G be a finite group such that G_{S} ≃ ker [n]_{A}. Let L := Γ(S, O_{S}). If χ : G → L^{∗} is a character of G we let ([n]_{∗}O_{A})_{χ} be the locally free subbundle of [n]_{∗}O_{A} which is the largest subbundle S of [n]_{∗}O such that the action of G on S is given by multiplication by χ. Since G_{L} is a diagonalisable group scheme over L, this gives a direct sum decomposition

[n]_{∗}O_{A} = ⊕_{χ}([n]_{∗}O_{A})_{χ}.  (19)

Now we use the equivariant form of Bismut’s relative curvature formula (see [7]). Let a ∈ G be any non-zero element. Since the fixed point scheme of a on A is empty, we get

ch_{a}([n]_{∗}(\overline{O}_{A})) = 0.  (20)

Here ch_{a}(·) is the equivariant Chern character form associated to the action of a. See [7] for details. The non-equivariant relative curvature formula gives

ch([n]_{∗}(\overline{O}_{A})) = rk([n]_{∗}O_{A}).  (21)

From the equations (20) and (21) and the fact that rk([n]_{∗}O_{A}) = #G, we deduce (by finite Fourier theory) that all the ([n]_{∗}O_{A})_{χ} are of rank 1 and that c_{1}(([n]_{∗}\overline{O}_{A})_{χ}) = 0, where ([n]_{∗}\overline{O}_{A})_{χ} is endowed with the metric induced by [n]_{∗}\overline{O}_{A}. A completely similar computation using the relative geometric fixed point formula (see [1]) shows that each line bundle ([n]_{∗}O_{A})_{χ} is actually a torsion line bundle. Notice also that since G acts by isometries on O_{A}, the direct sum decomposition (19) is an orthogonal direct sum.

Now notice that since [n] is finite and flat there is an isometric equivariant isomorphism

ϵ^{∗}([n]_{∗}\overline{O}_{A}) ≃ \overline{O}_{A}^{G}

where O_{A}^{G} is endowed with the G-action induced by the action of G on itself.

This shows that there exist isometric rigidifications ϵ^{∗}(([n]_{∗}\overline{O}_{A})_{χ}) ≃ \overline{O}_{S}.

Summing up the discussion of the previous paragraphs, we see that there exists an isometric isomorphism of vector bundles

[n]_{∗}\overline{O}_{A} = ⊕_{χ}([n]_{∗}\overline{O}_{A})_{χ}  (22)

where the direct sum is orthogonal and where each ([n]_{∗}\overline{O}_{A})_{χ} is a torsion line bundle, which is isometrically rigidified and which carries a hermitian metric, whose curvature form vanishes.

Now consider the isomorphism of vector bundles

⨁_{χ}([n]_{∗}O_{A})_{χ} ≃ ⨁_{M∈A^{∨}[n](S)} M  (23)

induced by the isomorphisms (18) and (22).

We shall need the

Claim. Let M and L be torsion line bundles on A. If there is a non-zero morphism of line bundles (without rigidification) L → M then L and M are isomorphic.

This follows from the fact that the Picard functor of A/S is representable by a scheme, which is separated over S, together with the fact that a non-trivial torsion bundle on an abelian variety has no global sections. Details are left to the reader.

The claim shows that the isomorphism (23) send each ([n]_{∗}O_{A})_{χ} into exactly one M ∈ A^{∨}[n](S) and that this morphism is an isomorphism of line bundles. After possibly rescaling the isomorphism ([n]_{∗}O_{A})_{χ} ≃ M by an element of Γ(S, O_{S}^{∗}), we obtain an isomorphism ([n]_{∗}O_{A})_{χ} ≃ M of rigidified line bundles. Since both ([n]_{∗}O_{A})_{χ} and M are endowed with the unique metrics, whose curvature form is translation invariant on the fibres of A(ℂ)/S(ℂ) and which are compatible with the given rigidifications, this implies that this is an isometric isomorphism. This completes the proof of the lemma. □

As at the beginning of the proof of assertion 1 in Theorem 1.2, we have a diagram

\begin{aligned}A ×_{S} A^{∨} & \overline{[n] × Id>} & A ×_{S} A^{∨} & \overline{>} & A^{∨} \\ A ×_{S} A^{∨} & \begin{matrix}Id × [n] \\ ∨ \end{matrix} & \overline{>} & A^{∨} & \begin{matrix}[n] \\ ∨ \end{matrix} \end{aligned}

such that

(Id × [n])^{∗}P ≃ ([n] × Id)^{∗}P

and such that the outer square is cartesian. Write q := p_{2} ∘([n]×Id). To clarify further computations, write R^{•}p_{2,∗}(V^{∨} ⊗ P) for H. We now compute

(−1)^{g}[n]^{∗}𝔤_{A}^{0} = [n]^{∗}[(−1)^{g+1}T (R^{•}p_{2,∗}(V^{∨} ⊗ P))  (24)

− ∑_{r⩾0}(−1)^{r}T (λ, \overline{V} _{r}^{∨} ⊗ \overline{P}) + ∫_{p_{2}} Td(\overline{T}p_{2}) ch(\overline{P})ηV̄^{∨}]

