2.1 The Bloch–Kato–Fontaine–Perrin-Riou conjecture [Reference Fontaine15, Reference Fontaine and Perrin-Riou16]

The motive $M= h^{j}(V_{0})(r)$ has a Betti realisation $H^{j}_{B}(V {}_{\mathbb C}, \mathbb Z(r))$ , a de Rham realisation $H^{j}_{dR} (V_{\mathbb C}, \mathbb C(r))$ , an action of complex conjugation c on $H^{j}_{B}(V {}_{\mathbb C}, \mathbb Z(0)) = H^{j}_{B}(V_{\mathbb C}, \mathbb Z)$ , a Hodge filtration F on $H^{j}_{dR} (V_{\mathbb C}, \mathbb C(r))$ and a period map

$$ \begin{align*} \theta_{j,r}: H^{j}(V_{\mathbb C}, \mathbb C(r)) \xrightarrow{\sim} H^{j}_{dR}(V_{\mathbb C}, \mathbb C(r)).\end{align*} $$

As abelian groups, $H^{j}_{B}(V_{\mathbb C}, \mathbb Z(r)) = H^{j}_{B}(V_{\mathbb C}, \mathbb Z)$ and $H^{j}_{B}(V_{\mathbb C}, \mathbb C(r)) = H^{j}_{B}(V_{\mathbb C}, \mathbb C)$ ; also, $H^{j}_{B}(V_{\mathbb C}, \mathbb C(r)) =H^{j}_{B}(V_{\mathbb C}, \mathbb Z(r))\otimes _{\mathbb Z}\mathbb C$ . The map $\theta _{j,0}$ is the classical period map from $H^{j}_{B}(V_{\mathbb C}, \mathbb C)$ to $H^{j}_{dR} (V_{\mathbb C}, \mathbb C)$ . We define

$$ \begin{align*}H^{j}(V_{\mathbb C}, \mathbb Z(r))^{+} &= \{x \in H^{j}(V_{\mathbb C}, \mathbb Z)~|~ c(x) =~(-1)^{r}\cdot x\}, \\ H^{j}(V_{\mathbb C}, \mathbb C(r))^{+} &= \{x \in H^{j}(V_{\mathbb C}, \mathbb C)~|~ c(x) =~(-1)^{r}\cdot x\}.\end{align*} $$

Finally, the map $\theta _{j,r}$ defined as $(2\pi i)^{r}\cdot \theta _{j,0}$ induces a map

(3) $$ \begin{align} \alpha_{M}: H^{j}(V_{\mathbb C}, \mathbb C(r))^{+} \to M_{B} \xrightarrow{\sim} M_{dR} \to t_{M}= \frac{M_{dR}}{F^{0}}.\end{align} $$

Fontaine–Perrin-Riou [Reference Fontaine and Perrin-Riou16, Reference Fontaine15] introduced certain vector spaces $H^{0}_{f}(M)$ and $H^{1}_{f}(M)$ that are conjecturally finite-dimensional $\mathbb Q$ -vector spaces. Further, they conjectured the existence of the fundamental exact sequence

(4) $$ \begin{align} 0 &\to H^{0}_{f}(M)\otimes{\mathbb R} \xrightarrow{c} \textrm{Ker}(\alpha_{M}) \to H^{1}_{f}(M^{*}(1))^{*}\otimes{\mathbb R} \xrightarrow{h} H^{1}_{f}(M)\otimes{\mathbb R} \xrightarrow{r} \textrm{Coker}(\alpha_{M}) \nonumber \\[3pt] & \to H^{0}_{f}(M^{*}(1))^{*}\otimes{\mathbb R} \to 0, \end{align} $$

where c is a cycle class map, h is a (Arakelov) height pairing and r is the Beilinson regulator. They used it to reformulate the Deligne–Beilinson conjecture $C_{DB}(M)$ [Reference Fontaine15] which predicts the special value $L^{*}(M,0) \in {\mathbb R^{\times }}/{\mathbb Q^{\times }}$ at $s=0$ of the L-function $L(M,s)$ .

To predict $L^{*}(M,0) \in \mathbb R^{\times }$ , they introduced similar exact sequences for each prime p; this is the conjecture $C_{BK}(M)$ [Reference Fontaine15] for p. One obtains a prediction for the special value of $\zeta _{HW}(V_{0},s)$ at $s=r$ by combining $C_{BK}(M)$ for the motives $h^{j}(V_{0})(r)$ for all j. For more details, see [Reference Bloch and Kato2, Reference Fontaine and Perrin-Riou16, Reference Kings23, Reference Kings24, Reference Fontaine15, Reference Flach10, Reference Flach11, Reference Flach and Morin12].

But our approach, following [Reference Lichtenbaum27], to $\zeta ^{*}(V,r)$ is different: we endow the vector spaces in (4) with integral structures using Weil-étale motivic cohomology groups and the de Rham cohomology groups of V. The determinants of the maps in (4) with respect to these integral structures will be used for a description of $\zeta ^{*}(V,r)$ .

2.2 Weil-étale motivic cohomology groups

These groups $H^{i}_{W}(V, \mathbb Z(r))$ [Reference Lichtenbaum27, §2.1] are defined as étale motivic cohomology $H^{i}_{et}(V, \mathbb Z(r))$ for $i \le 2r$ and $r \ge 0$ and then, for $i>2r$ , as the dual of étale motivic cohomology

$$ \begin{align*}H^{i}(\textrm{RHom}(R\Gamma_{et}(V, \mathbb Z(d-r)), \mathbb Z [-2d-1]));\end{align*} $$

thereby, for $i>2r$ , they sit in an exact sequence

$$ \begin{align*}0 \to \textrm{Ext}^{1}(H^{2d+2-i}_{et}(V, \mathbb Z(d-r)), \mathbb Z) \to H^{i}_{W}(V, \mathbb Z(r)) \to \textrm{Hom}(H^{2d+1-i}_{et}(V, \mathbb Z(d-r)), \mathbb Z) \to 0.\end{align*} $$

We define $H^{i}_{W}(X, \mathbb Z(r))$ to be zero when $r<0$ and $i\le 2r$ .

The groups $H^{i}_{W}(V, \mathbb Z(r))$ are conjectured to be finitely generated.

2.3 Integral structures and complexes

We recall the well-known theory of determinants [Reference Kings24, Lecture 1, §5].

A lattice L in a finite-dimensional vector space C is the $\mathbb Z$ -submodule generated by a basis of C. Write $\Lambda L \cong \mathbb Z$ (noncanonically) for the highest exterior power of L and $\Lambda C \cong \mathbb C$ (noncanonically) for the highest exterior power of C and $(\Lambda C)^{-1}$ for its dual. Given a finite complex of finite-dimensional complex vector spaces $C_{i}$ and lattices $L_{i} \subset C_{i}$

$$ \begin{align*}C_{\bullet} :\qquad 0 \to \cdots C_{i} \xrightarrow{f_{i}} C_{i+1} \xrightarrow{f_{i+1}} C_{i+2} \to \cdots, \end{align*} $$

one has a canonical map $g: \Lambda (L_{\bullet }) \to \Lambda (C_{\bullet })$ where

$$ \begin{align*} \Lambda (L_{\bullet}) = \otimes_{j}(\Lambda L_{j})^{(-1)^{j}} \cong \mathbb Z, \qquad \Lambda(C_{\bullet}) = \otimes_{i} (\Lambda C_{i})^{(-1)^{i}}.\end{align*} $$

If $C_{\bullet }$ is exact, then one has a canonical isomorphism

$$ \begin{align*}h: \Lambda(C_{\bullet}) \xrightarrow{\sim} \mathbb C.\end{align*} $$

We define

$$ \begin{align*} det(C_{\bullet},L_{\bullet}) = h(g(e))\end{align*} $$

as the image of a generator e of $\Lambda (L_{\bullet }) \cong \mathbb Z$ under the composite map

$$ \begin{align*}\Lambda (L_{\bullet}) \xrightarrow{g} \Lambda (C_{\bullet}) \xrightarrow{h} \mathbb C;\end{align*} $$

$det(C_{\bullet },L_{\bullet })$ is well-defined as an element of

$$ \begin{align*} {\mathbb C^{\times}}/{\{\pm 1\}}.\end{align*} $$

2.3.2 Integral structures on complexes

Given a finite complex $C_{\bullet }$ of finite-dimensional complex vector spaces, an integral structure L on $C_{\bullet }$ is a collection $L_{\bullet } = (L_{i}, t_{i})$ where $L_{i}$ is a lattice in $C_{i}$ and $t_{i}$ is a positive rational number. If $C_{\bullet }$ is exact, we define

$$ \begin{align*}\chi (C_{\bullet}, L_{\bullet}) = \frac{det (C_{\bullet}, L_{\bullet} )}{T} \in {\mathbb C^{\times}}/{\{\pm 1\}}, \qquad T = \frac{t_{0}\cdot ~\cdot~t_{2}~.\cdots}{t_{1}~\cdot~t_{3}~\cdot~\cdots} \in \mathbb Q_{>0}.\end{align*} $$

Here, T is a multiplicative Euler characteristic.

Example 6. A collection $\phi $ of homomorphisms $\phi _{i}: A_{i} \to C_{i}$ where, for all i, $A_{i}$ is a finitely generated abelian group, $\textrm {Ker}(\phi _{i})$ is finite and $\textrm {Im}(\phi _{i})$ is a lattice in $C_{i}$ gives an integral structure $L_{\bullet }= (L_{i}, t_{i}) $ where $L_{i}= \phi _{i}(A_{i})$ and $t_{i} =[\textrm {Ker}(\phi _{i})]$ ; in this case, we write $T= \chi (A_{tor})$ . If $C_{\bullet }$ is exact, then we have

$$ \begin{align*}\chi (C_{\bullet}, A_{\bullet}) = \frac{det (C_{\bullet}, L_{\bullet} )}{ \chi (A_{tor})} \in {\mathbb C^{\times}}/{\{\pm 1\}}.\end{align*} $$

We often refer to $A_{\bullet }$ as an integral structure on $C_{\bullet }$ .

If there is a canonical integral structure $L_{\bullet }$ on $C_{\bullet }$ , then we write $det(C_{\bullet })$ and $\chi (C_{\bullet })$ for $det(C_{\bullet },L_{\bullet })$ and $\chi (C_{\bullet },L_{\bullet })$ .

2.3.3 Integral structures on $\mathcal O_{F}$ -modules

Let M be a finitely generated $\mathcal O_{F}$ -module of rank d; there are two natural integral structures $i_{M}$ and $j_{M}$ on $V= M\otimes _{\mathbb Z}\mathbb C $ . Let $\phi : M \to V$ be the canonical map. The integral structure $i_{M}$ is obtained by entirely forgetting the $\mathcal O_{F}$ -structure:

(5) $$ \begin{align} i_{M} = (\phi(M), [M_{tor}]). \end{align} $$

While $i_{M}$ appears in the conjectural formula (1), anotherFootnote ¹ integral structure $j_{M}$ appears in Conjecture 20; $j_{M}$ uses the $\mathcal O_{F}$ -structure on M. Recall the standard isomorphism

(6) $$ \begin{align} \psi: \mathcal O_{F} \otimes_{\mathbb Z}\mathbb C \xrightarrow{\sim} \mathbb C^{r_{1}+2r_{2}} = \mathbb Z^{r_{1}+2r_{2}}\otimes\mathbb C \end{align} $$

that sends $x\otimes 1\in \mathcal O_{F}\otimes \mathbb C$ to the element of $\mathbb C^{r_{1}+2r_{2}}$ given by the collection of $\sigma (x)$ as $\sigma \in \tilde {S}$ runs through the embeddings of F in $\mathbb C$ . Define $M_{\sigma } = M\otimes _{\mathcal O_{F}} \mathbb C$ using $\sigma : F\to \mathbb C$ .