= (−1)^{g+1}T (R^{•}q_{∗}(V^{∨} ⊗ (Id × [n])^{∗}P))  (25)

− ∑_{r⩾0}(−1)^{r}T (λ, \overline{V} _{r}^{∨} ⊗ (Id × [n])^{∗}\overline{P})

+ ∫_{q} Td(\overline{T}q) ch((Id × [n])^{∗}\overline{P})ηV̄^{∨}

= (−1)^{g+1}T (R^{•}q_{∗}(V^{∨} ⊗ ([n] × Id)^{∗}P))  (26)

− ∑_{r⩾0}(−1)^{r}T (λ, \overline{V} _{r}^{∨} ⊗ ([n] × Id)^{∗}\overline{P})

+ ∫_{q} Td(\overline{T}q) ch(([n] × Id)^{∗}\overline{P})ηV̄^{∨}

In view of Lemma 4.1, we may replace \overline{V} by [n]^{∗}\overline{V}_{A} and λ by [n]^{∗}λ_{A} in the string of equalities (27) without changing its truth-value. Thus

(−1)^{g}[n]^{∗}𝔤_{A}^{0} = (−1)^{g+1}T (R^{•}q_{∗}(([n] × Id)^{∗}(V^{∨} ⊗ P)))

− ∑_{r⩾0}(−1)^{r}T ([n]^{∗}λ, ([n] × Id)^{∗}(\overline{V} _{r}^{∨} ⊗ \overline{P}))

+ ∫_{q} Td(\overline{T}q)([n] × Id)^{∗}(ch(\overline{P})ηV̄^{∨})

=^{(∗)} (−1)^{g+1}T (R^{•}p_{2,∗}(([n] × Id)_{∗}(O) ⊗ V^{∨} ⊗ P))

− ∑_{r⩾0}(−1)^{r}T (([n] × Id)_{∗}(\overline{O}) ⊗ \overline{V} _{r}^{∨} ⊗ \overline{P})

+ ∫_{p_{2}} Td(\overline{T}p_{2})([n] × Id)_{∗}(1) ch(\overline{P})ηV̄^{∨}

=^{(∗∗)} ∑_{τ∈A^{∨}(S)} [(−1)^{g+1}T (R^{•}p_{2,∗}((Id × (τ ∘ π))^{∗}(P) ⊗ V^{∨} ⊗ P))

− ∑_{r⩾0}(−1)^{r}T ((Id × (τ ∘ π))^{∗}(\overline{P}) ⊗ \overline{V} _{r}^{∨} ⊗ \overline{P})

+ ∫_{p_{2}} Td(\overline{T}p_{2}) ch((Id × (τ ∘ π))^{∗}(\overline{P}) ⊗ \overline{P})ηV̄^{∨}]

= ∑_{τ∈A^{∨}(S)} [(−1)^{g+1}τ_{∗}(T (R^{•}p_{2,∗}(V^{∨} ⊗ \overline{P}))) − τ_{∗}[∑_{r⩾0}(−1)^{r}T (\overline{V} _{r}^{∨} ⊗ \overline{P})]

+ τ_{∗}[∫_{p_{2}} Td(\overline{T}p_{2})(\overline{P})ηV̄^{∨}]]

= (−1)^{g} ∑_{τ∈A^{∨}(S)} τ_{∗}𝔤_{A}^{0}.

The equality (*) is justified by the following Proposition 4.3. The equality

T ([n]^{∗}λ, ([n] × Id)^{∗}(\overline{V} _{r}^{∨} ⊗ \overline{P})) = T (([n] × Id)_{∗}(\overline{O}) ⊗ \overline{V} _{r}^{∨} ⊗ \overline{P}) is verified for any r. Proof. (of Proposition 4.3). This is a direct consequence of [35, Intro., Th. 0.1]. □ For the equality (**), we used Lemma 4.2. We may now compute

[n]_{∗}[n]^{∗}𝔤_{A}^{0} = n^{2g}𝔤_{A}^{0} = [n]_{∗}( ∑_{τ∈A^{∨}(S)} τ_{∗}𝔤_{A}^{0}) = n^{2g}[n]_{∗}𝔤_{A}^{0}

i.e.

𝔤_{A}^{0} = [n]_{∗}𝔤_{A}^{0}.

Documenta Mathematica 20 (2015) 631–668

We have thus proven that 𝔤_{A}^{0} satisfies the conditions (a), (b), (c) in Theorem 1.1. Thus 𝔤_{A}^{0} = 𝔤_{A} and looking at equation (16), we see that we have almost concluded the proof of Theorem 1.3.1. To finish, we quote [27], where it is shown that T (λ, \overline{P}^{0}) = Td^{−1}(ϵ^{∗}\overline{Ω})γ for some real differential form γ of type (g − 1, g − 1) on A^{∨}\S_{0}^{∨}. From the above, we see that γ = (−1)^{g+1}𝔤_{A} and we are done. Proof of Theorem 1.2.5. We revert to the hypotheses of the introduction and of Theorem 1.2.5 (in particular, we do not suppose anymore that for some n ⩾ 2, the group scheme ker [n] is the constant group scheme). To verify the equation 𝔤_{A×_{S}B} = π_{A^{∨}}^{∗}(𝔤_{A}) ∗ π_{B^{∨}}^{∗}(𝔤_{B}), it is sufficient to check that the class of currents π_{A^{∨}}^{∗}(𝔤_{A}) ∗ π_{B^{∨}}^{∗}(𝔤_{B}) satisfies the axioms (a), (b), (c) in Theorem 1.1. The fact that the elements of π_{A^{∨}}^{∗}(𝔤_{A}) ∗ π_{B^{∨}}^{∗}(𝔤_{B}) are Green currents for the unit section of A^{∨} ×_{S} B^{∨} follows immediately from the definitions. This settles (a). To verify (b), we consider the commutative diagram