The isomorphism (6) shows

$$ \begin{align*}V= M\otimes_{\mathbb Z}\mathbb C \cong M \otimes_{\mathcal O_{F}} (\mathcal O_{F}\otimes_{\mathbb Z} \mathbb C) \cong M\otimes_{\mathcal O_{F}}\mathbb C^{r_{1}+2r_{2}} \cong \prod_{\sigma \in \tilde{S}} M_{\sigma}.\end{align*} $$

Suppose $B=\{x_{1}, \cdots , x_{d}\} \subset M$ is linearly independent over $\mathcal O_{F}$ . Let $\{\epsilon _{j}\}$ be the standard basis of $ \mathbb C^{r_{1}+2r_{2}}$ . Define $\tilde {B} = <x_{i}\otimes \epsilon _{j}>$ to be the $\mathbb Z$ -span of $x_{i}\otimes \epsilon _{j}$ inside V. Consider the integral structure $i(B)= (\tilde {B}, b)$ where b is the order of the quotient of M by the $\mathcal O_{F}$ -submodule $<x_{1}, \cdots , x_{d}>$ generated by B.

Proposition 7. For any two such subsets B and $B^{\prime }$ of M, $i(B)$ and $i(B^{\prime })$ are Euler-equivalent.

Proof. Let $B^{\prime } =\{c_{1}x_{1}, c_{2}x_{2}, \cdots , c_{d}x_{d}\}$ and $C = c_{1}.c_{2}\cdots c_{d}$ and consider $i(B^{\prime })$ . As b and $b^{\prime }$ are related by $b^{\prime } = C.b$ and the determinant of the identity map of V with respect to $i(B)$ and $i(B^{\prime })$ is C, $i(B)$ and $i(B^{\prime })$ are Euler-equivalent. Given B and $B^{\prime }$ , one can find $B^{\prime \prime }$ such that the previous argument applies to the pairs $B, B^{\prime \prime }$ and $B^{\prime }, B^{\prime \prime }$ .

Definition 8. For any finitely generated $\mathcal O_{F}$ -module M, the $\mathcal O_{F}$ -integral structure $j_{M}$ on $V =M\otimes _{\mathbb Z}\mathbb C$ is the Euler equivalence class in Proposition 7.

2.3.5 Pairings and determinants

Let N be a finitely generated abelian group and $\psi : N/{N_{tor}} \times N/{N_{tor}} \to \mathbb R$ be a nondegenerate symmetric bilinear form on N. One defines $\Delta (N) \in \mathbb C^{\times }/{\{\pm 1\}}$ as the determinant of the matrix $\psi (b_{i}, b_{j})$ divided by $(N: N_{0})^{2}$ where $N_{0}$ is the subgroup of finite index generated by a maximal linearly independent subset $b_{i}$ of N; therefore, $\Delta (N)$ is independent of the choice of $b_{i}$ s and incorporates the order of the torsion subgroup of N. We can rewrite $\psi $ as the integral structure $(C_{\bullet }, L_{\bullet })$ with $L_{0}=N, L_{1}= \textrm {Hom}(N, \mathbb Z)$ :

$$ \begin{align*}C_{0}= N\otimes\mathbb C \xrightarrow{\psi_{*}} \textrm{Hom}(N, \mathbb Z)\otimes\mathbb C= C_{1}, \qquad \Delta(N) = \frac{det(\psi_{*}) }{[N_{tor}]^{2}}; \end{align*} $$

here $det(C_{\bullet }, L_{\bullet })$ is the determinant $\det (\psi _{*})$ of $\psi $ computed with the bases $L_{0}$ and $L_{1}$ . Given a short exact sequence of finitely generated groups $ 0 \to N^{\prime } \to N \to N^{\prime \prime } \to 0$ which splits over $\mathbb Q$ as an orthogonal direct sum

$$ \begin{align*}N_{\mathbb Q} \cong N^{\prime}_{\mathbb Q} \oplus N^{\prime\prime}_{\mathbb Q}\end{align*} $$

with respect to a definite pairing $\psi $ on N, one has the following standard relation:

(7) $$ \begin{align} \Delta(N) = \Delta(N^{\prime})~\cdot~\Delta(N^{\prime\prime}). \end{align} $$

2.4 Motives

We start with the integral structures underlying the Betti and de Rham realisations of the motives $M= h^{j}(V_{0})(r)$ for $0\le j \le 2(d-1)$ .

Definition 10. The integral structure $H^{j}_{B}(V_{\mathbb C}, \mathbb Z(r))^{+}$ is defined as the complex vector space $H^{j}_{B}(V_{\mathbb C}, \mathbb C(r))^{+}$ together with the homomorphism

$$ \begin{align*}H^{j}_{B}(V_{\mathbb C}, \mathbb Z)^{+} \to H^{j}_{B}(V_{\mathbb C}, \mathbb C)^{+}\xrightarrow{\times (2\pi i)^{r}} H^{j}_{B}(V_{\mathbb C}, \mathbb C(r))^{+} ;\end{align*} $$

the first map is the natural map and the second map is multiplication by $(2\pi i)^{r}$ .

2.5 de Rham realisation of $M= h^{j}(V_{0})(r)$

The de Rham realisation $M_{dR} = H^{j}_{dR}(V_{\mathbb C}, \mathbb C(r))$ is $H^{j}_{dR}(V_{\mathbb C}, \mathbb C)$ with the Hodge filtration shifted by r; that is, the Tate twist in de Rham realisation shifts the Hodge filtration but does not change the vector space. The tangent space $(t_{M})_{\mathbb C}$ of M is defined as follows:

$$ \begin{align*}(t_{M})_{\mathbb C} = \frac{H^{j}_{dR}(V_{\mathbb C}, \mathbb C(r))}{F^{0}} = \prod_{k<r} H^{j-k}(V_{\mathbb C}, \Omega^{k}_{V_{\mathbb C}}) = \prod_{\sigma \in \tilde{S} }~ \prod_{k<r} H^{j-k}(V_{\sigma}, \Omega^{k}_{V_{\sigma}}).\end{align*} $$

Let $\lambda ^{k}\Omega _{V/S}$ denote the kth derived exterior power of the sheaf $\Omega _{V/S}$ on V; see the appendix of [Reference Lichtenbaum27].

Definition 12. The integral structure on the complex vector space $(t_{M})_{\mathbb C}$ is given by the map

$$ \begin{align*}t_{M}:= \prod_{k<r} H^{j-k}(V, \lambda^{k} \Omega_{V/S}) \to \prod_{\sigma \in \tilde{S}}~ \prod_{k<r} H^{j-k}(V_{\sigma}, \Omega^{k}_{V_{\sigma}}) = (t_{M})_{\mathbb C}.\end{align*} $$

As in (5), one views $H^{j-k}(V, \lambda ^{k} \Omega _{V/S}) $ as an abelian group; its structure as a module over $\mathcal O_{F}$ is not used.

Remark 13. We can use the Hodge decomposition of $(t_{M})_{\mathbb C}$ to decompose the period map $\alpha _{M}$ of (3) as a sum of $\alpha _{k}$ where $\alpha _{k}$ is the composite map

$$ \begin{align*}H^{j}(V_{\mathbb C}, \mathbb C(r))^{+} \xrightarrow{\alpha} (t_{M})_{\mathbb C} = \prod_{k<r} H^{j-k}(V_{\mathbb C}, \Omega^{k}_{V_{\mathbb C}}) \to H^{j-k}(V_{\mathbb C}, \Omega^{k}_{V_{\mathbb C}}).\end{align*} $$

The enhanced period map $\gamma _{M}$ of [Reference Lichtenbaum27, §2.3],

$$ \begin{align*}\gamma_{M}: H^{j}(V_{\mathbb C}, \mathbb C(r))^{+} \to (t_{M})_{\mathbb C},\end{align*} $$

likewise decomposes as a sum of $\gamma _{k}:= \Gamma ^{*}(r-k)~\cdot ~\alpha _{k}$ . Here $\Gamma $ is the classical gamma function and $\Gamma ^{*}(m)$ denotes the special value at $s=m$ .

2.6 Integral structures on certain motivic exact sequences

The conjectural formula (1) from [Reference Lichtenbaum27, Conjecture 3.1] predicts that the special value $\zeta ^{*}(V,r)$ of $\zeta (V,s)$ at $s=r$ is equal to $ \pm \chi (V,r)$ (up to powers of 2) defined as follows:

$$ \begin{align*} \chi(V,r)= \frac{\chi_{A,C}(V,r)}{\chi_{B}(V,r)} = \frac{\chi(C(r))~\cdot~\chi_{A}(V,r)~\cdot~\chi_{A^{\prime}}(V,r)}{\chi_{B}(V,r)}.\end{align*} $$

In more detail,

• • $M_{j}$ is the motive $h^{j}(V_{0}, \mathbb Z(r))$ with $0\le j \le 2d-2$ with the enhanced period map

  $$ \begin{align*}\gamma_{M_{j}} : H^{j}_{B}(V_{\mathbb C}, \mathbb Z(r))^{+}_{\mathbb C} \to t_{M_{j}}.\end{align*} $$

• • We fix an integral structure on $ \textrm {Ker}(\gamma _{M_{j}})$ and $ \textrm {Coker}(\gamma _{M_{j}})$ for all j and use them implicitly in all that follows. Under this condition, the final conjecture is actually independent of the choice of these integral structures.

• • The exact sequences $B(j,r)$ with integral structures

  (8) $$ \begin{align} 0 \to \textrm{Ker}(\gamma_{M_{j}}) \to H^{j}_{B}(V_{\mathbb C}, \mathbb Z(r))^{+}_{\mathbb C} \to t_{M_{j}} \to \textrm{Coker}(\gamma_{M_{j}}) \to 0. \end{align} $$
  We write
  $$ \begin{align*}\chi_{B}(V,r) =\prod_{j} (\chi(B(j,r)))^{(-1)^{j}} = \frac{\chi(B(0,r))~\cdot~\chi(B(2,r))~\cdots}{\chi(B(1,r))~\cdot~\chi(B(3,r)) \cdots}.\end{align*} $$

• • The integral structures $A(j,r)$ for $0\le j\le \textrm {min}(2d-1, 2r-3)$ and $A^{\prime }(j,r)$ for $2d+1\ge j\ge \textrm {max}(0, 2r+1)$ are defined as

  (9) $$ \begin{align} A(j,r): H^{j+1}_{W}(V, \mathbb Z(r))_{\mathbb C} \to \textrm{Coker}(\gamma_{M_{j}}), \qquad A^{\prime}(j,r): \textrm{Ker}(\gamma_{M_{j}}) \to H^{j+2}_{W}(V,\mathbb Z(r))_{\mathbb C}. \end{align} $$
  We write
  (10) $$ \begin{align} \chi_{A}(V,r) &= \prod_{j=0}^{\textrm{min}(2d-1, 2r-3)} (\chi(A(j,r)))^{(-1)^{j}}, \nonumber\\ \chi_{A^{\prime}}(V,r) &= \prod_{j=\textrm{max}(0, 2r+1) }^{j=2d-1} (\chi(A^{\prime}(j,r)))^{(-1)^{j}}. \end{align} $$
  The map in $A(j,r)$ is Beilinson’s regulator from (4), usually stated as a map from algebraic K-theory of $V_{0}$ to the Deligne cohomology of $V_{0}$ ; it is conjectured to be an isomorphism. The group $\textrm {Ker}(M_{2d-2-j, d-r})$ can be identified with the dual of $\textrm {Coker}(M_{j,r})$ - see [Reference Lichtenbaum27, §2] or [Reference Fontaine15, §6.9]. The integral structures $A^{\prime }(j,r)$ are defined as the dual of $A(2d-2-j,d-r)$ in [Reference Lichtenbaum27].