A × B × A^{∨} × B^{∨}

A × A^{∨} < q_{AA^{∨}} A × A^{∨} × B^{∨} p_{AA^{∨}B^{∨}} ; \underline{<}  A^{∨} × B × B^{∨} \overline{\begin{matrix}r_{BB^{∨}} \\ > \end{matrix}}^{p_{A^{∨}BB^{∨}} \overline{>}}  B × B^{∨}

A^{∨} × B^{∨} \begin{aligned}p_{A^{∨}B^{∨}} \\ & ∨ \end{aligned} r_{A^{∨}B^{∨}} ; \underline{<}^{q_{A^{∨}B^{∨}}  \overline{>}}

where the morphisms are the obvious ones. Notice that in this diagram, the square is cartesian. We compute

(−1)^{g_{A×B}} p_{A^{∨}B^{∨},∗}(ĉh(\overline{P}_{A×B}))^{g_{A×B}}
= (−1)^{(g_{A}+g_{B})}p_{A^{∨}B^{∨},∗}(ĉh(\overline{P}_{A×B}))^{(g_{A}+g_{B})}=(−1)^{(g_{A}+g_{B})}p_{A^{∨}B^{∨},∗}(ĉh(\overline{P}_{A×B}))
= (−1)^{(g_{A}+g_{B})}p_{A^{∨}B^{∨},∗}(ĉh(p_{AA^{∨}B^{∨}}^{∗}q_{AA^{∨}}^{∗} \overline{P}_{A}) · ĉh(p_{A^{∨}BB^{∨}}^{∗}r_{BB^{∨}}^{∗} \overline{P}_{B}))
= (−1)^{(g_{A}+g_{B})}r_{A^{∨}B^{∨},∗}(p_{A^{∨}BB^{∨},∗}(ĉh(p_{AA^{∨}B^{∨}}^{∗}q_{AA^{∨}}^{∗} \overline{P}_{A}) · ĉh(p_{A^{∨}BB^{∨}}^{∗}r_{BB^{∨}}^{∗} \overline{P}_{B})))
= (−1)^{(g_{A}+g_{B})}r_{A^{∨}B^{∨},∗}(p_{A^{∨}BB^{∨},∗}(ĉh(p_{AA^{∨}B^{∨}}^{∗}q_{AA^{∨}}^{∗} \overline{P}_{A})) · ĉh(r_{BB^{∨}}^{∗} \overline{P}_{B}))
= (−1)^{(g_{A}+g_{B})}r_{A^{∨}B^{∨},∗}(r_{A^{∨}B^{∨}}^{∗}(q_{A^{∨}B^{∨},∗}(ĉh(q_{AA^{∨}}^{∗} \overline{P}_{A}))) · ĉh(r_{BB^{∨}}^{∗} \overline{P}_{B}))
= (−1)^{(g_{A}+g_{B})}r_{A^{∨}B^{∨},∗}(ĉh(r_{BB^{∨}}^{∗} \overline{P}_{B})) · q_{A^{∨}B^{∨},∗}(ĉh(q_{AA^{∨}}^{∗} \overline{P}_{A}))
= (−1)^{g_{A}}π_{A^{∨}}^{∗}(ĉh(\overline{P}_{A})) · (−1)^{g_{B}} π_{B^{∨}}^{∗}(ĉh(\overline{P}_{B})) = π_{A^{∨}}^{∗}((S_{0}^{∨,A}, 𝔤_{A})) · π_{B^{∨}}^{∗}((S_{0}^{∨,B}, 𝔤_{B}))
= (S_{0}^{∨,A×B}, π_{A^{∨}}^{∗}(𝔤_{B}) ∗ π_{B^{∨}}^{∗}(𝔤_{B}))

Here we have used the projection formula repeatedly, as well as the fact that direct images in arithmetic Chow theory are compatible with base-change. We also used Theorem 1.2.2. (in the second equation before last) and the definition of the intersection product in arithmetic Chow theory (in the last equation). This settles (b). To verify (c) we revert to the hypothesis made at the beginning of this subsection. In particular, we suppose that for some n ⩾ 2, the scheme ker [n] is the constant group scheme in A. As explained at the beginning of this subsection, this does not restrict generality. First notice that by the definition of the symbols [n]_{∗} and [n]^{∗}, we have

[n]^{∗}[n]_{∗}𝔤_{A} = ∑_{τ∈A^{∨}[n](S)} τ^{∗}(𝔤_{A})

and thus by Theorem 1.1(c),

[n]^{∗}𝔤_{A} = ∑_{τ∈A^{∨}[n](S)} τ^{∗}(𝔤_{A})  (27)