• • The exact sequence $C(r)_{\mathbb C}$ with integral structure (for $0 \le r\le d$ )

  (11) $$ \begin{align} 0 &\to H^{2r-1}_{W}(V, \mathbb Z(r))_{\mathbb C} \to \textrm{Coker}(\gamma_{M_{2r-2}}) \to H^{2r+1}_{W}(V, \mathbb Z(r))_{\mathbb C} \xrightarrow{h}\nonumber\\[3pt] &\to H^{2r}_{W}(V, \mathbb Z(r))_{\mathbb C} \to \textrm{Ker}(\gamma_{M_{2r}}) \to H^{2r+2}_{W}(V, \mathbb Z(r))_{\mathbb C} \to 0.\end{align} $$
  The maps here are integral versions of the maps from (4): the first is Beilinson’s regulator, the second is the dual of the cycle map, the third is the Arakelov intersection/height pairing map, the fourth map is the cycle class map and the fifth is the dual of Beilinson’s regulator. Beilinson’s conjectures [Reference Lichtenbaum27, Conjectures 2.3.4-2.3.7] imply that $C(r)_{\mathbb C}$ is exact; we refer to [Reference Lichtenbaum27, §2] for more details. Integral versions of (4) have also been introduced by Flach–Morin [Reference Flach and Morin12, Conjecture 2.9 and (28), (29)].

Remark 14. (see Table 1) (i) By [Reference Fontaine15, §7.1] or [Reference Nekovář34, §2.1 and (2.2.1)], if the weight of M is positive (so $j>2r$ ), then one has $\textrm {Coker}(\gamma _{M}) =0$ . Dually, if $j-2r <0$ , then one has $\textrm {Ker}(\gamma _{M}) =0$ . It remains to consider the case $M= h^{2r}(V_{0}, \mathbb Z(r))$ . Since $N = h^{2r}(V_{0}, \mathbb Z(r+1))$ has negative weight, it follows that $\textrm {Ker}( \gamma _{N})=0$ . Hence, $\textrm {Coker}(\gamma _{M_{2r}}) =0$ as it is dual to $\textrm {Ker}( \gamma _{N})$ .

(ii) Note that each group $H^{i}_{W}(V,\mathbb Z(r))$ appears exactly once.

All of the exact sequences above are variants of motivic exact sequences (4) but equipped with an integral structure. Our convention is that the terms $ \textrm {Ker}(\gamma _{M})$ have even degree and $\textrm {Coker}(\gamma _{M})$ have odd degree. If $j=2r-1$ , then $\textrm {Ker}(\gamma _{M_{j}})=0 = \textrm {Coker}(\gamma _{M_{j}})$ ; if $j \neq 2r-1$ , these terms occur exactly twice in the conjecture, once in the B-complexes and once in A, $A^{\prime }$ and C-complexes. Since $\chi _{B}$ is the denominator and the others ( $\chi _{A}$ , $\chi _{A^{\prime }}$ and $\chi (C)$ ) are in the numerator of $\chi (V,r)$ , the conjecture is independent of the choice of integral structures on the terms $ \textrm {Ker}(\gamma _{M})$ and $\textrm {Coker}(\gamma _{M})$ .

(iii) The conjectural formula (1) from [Reference Lichtenbaum27, Conjecture 3.1] tacitly assumes the generalised Beilinson–Soulé conjecture from [Reference Lichtenbaum27] which predicts

$$ \begin{align*} \textrm{if}~r<0,~\textrm{then}~H^{i}_{W}(X, \mathbb Z(r)) = \textrm{finite~2-group}~\textrm{for}~i\le 1.\end{align*} $$

If this is not assumed, then (10) has to be replaced with

$$ \begin{align*}\chi_{A}(V,r) = \prod_{j=0}^{2r-3} (\chi(A(j,r)))^{(-1)^{j}}, \qquad \chi_{A^{\prime}}(V,r) = \prod_{j=2r+1}^{j=2d-1} (\chi(A^{\prime}(j,r)))^{(-1)^{j}}\end{align*} $$

and the limits on j in the first two lines of Table 1 have to be replaced with $2d+2> j>2r+2$ (in the first line) and $0< j< 2r-1$ (in the second line). We refer to [Reference Lichtenbaum27] for more details.

Table 1 Tabulation of invariants in the various integral structures for the motives $M_{j} =h^{j}(V_{0})(r)$

(iv) We refer to [Reference Lichtenbaum27] for the conjectures such as the finite generation of $H^{*}_{W}(V, \mathbb Z(r))$ which are implicit in the formulation of (1).

It is instructive to consider the conjecture [Reference Lichtenbaum27, Conjecture 3.1] in the case of number fields before delving into the case of arithmetic surfaces.

2.7 The case of number fields: $V=S$ , $d=1$ and $r=0$ and $r=1$

This example is worked out in [Reference Lichtenbaum27, §7].

2.7.1 The case $r=0$

The motives are $M_{j} =h^{j}(S)(0)$ .

We know $H^{j}_{et}(S, \mathbb Z(1)) =0$ if $j <1$ and that $H^{1}_{et}(S, \mathbb Z(1)) = \mathcal O_{F}^{\times }$ , $H^{2}_{et}(S, \mathbb Z(1)) = \textrm {Pic}(S)$ , $H^{3}_{et}(S, \mathbb Z(1)) =0$ (up to a finite 2-group) and $H^{0}_{et}(S, \mathbb Z(0)) = \mathbb Z$ .

It immediately follows from the definitions that $H^{0}_{W}(S, \mathbb Z(0)) =\mathbb Z$ , $H^{1}_{W}(S, \mathbb Z(0)) =0$ (up to a finite $2$ -group) and $H^{3}_{W}(S, \mathbb Z(0)) = \mu _{F}^{\vee }$ , the dual of the roots of unity in F.