Of course, similar equations hold for 𝔤_{B}. Notice also that equation (27) implies that [n]_{∗}𝔤_{A} = 𝔤_{A}, as can be see by applying [n]_{∗} to both sides of equation (27) (see the calculation made after Proposition 4.3). Thus the equation (27) is actually equivalent to the equation [n]_{∗}𝔤_{A} = 𝔤_{A}. Now we may compute

[n]_{A^{∨}×B^{∨}}^{∗}(π_{A^{∨}}^{∗}(𝔤_{A}) ∗ π_{B^{∨}}^{∗}(𝔤_{B})) = π_{A^{∨}}^{∗}([n]_{A^{∨}}^{∗}(𝔤_{A})) ∗ π_{B^{∨}}^{∗}([n]_{B^{∨}}^{∗}(𝔤_{B}))

= π_{A^{∨}}^{∗}( ∑_{τ∈A^{∨}[n](S)} τ^{∗}(𝔤_{A})) ∗ π_{B^{∨}}^{∗}( ∑_{τ∈B^{∨}[n](S)} τ^{∗}(𝔤_{B}))
= ∑_{τ∈A^{∨}[n](S)×B^{∨}[n](S)} τ^{∗}(π_{A^{∨}}^{∗}(𝔤_{A}) ∗ π_{B^{∨}}^{∗}(𝔤_{B}))

which settles (c). Here we used in the first line the fact that the ∗-product is naturally compatible with finite étale pull-back.

4.2 Proof of 1.3.2

We shall apply the Adams-Riemann-Roch theorem in Arakelov geometry proven in [42, Th. 3.6]. Let \overline{M} be the rigidified hermitian line bundle on A corresponding to σ. By assumption, there is an isomorphism \overline{M}^{⊗n} ≃ \overline{O}_{A} of rigidified hermitian line bundles. Let k, l be two positive integers such that k = l (mod n). Let \overline{Ω} := \overline{Ω}_{A/S}. The theorem [42, Th. 3.6] implies that the identity

ψ^{k}(R^{•}π_{∗}\overline{M}) − ψ^{k}(T (λ, \overline{M}))

= θ^{k}(ϵ^{∗}\overline{Ω})^{−1}R^{•}π_{∗}(\overline{M}^{⊗k}) − ch(θ^{k}(ϵ^{∗}\overline{Ω})^{−1})T (λ, \overline{M}^{⊗k}) (28)

holds in K̂_{0}(S)[\frac{1}{k}] and that the identity

ψ^{l}(R^{•}π_{∗}\overline{M}) − ψ^{l}(T (λ, \overline{M}))

= θ^{l}(ϵ^{∗}\overline{Ω})^{−1}R^{•}π_{∗}(\overline{M}^{⊗l}) − ch(θ^{l}(ϵ^{∗}\overline{Ω})^{−1})T (λ, \overline{M}^{⊗l}) (29) holds in K̂_{0}(S)[\frac{1}{l}]. Here the symbols ψ^{∗}(·) refer to the Adams operations acting on arithmetic K_{0}-theory; see [42, sec. 2 and before Th. 3.6] for the exact definition. For the definition of the symbol θ^{∗}(·), see [42, sec. 2]. In the following computations, we shall need the following properties of these symbols. Define ϕ^{t}(·) to be the additive operator, which sends a differential form η of type (r, r) to the differential form t^{r} · η. It is proven in [42, Prop. 4.2] that θ^{k}(ϵ^{∗}\overline{Ω}) is a unit in K̂_{0}(S)[\frac{1}{k}] and we have

ch(θ^{k}(ϵ^{∗}\overline{Ω})) = k^{rk(Ω)} Td(ϵ^{∗}\overline{Ω}^{∨})ϕ^{k}(\overline{Td}^{−1}(ϵ^{∗}\overline{Ω}^{∨}))  (30)

(and similarly for l instead of k). See [42, Lemma 6.11] for this. Secondly, if η ∈ Ã(S_{ℝ}), then we have

ψ^{k}(η) = k · ϕ^{k}(η)

in K̂_{0}(S). In view of the fact that θ^{k}(ϵ^{∗}\overline{Ω}) is a unit in K̂_{0}(S)[\frac{1}{k}], we get the equation

ch(θ^{k}(ϵ^{∗}\overline{Ω}))ψ^{k}(R^{•}π_{∗}\overline{M}) − ch(θ^{k}(ϵ^{∗}\overline{Ω}))ψ^{k}(T (λ, \overline{M}))

= R^{•}π_{∗}(\overline{M}^{⊗k}) − T (λ, \overline{M}^{⊗k})

in K̂_{0}(S)[\frac{1}{k}] from equation (28). Similarly, we get

ch(θ^{l}(ϵ^{∗}\overline{Ω}))ψ^{l}(R^{•}π_{∗}\overline{M}) − ch(θ^{l}(ϵ^{∗}\overline{Ω}))ψ^{l}(T (λ, \overline{M})) = R^{•}π_{∗}(\overline{M}^{⊗l}) − T (λ, \overline{M}^{⊗l})

in K̂_{0}(S)[\frac{1}{l}]. Since k = l (mod n), we obtain the equation

ch(θ^{k}(ϵ^{∗}\overline{Ω}))ψ^{k}(R^{•}π_{∗}\overline{M}) − ch(θ^{k}(ϵ^{∗}\overline{Ω}))ψ^{k}(T (λ, \overline{M}))