There is also an exact sequence

$$ \begin{align*}0 \to \textrm{Pic}(S)^{\vee} \to H^{2}_{W}(S, \mathbb Z(0)) \to \textrm{Hom}(\mathcal O_{F}^{\times}, \mathbb Z) \to 0.\end{align*} $$

Note that $t_{M_{j}}=0$ for all j.

$\chi _{B}(S,0)=1$ : The sequence $B(j,0)$ is zero for all $j \neq 0$ and we can arrange $\chi (B(0,0)) =1$ by using the isomorphism

$$ \begin{align*} B(0,0):\qquad \textrm{Ker} (\gamma_{M_{0}}) \xrightarrow{\sim} H^{0}_{B}(S_{\mathbb C}, \mathbb Z(0))_{\mathbb C}^{+}\end{align*} $$

to transfer the integral structure $H^{0}_{B}(S_{\mathbb C}, \mathbb Z(0))^{+}$ to $\textrm {Ker}(\gamma _{M_{0}})$ .

Since $A(j,r)$ contributes only for $j \le 2r-3 =-3$ , it follows that there is no contribution from any of the $A(j,0)$ -terms. We see that $A^{\prime }(j,0) =0$ unless $j=1$ , in which case $A^{\prime }(1,0) = \mu _{F}^{\vee }$ in degree 2. Thus, we obtain $\chi (A^{\prime }(1,0)) = w$ and

$$ \begin{align*}\chi_{A^{\prime}}(S,0) = \frac{1}{w}.\end{align*} $$

The sequence $C(0)_{\mathbb C}$ for $V=S$

$$ \begin{align*}0\to H^0_W(S, \mathbb Z(0))_{\mathbb C} \to \textrm{ker}(\gamma_{M_0}) \to H^2_W(S, \mathbb Z(0))_{\mathbb C} \to 0\end{align*} $$

becomes the sequence (our convention is that $\textrm {Ker}(\gamma )$ has even degree, say, degree 2)

(12) $$ \begin{align} 0 \to \underset{\textrm{degree 1}}{\mathbb C} \to \underset{\textrm{degree 2}}{\mathbb C^{r_{1}+r_{2}}} \to \underset{\textrm{degree 3}}{\textrm{Hom}(\mathcal O_{F}^{\times}, \mathbb C)} \to 0, \end{align} $$

with $H^{2}_{W}(S, \mathbb Z(0))$ providing the standard integral structure $\textrm {Hom}(\mathcal O_{F}^{\times }, \mathbb Z)$ on the last term and the second map being the dual of the classical regulator. So $\chi (C(0)) = hR$ and so

$$ \begin{align*} \chi(S,0) = \frac{\chi_{A,C}(S,0)}{\chi_{B}(S,0)} = \frac{\chi(C(0))~\cdot~\chi_{A^{\prime}}(S,0)}{1} = \frac{hR}{w}.\end{align*} $$

As is well-known,

(13) $$ \begin{align} \zeta^{*}(S,0) = -hR/{w}.\end{align} $$

So the conjectural formula (1) from [Reference Lichtenbaum27, Conjecture 3.1] is valid in this case.

2.7.2 The case $r=1$

The motives are $M_{j}=h^{j}(S)(1)$ .

As $A(j,1)$ contribute only for $j <-1$ , we have $\chi _{A}(S,1) =1$ . The complex $C(1)_{\mathbb C}$ (our convention is that $\textrm {Coker}(\gamma )$ has odd degree, say 1)

$$ \begin{align*}0 \to H^{1}_{W}(S, \mathbb Z(1))_{\mathbb C} \to \textrm{Coker}(\gamma_{M_{0}}) \to H^{3}_{W}(S, \mathbb Z(1))_{\mathbb C} \to 0 \to 0 \cdots \end{align*} $$

becomes

(14) $$ \begin{align}0 \to \underset{\textrm{degree 0 }}{\mathcal O_{F}^{\times}\otimes\mathbb C} \to \underset{\textrm{degree 1}}{\mathbb C^{r_{1}+r_{2}}} \to \underset{\textrm{degree 2 }}{\mathbb C} \to 0,\end{align} $$

with the last two terms getting the standard basis and the first term a basis from $\mathcal O_{F}^{\times }$ . This is exactly as in [Reference Lichtenbaum27, §7]. Thus, $det(C(1))$ is the classical regulator R. One checks that $H^{i}_{W}(S, \mathbb Z(1)) =0$ for $i> 3$ and that $H^{3}_{W}(S, \mathbb Z(1)) = \textrm {Hom}(H^{0}_{et}(S, \mathbb Z(0)), \mathbb Z)$ . As $H^{i}_{W}(S,\mathbb Z(1)) = H^{i}_{et}(S, \mathbb Z(1))$ for $i \le 2$ , this completes the computation of $H^{*}_{W}(S, \mathbb Z(1))$ . From this, we see that

$$ \begin{align*}\chi (C(1)) = \frac{hR}{w}.\end{align*} $$

We need to consider $A^{\prime }(j,1)$ only when $j \ge 3$ . Since $t_{M_{j}} =0$ for $j>0$ and $H^{j+2}_{W}(S, \mathbb Z(1)) =0$ for $j\ge 3$ , we see $A^{\prime }(j,1) =0$ for $j\ge 3$ . As a result, $\chi _{A^{\prime }}(S,1)=1$ .

Finally, $B(j,1) =0$ if $j \neq 0$ and $B(0,1)$ given by

$$ \begin{align*}0\to H^0_B(S_{\mathbb C}, \mathbb Z(1))^{+}_{\mathbb C} \to t_{M_0} \to \textrm{Coker}(\gamma_{M_0}) \to 0\end{align*} $$

becomes (our convention is that $\textrm {Coker}(\gamma )$ has odd degree)

$$ \begin{align*}0 \to \underset{\textrm{degree 1 }}{\mathbb C^{r_{2}}} \to \underset{\textrm{degree 2 }}{\mathcal O_{F}\otimes_{\mathbb Z}\mathbb C} \to \underset{\textrm{degree 3 }}{\mathbb C^{r_{1}+r_{2}}} \to 0.\end{align*} $$

The first map is the composition of the natural inclusion of $\mathbb C^{r_{2}}$ in $\mathbb C^{r_{1}+2r_{2}}$ (multiplied by $2\pi i$ ) with the inverse of $\psi $ of (6). The integral structure on $\textrm {Coker}(\gamma _{M_{0}}) \cong \mathbb C^{r_{1}+r_{2}}$ is defined by $H^{0}_{B}(S, \mathbb Z(1))^{-} \cong \mathbb Z^{r_{1}+r_{2}}$ . Since the determinant of $\psi $ with respect to the usual bases is $\sqrt {d}_{F}$ and the integral structures are torsion-free, we obtain

$$ \begin{align*} det (B(0,1)) = \frac{\sqrt{d_{F}}}{(2\pi i)^{r_{2}}} = \chi(B(0,1)).\end{align*} $$

Hence, we obtain

$$ \begin{align*}\chi(S,1) =\frac{\chi(C(1))}{\chi(B(0,1))} = \frac{hR(2\pi i)^{r_{2}}}{w\sqrt{d_{F}}};\end{align*} $$

as the sign of $d_{F}$ is $(-1)^{r_{2}}$ , this is equal to the usual formula (up to a power of 2)

(15) $$ \begin{align} \zeta^{*}(S,1) = \frac{2^{r_{1}}h R(2\pi)^{r_{2}}}{w\sqrt{|d_{F}|}}.\end{align} $$

So the conjectural formula (1) from [Reference Lichtenbaum27, Conjecture 3.1] is valid in this case, too.

2.8 The case of arithmetic surfaces: $V=X$ and $r=1$

We now turn to an explicit description of the terms that enter into the description of $\zeta ^{*}(X,1)$ ; this uses the motives $M_{j} = h^{j}(X_{0})(1)$ for $j=0,1,2$ of the algebraic curve $X_{0}$ over F and $t_{M_{j}} = H^{j}(X_{0}, \mathcal O_{X})$ . We begin with the Weil-étale motivic cohomology groups of X.

2.8.1 The groups $H^{*}_{W}(X,\mathbb Z(1))$

Fix an arithmetic surface $\pi :X\to S$ .

Since the motivic complex $\mathbb Z(1)$ is $ \mathbb G_{m}[-1]$ , one has the identifications

$$ \begin{align*}H^{j}_{et}(X, \mathbb Z(1)) = H^{j-1}_{et}(X, \mathbb G_{m}),\end{align*} $$

the Picard group $\textrm {Pic}(X) = H^{2}_{et}(X, \mathbb Z(1))$ and the Brauer group $\textrm {Br}(X) = H^{3}_{et}(X, \mathbb Z(1))$ .

Recall that $H^{i}_{W}(X, \mathbb Z(1))$ [Reference Lichtenbaum27, §2.1] are defined as étale motivic cohomology $H^{i}_{et}(X, \mathbb Z(1))$ for $i \le 2$ and, for $i>2$ , as the dual of étale motivic cohomology

$$ \begin{align*}H^{i}(\textrm{RHom}(R\Gamma_{et}(X, \mathbb Z(1)), \mathbb Z [-3])),\end{align*} $$

thereby sitting in an exact sequence (for $i>2$ )

$$ \begin{align*}0 \to \textrm{Ext}^{1}(H^{6-i}_{et}(X, \mathbb Z(1)), \mathbb Z) \to H^{i}_{W}(X, \mathbb Z(1)) \to \textrm{Hom}(H^{5-i}_{et}(X, \mathbb Z(1)), \mathbb Z) \to 0.\end{align*} $$

As $H^{i}_{W}(X, \mathbb Z(1)) =0$ for $i=0$ and $i \ge 6$ , the following is a complete description of $H^{*}_{W}(X, \mathbb Z(1))$ :

(16) $$ \begin{align} H^{1}_{W}(X, \mathbb Z(1)) = {{\mathcal O}_{F}}^{\times}, \quad H^{2}_{W}(X, \mathbb Z(1)) = \textrm{Pic}(X), \quad H^{5}_{W}(X, \mathbb Z(1)) = \textrm{Ext}^{1}({\mathcal O_{F}}^{\times}, {\mathbb Z}), \end{align} $$

(17) $$ \begin{align} 0 \to \textrm{Ext}^{1}(\textrm{Br}(X), {\mathbb Z}) \to H^{3}_{W}(X, \mathbb Z(1)) \to \textrm{Hom}(\textrm{Pic}(X), \mathbb Z) \to 0, \end{align} $$

(18) $$ \begin{align} 0 \to \textrm{Ext}^{1}(\textrm{Pic}(X), {\mathbb Z}) \to H^{4}_{W}(X, \mathbb Z(1)) \to \textrm{Hom}(\mathcal O_{F}^{\times}, \mathbb Z) \to 0.\end{align} $$

Remark. As the finite generation of $\textrm {Pic}(X)$ is well-known (theorem of Mordell–Weil–Roquette), the finite generation of $H^{*}_{W}(X,\mathbb Z(1))$ reduces to the finiteness of $\textrm {Br}(X)$ .

2.8.2 The complexes $A(j,1)$ and $A^{\prime }(j,1)$

In our case, only $A^{\prime }(3,1)$ could be nonzero.

In our case, $r=1$ . For $A(j,1)$ to intervene, one needs $j\le 2r-3 =-1$ as $2d-1=3$ . Since $H^{0}_{W}(X,\mathbb Z(1)) =0$ , we obtain $\chi (A(j,1)=1)$ for all j.