= ch(θ^{l}(ϵ^{∗}\overline{Ω}))ψ^{l}(R^{•}π_{∗}\overline{M}) − ch(θ^{l}(ϵ^{∗}\overline{Ω}))ψ^{l}(T (λ, \overline{M}))

in K̂_{0}(S)[\frac{1}{kl}]. In view of equation (30) and of the fact that R^{r}π_{∗}M = 0 for all r ⩾ 0, this translates to the equation

k^{g} Td(ϵ^{∗}\overline{Ω}^{∨})ϕ^{k}(\overline{Td}^{−1}(ϵ^{∗}\overline{Ω}^{∨}))ψ^{k}(T (λ, \overline{M}))

= l^{g} Td(ϵ^{∗}\overline{Ω}^{∨})ϕ^{l}(\overline{Td}^{−1}(ϵ^{∗}\overline{Ω}^{∨}))ψ^{l}(T (λ, \overline{M}))

in K̂_{0}(S)[\frac{1}{kl}]. Recall that in [27] it is shown that T (λ, \overline{M}) = Td^{−1}(ϵ^{∗}\overline{Ω})γ, where γ is a real differential form of type (g −1, g −1) on S. So we may rewrite

k^{g+1} Td(ϵ^{∗}\overline{Ω}^{∨})ϕ^{k}(\overline{Td}^{−1}(ϵ^{∗}\overline{Ω}^{∨}))ϕ^{k}(Td^{−1}(ϵ^{∗}\overline{Ω}))ϕ^{k}(γ)

= l^{g+1} Td(ϵ^{∗}\overline{Ω}^{∨})ϕ^{l}(\overline{Td}^{−1}(ϵ^{∗}\overline{Ω}^{∨}))ϕ^{l}(Td^{−1}(ϵ^{∗}\overline{Ω}))ϕ^{l}(γ) (31)

Furthermore

Lemma 4.4. We have Td^{−1}(ϵ^{∗}\overline{Ω}^{∨} ⊕ ϵ^{∗}\overline{Ω}) = 1.

Proof. We may (and do) assume that R = ℂ. Consider the relative Hodge extension

0 → R^{0}π_{∗}(Ω) → H_{dR}^{1}(A/S) → R^{1}π_{∗}(O_{A}) → 0  (32)

where H_{dR}^{1}(A/S) := R^{1}π_{∗}(Ω_{A/S}^{•}) is the first relative de Rham cohomology sheaf. The sequence (32) is the expression of the filtration on R^{1}π_{∗}(Ω_{A/S}^{•}), which comes from the relative Hodge to de Rham spectral sequence. This spectral sequence is the first hypercohomology spectral sequence of the relative de Rham complex Ω_{A/S}^{•} and it degenerates by [13, Prop. 5.3]. The relative form of the GAGA theorem shows that there is an isomorphism of holomorphic vector bundles H_{dR}^{1}(A/S)(ℂ) ≃ (R^{1}π(ℂ)_{∗}ℂ) ⊗_{ℂ} O_{S(ℂ)} (see [14, p. 31]) and via this isomorphism we endow H_{dR}^{1}(A/S)(ℂ) with the fibrewise Hodge metric, whose formula is given in [38, before Lemma 2.7]. Since the metric on H_{dR}^{1}(A/S)(ℂ) is locally constant by construction, the curvature matrix of the hermitian vector bundle H_{dR}^{1}(A/S)(ℂ) ≃ (R^{1}π(ℂ)_{∗}ℂ) ⊗_{ℂ} O_{S(ℂ)} vanishes. Now the formula in [38, Lemma 2.7] shows that in the sequence (32), the L^{2}-metric on the first term corresponds to the induced metric and the L^{2}-metric on the end term corresponds to the quotient metric. We now view the sequence (32) as a sequence of hermitian vector bundles with the metrics described above. Using [36, Th. 3.4.1], we see that the secondary class T̃d of the sequence (32) is dd^{c}-closed. Thus

Td(R^{1}π_{∗}(O_{A}, L^{2}) ⊕ R^{0}π_{∗}(Ω, L^{2})) = Td((H_{dR}^{1}(A/S), Hodge metric)) = 1.

Now notice that by relative Lefschetz duality for Hodge cohomology (see [13, Lemme 6.2]) and Grothendieck duality, there is an isomorphism of O_{S}-modules ϕ_{λ} : R^{1}π_{∗}(O_{A}) ⟶^{∼} R^{0}π_{∗}(Ω)^{∨}, which is dependent on λ. To describe it, let S = Spec ℂ. Under the Hodge-de Rham splitting of the sequence (32), the morphism ϕ_{λ} is given by the formula

ω ↦ ω ∧ λ^{g−1} ↦ ∫_{A}(ω ∧ λ^{g−1}) ∧ (·)

(notice that this formula does actually not depend on the splitting). Now comparing the last formula with the formula for the Hodge metric in [38, before Lemma 2.7], we see that, up to a constant factor, ϕ_{λ} induces an isometry between R^{1}π_{∗}(O_{A}, L^{2}) and the dual of the hermitian vector bundle R^{0}π_{∗}(Ω, L^{2}). To complete the proof, notice that since the volume of the fibres of π(ℂ) is locally constant (by the assumption on λ), the natural isomorphism of vector bundles R^{0}π_{∗}(Ω, L^{2}) ≃ ϵ^{∗}\overline{Ω} is an isometry up to a locally constant factor. Hence the Chern forms of R^{0}π_{∗}(Ω, L^{2}) and ϵ^{∗}\overline{Ω} are the same. This completes the proof. □ Together with Lemma 4.4, we see that (31) gives the identity

k^{2g} Td(\overline{Ω}^{∨})γ = l^{2g} Td(\overline{Ω}^{∨})γ

or in other words

(k^{2g} − l^{2g})T (λ, \overline{M}) = 0  (33)

Documenta Mathematica 20 (2015) 631–668

in K̂_{0}(S)[\frac{1}{kl}]. We shall now show that equation (33) implies the identity

2g · n · N_{2g} · T (λ, \overline{M}) = 0  (34)

in K̂_{0}(S). For this consider the following combinatorial lemma.

Lemma 4.5. Let G be an abelian group, written additively. Let c ⩾ 1 and let α ∈ G. Suppose that for all k, l > 0 such that k = l (mod n), we have

(l^{c} − k^{c}) · α = 0

in G[\frac{1}{kl}]. Then

order(α) | 2 · n · c · [ ∏_{{p prime, \atop p∤n, (p−1)|c}} p]

Proof. Rephrasing the hypotheses of the lemma, we find that for any k, l > 0 such that k = l (mod n), there exist integers a, b ⩾ 0 (depending on the couple (k, l)) such that:

k^{a}l^{b}(l^{c} − k^{c}) · g = 0.  (35)

For any (k, l) as above, we will denote by γ the integer such that l = k + γn. In what follows, p will be a prime dividing order(α) and we will write δ_{p} (or in short δ if no confusion can occur) the positive valuation v_{p}(order(α)). We deduce from equation (35) that:

p^{δ} | k^{a}l^{b}(l^{c} − k^{c}).

From now on, k and l will be chosen such that k, l ≠ 0 (mod p), this implying in particular that the classes of k and l in ℤ/p^{δ}ℤ are invertible. We thus get:

p^{δ} | (l^{c} − k^{c})

or equivalently:

(l/k)^{c} = 1 (mod p^{δ})

in (ℤ/p^{δ}ℤ)^{∗}. We will denote by C ⊂ (ℤ/p^{δ}ℤ)^{∗} the set of the classes (l/k) with l and k as above. The restriction of the map φ : x ↦ x^{c} to C is then identically equal to 1. In what follows, in order to bound δ, we determine the set C. We must distinguish between two cases.

Case 1. The prime p doesn’t divide n. Taking k = n, we have (l/k) = (k + γn)/k = 1 + γ for all the integers γ satisfying (1 + γ) ∧ p = 1. We deduce from this that necessarily:

C = (ℤ/p^{δ}ℤ)^{∗}.

We now need the following well-known group isomorphisms, which we recall for the sake of the exposition (the group laws are multiplicative on the left-hand side and additive on the right-hand side):

• If p ≠ 2, (ℤ/p^{δ}ℤ)^{∗} ≃ ℤ/p^{δ−1}(p − 1)ℤ.

• If p = 2, (ℤ/2ℤ)^{∗} ≃ {0}, (ℤ/4ℤ)^{∗} ≃ ℤ/2ℤ and for δ ⩾ 3, (ℤ/2^{δ}ℤ)^{∗} ≃ (ℤ/2ℤ) × (ℤ/2^{δ−2}ℤ), the projection on the first factor being the obvious reduction map to (ℤ/4ℤ)^{∗} ≃ ℤ/2ℤ.

Let’s then discuss the two subcases p ≠ 2 and p = 2 separately. If p ≠ 2, the restriction of φ to C = (ℤ/p^{δ}ℤ)^{∗} ≃ ℤ/p^{δ−1}(p − 1)ℤ is equal to 1 identically. On the righten side, φ is the multiplication by \overline{c} the class of c in ℤ/p^{δ−1}(p − 1)ℤ. One must then have \overline{c} = 0, and thus:

(p − 1)p^{δ−1} | c,

i.e. the two conditions p − 1 | c and δ_{p} ⩽ 1 + v_{p}(c). If p = 2, using the same argument as above, the discussion falls into three different subsubcases:

• δ_{2} = 1, no condition.

• δ_{2} = 2, we find that 2 | c.

• δ_{2} ⩾ 3, we get that 2 | c and 2^{δ_{2}−2} | c, i.e. δ_{2} ⩽ 2 + v_{2}(c).

Summing up those three subsubcases, we finally conclude that if c is odd then δ_{2} = 1 and if c is even then δ_{2} ⩽ 2 + v_{2}(c).

Let’s now come to the:

Case 2. The prime p divide n. Let’s define β := v_{p}(n) ⩾ 1 and n′ := n/p^{β} and let’s suppose in addition that δ > β. We compute in (ℤ/p^{β}ℤ)^{∗}

l/k = (k + γn)/k = 1 + γn/k = 1 + (γn′/k)p^{β} = 1

and thus the set C is contained in the kernel K of the reduction morphism:

(ℤ/p^{δ}ℤ)^{∗} ⟶ (ℤ/p^{β}ℤ)^{∗}.

The integer n′ being prime to p, its class in (ℤ/p^{δ}ℤ) is invertible and one can take k = n′. We find that for any integer γ:

l/k = 1 + γn/n′ = 1 + γp^{β}

in (ℤ/p^{δ}ℤ)^{∗}, this implying the set equality:

C = K.

We are thus left to determine K. Again, we will distinguish between two subcases:

On a Canonical Class of Green Currents … 663

If p ≠ 2 we have:

#K = #(ℤ/p^{δ}ℤ)^{∗}/#(ℤ/p^{β}ℤ)^{∗} = p^{δ−1}(p − 1)/p^{β−1}(p − 1) = p^{δ−β} from which we deduce first that K ≃ ℤ/p^{δ−β}ℤ and then immediately that p^{δ−β} | c, i.e.

δ ⩽ β + v_{p}(c) = v_{p}(n) + v_{p}(c).

If p = 2, the discussion now falls into five different subsubcases.

• β = 1 and δ_{2} = 1, no condition.

• β = 1 and δ_{2} = 2, we find that 2 | c, i.e. c is even.

• β = 1 and δ_{2} ⩾ 3, then we get 2 | c and 2^{δ_{2}−2} | c, i.e. δ_{2} ⩽ 2 + v_{2}(c) with c being necessary even. To summarize those three first cases let’s write that when β = 1, if c odd then δ_{2} = 1 and if c is even then δ_{2} ⩽ 2 + v_{2}(c).

• β = 2 and δ_{2} = 2, no condition.

• In all other cases, the kernel K is contained in (ℤ/2^{δ_{2}−2}ℤ) and as a

consequence is cyclic. We thus get:

#K = #(ℤ/2^{δ_{2}}ℤ)^{∗}/#(ℤ/2^{β}ℤ)^{∗} = 2^{δ_{2}−1}/2^{β−1} = 2^{δ_{2}−β} and so 2^{δ_{2}−β} | c, from which we deduce:

δ_{2} ⩽ β + v_{2}(c) = v_{2}(n) + v_{2}(c).

In conclusion of the subcase p = 2, we find that:

δ_{2} ⩽ v_{2}(n) + v_{2}(c) + w_{2},

with w_{2} = 1 if v_{2}(n) = 1 and c is even, and w_{2} = 0 otherwise. Putting everything together, we have finally proven that:

order(α) | F_{2} × ∏_{p prime, p≠2, p doesn’t divide n, (p−1)|c} p^{1+v_{p}(c)} × ∏_{{p prime, \atop p≠2, p|n}} p^{v_{p}(n)+v_{p}(c)}

where the F_{2} factor is given by the following rules:

• if n and c are odd then F_{2} = 2,

• if n is odd and c is even then F_{2} = 2^{2+v_{2}(c)},

• if n is even then F_{2} = 2^{v_{2}(n)+v_{2}(c)+w_{2}} ; with w_{2} = 1 if v_{2}(n) = 1 and c is

even, and w_{2} = 0 otherwise.

Documenta Mathematica 20 (2015) 631–668

The lemma’s statement is then a direct consequence of this (more precise) assertion. □ In view of Lemma 4.5, the set of identities (33) implies that the order of T (λ, \overline{M}) in K̂_{0}(S) divides 2 · n · 2g · [∏_{p prime, p∤n, (p−1)|2g} p]. Now use the notations of the last lemma. It is shown in [40, Appendix B] that the equality

2 · denominator[(−1)^{\frac{c+2}{2}} B_{c}/c] = 2 · ∏_{{p prime, \atop (p−1)|c}} p^{ord_{p}(c)+1}  (36)

holds if c is even. This proves the identity (34) . To conclude the proof of Theorem 1.3.2, recall that there is an exact sequence

K_{1}(A^{∨}) ⟶^{−2reg_{an}} ⊕_{p⩾0}Ã^{p,p}(A_{ℝ}^{∨}) →^{a} K̂_{0}(A^{∨}) → K_{0}(A^{∨}) → 0

(see [21, Th. 6.2 (i)] for this).

5 The case of elliptic schemes

In this last section, we shall consider elliptic schemes and compare the conclusions of Theorems 1.1, 1.2 and 1.3 with classical results on elliptic units. So suppose that A is of relative dimension 1, i.e. that A is an elliptic scheme over S. Suppose also that the structural morphism S → Spec R is the identity on Spec R. Let σ ∈ Σ be an embedding of R into ℂ. There exists an isomorphism of complex Lie groups

A(ℂ)_{σ} := (A ×_{R,σ} ℂ)(ℂ) = ℂ/(ℤ + ℤ · τ_{σ})  (37)

where τ_{σ} ∈ ℂ lies in the upper half plane.

Proposition 5.1. (a) The restriction of 𝔤_{A^{∨}} to A(ℂ)_{σ} = ℂ/(ℤ + ℤ · τ_{σ})\{0} is given by the function

ϕ(z) = ϕ_{A,σ}(z) := −2log|e^{−z·η(z)/2}sigma(z)Δ(τ_{σ})^{\frac{1}{12}} |

(b) Endow ℂ/(ℤ+ℤ·τ_{σ}) with its Haar measure of total measure 1. The function ϕ then defines an L^{1}-function on ℂ/(ℤ + ℤ · τ_{σ}) and the restriction of 𝔤_{A^{∨}} to A(ℂ)_{σ} is the current [ϕ] associated with ϕ.

Here Δ(•) is the discriminant modular form, sigma(z) is the Weierstrass sigmafunction associated with the lattice [1, τ_{σ}] and η is the quasi-period map associated with the lattice [1, τ_{σ}], extended ℝ-linearly to all of ℂ (see [43, I, Prop. 5.2] for the latter). Proof. The formula (a) follows from Theorem 1.3.1 and the formula for the Ray-Singer analytic torsion of a flat line bundle on an elliptic curve given in [41, Th. 4.1].

For statement (b), notice that there exists a Green form η of log type along 0 (see [19, Th. 1.3.5, p.106]). In this situation η is a real-valued C^{∞} function on A(ℂ)_{σ}\{0}, which is locally and hence globally L^{1} on A(ℂ)_{σ} (because A(ℂ)_{σ} is compact). By definition the current [η] associated with η is a Green current for 0 and by [19, Lemma 1.2.4], there exists a C^{∞} real-valued function f on A(ℂ)_{σ}, such that 𝔤_{A^{∨}}|_{A(ℂ)_{σ}} = [η] + [f] = [η + f]. Now by construction the restriction of 𝔤_{A^{∨}} to A(ℂ)_{σ}\{0} is given by the current associated with the restriction to A(ℂ)_{σ}\{0} of the locally L^{1}-function η + f. Since η + f and ϕ are both C^{∞} on A(ℂ)_{σ}\{0}, they must actually coincide on A(ℂ)_{σ}\{0}. This proves (b). □ The distribution relations for 𝔤_{A^{∨}} (i.e. Theorem 1.1.3) imply that the function ϕ(z) has the property that

∑_{{w∈ℂ/(ℤ+ℤ·τσ), \atop n·w=z}} ϕ_{A,σ}(w) = ϕ_{A,σ}(z).

for any n ⩾ 2 and z ∈ ℂ/(ℤ + ℤ · τ_{σ})\{0}. This also follows from the more precise distribution relations given in [28, Th. 4.1, p. 43]. Now suppose that R is a Dedekind ring. We suppose given \underline{z} ∈ A(S), whose image is disjoint from the unit section and such that n · \underline{z} = 0. Let \underline{z}_{σ} ∈ ℂ/(ℤ + ℤ · τ_{σ}) be the element corresponding to \underline{z}. We compute from the definition that N_{2} = 24. Theorem 1.3.2 now implies that there exists u ∈ R^{∗}, which does not depend on σ ∈ Σ, such that

log|σ(u)| = 24 · n · ϕ_{A,σ}(\underline{z}_{σ})  (38)

In particular the real number

exp(24 · n · ϕ_{A,σ}(\underline{z}_{σ}))

is an algebraic unit. If R is the ring of integers of a number field, n has at least two distinct prime factors and Σ = {σ, σ̄} then (38) is also a consequence of [28, Th. 2.2, p. 37]. Notice that (38) overlaps with part of the reciprocity law for elliptic units, if A\overline{Frac(R)} is assumed to have complex multiplication by the ring of integers of an imaginary quadratic field (see [31, chap. 19, par. 3, Th. 3]). Special instances of elliptic units were first (implicitly) constructed by Eisenstein in his analytic proof of cubic and quartic reciprocity laws (cf. [18] and [16], [17]). For a thorough historical discussion of Eisenstein’s contribution and additional references, see [33, §8]. Remark. Our proof of the fact that the real number exp(24 · n · ϕ_{A,σ}(\underline{z}_{σ})) is an algebraic unit shows that 24 naturally comes from a Bernoulli number via von Staudt’s theorem. Indeed, von Staudt’s theorem is the main tool in the proof of (36). The proof of the fact that the number exp(24 · n · ϕ_{A,σ}(\underline{z}_{σ})) (in fact even the number exp(12 · n · ϕ_{A,σ}(\underline{z}_{σ}))) is an algebraic unit, which is given in [28, Th. 2.2, p. 37], does not seem to establish such a link.

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V. Maillot Institut de Mathématiques de Jussieu Université Paris 7 Denis Diderot C.N.R.S. Case Postale 7012 2 place Jussieu F-75251 Paris Cedex 05 FRANCE vmaillot@math.jussieu.fr D. Rössler Institut de Mathématiques Équipe Emile Picard Université Paul Sabatier 118 Route de Narbonne 31062 Toulouse cedex 9 FRANCE rossler@math.univ-toulouse.fr