For $A^{\prime }(j,1)$ to intervene, one needs $j \ge 3$ . For $j=3$ , since $H^{5}_{W}(X,\mathbb Z(1))$ is a finite group of order w, we have $det(A^{\prime }(3,1)) =1$ and $\chi (A^{\prime }(3,1)) =w$ . If $j>3$ , then $H^{j+2}_{W}(X, \mathbb Z(1)) =0$ and $M_{j} =0$ ; so $\chi (A^{\prime }(j,1)) =1$ for $j>3$ . Thus,

(19) $$ \begin{align} \chi_{A}(X,1) =1, \qquad \chi_{A^{\prime}}(X,1) = \frac{1}{w}.\end{align} $$

2.8.3 The groups $H^{i}(X, \mathcal O_{X})$

Since $\pi :X \to S$ is proper, $H^{i}(X, \mathcal O_{X})$ are finitely generated $\mathcal O_{F}$ -modules.

Lemma 15. $H^{i}(X, \mathcal O_{X})$ is zero for $i\ge 2$ .

Proof. Since $X\to S$ is a relative curve, $H^{i}(X_{s}, \mathcal O_{X_{s}}) =0$ (for $i>1$ ) at all points $s\in S$ . Since S is a reduced noetherian scheme and $\pi : X\to S$ is proper and $\mathcal O_{X}$ is a coherent sheaf on X flat over S, by Grothendieck’s theorem on formal functions (see L. Illusie’s article [Reference Fantechi, Göttsche, Illusie, Kleiman, Nitsure and Vistoli9, Corollary 8.3.4]), $R^{i}\pi _{*}\mathcal O_{X} =0$ for $i>1$ .

Using Subsection 2.8.2 and Lemma 15, we find that

(20) $$ \begin{align} \chi(X,1) = \frac{{\chi (C(1))}}{\chi(A^{\prime}(3,1))}~\cdot~\frac{\chi(B(1,1))}{\chi(B(0,1))} =\frac{{\chi (C(1))}}{w}~\cdot~\frac{\chi(B(1,1))}{\chi(B(0,1))},\end{align} $$

where the exact sequence $B(j,1)$ is

(21) $$ \begin{align} 0 \to \textrm{Ker}(\gamma_{M_{j}}) \to H^{j}_{B}(X_{\mathbb C}, \mathbb Z(1))^{+}_{\mathbb C} \to t_{M_{j}} \to \textrm{Coker}(\gamma_{M_{j}}) \to 0 \end{align} $$

and $C(1)_{\mathbb C}$ is

(22) $$ \begin{align} 0 &\to H^{1}_{W}(X, \mathbb Z(1))_{\mathbb C} \to \textrm{Coker}(\gamma_{M_{0}}) \to H^{3}_{W}(X, \mathbb Z(1))_{\mathbb C} \to H^{2}_{W}(X, \mathbb Z(1))_{\mathbb C} \to \textrm{Ker}(\gamma_{M_{2}}) \nonumber\\[3pt] &\to H^{4}_{W}(X, \mathbb Z(1))_{\mathbb C} \to 0.\end{align} $$

So Conjecture 1 provides an intrinsic formula for $\zeta ^{*}(X,1)$ using only $B(0,1)$ , $B(1,1)$ , $C(1)$ and $H^{*}_{W}(X, \mathbb Z(1))$ . We compute $\chi (B(0,1))$ and explain how $\chi (B(2,1))$ can be assumed to be one.

2.8.4 $\chi (B(0,1))$

This has been computed ([Reference Lichtenbaum27, §7]) in Subsection 2.7. Here

(23) $$ \begin{align} \textrm{Ker}(\gamma_{M_{0}}) =0, \qquad \textrm{Coker}(\gamma_{M_{0}}) \cong \mathbb C^{r_{1}+r_{2}} ,\qquad H^{0}(X, \mathcal O) = \mathcal O_{F}. \end{align} $$

Our convention for $B(0,1)$ is that $\textrm {Coker}(\gamma _{M})$ is in odd degree, say, degree 3. As the integral structure

$$ \begin{align*}B(0,1):\qquad 0 \to \underset{\textrm{degree 1 }}{H^{0}_{B}(X_{\mathbb C}, \mathbb Z(1))^{+}_{\mathbb C}} \to \underset{\textrm{degree 2}}{(\mathcal O_{F})_{\mathbb C}} \to \underset{\textrm{degree 3}}{\textrm{Coker}(\gamma_{M_{0}})} \to 0\end{align*} $$

is torsion-free, we have (again using that the sign of $d_{F}$ is $(-1)^{r_{2}}$ )

(24) $$ \begin{align} \chi (B(0,1)) = det (B(0,1)) = \frac{\sqrt{d_{F}}}{(2\pi i)^{r_{2}}} = \frac{\sqrt{|d_{F}|}}{(2\pi)^{r_{2}}}.\end{align} $$

2.8.5 The sequence $B(2,1)$

By Lemma 15, $t_{M_{2}} = H^{2}(X_{0}, \mathcal O) =0$ ; so, in $B(2,1)$ ,

$$ \begin{align*} \textrm{Ker}(\gamma_{M_{2}}) \xrightarrow{\sim} H^{2}_{B}(X_{\mathbb C}, \mathbb Z(1))^{+}_{\mathbb C},\end{align*} $$

we use the isomorphism to transfer the integral structure from $H^{2}_{B}(X_{\mathbb C}, \mathbb Z(1))^{+}$ to $\textrm {Ker}(\gamma _{M_{2}})$ , thereby obtaining

(25) $$ \begin{align}\chi(B(2,1)) = 1.\end{align} $$

3.1 Statement of the conjecture [Reference Tate36, Reference Gross19]

Our presentation of this conjecture follows (almost verbatim) [Reference Gross19, § 2] where it is formulated in terms of Néron models.

Let A be an abelian variety of dimension g over $\text {Spec}~F$ . Let $L(A,s)$ be its L-series over F, which is defined – see (28) by a Euler product convergent in the half-plane Re $(s)> \frac {3}{2}$ . We assume that $L(A,s)$ has an analytic continuation to the entire complex plane. We write $L^{*}(A,1)$ for its special value at $s=1$ . Write

$$ \begin{align*} L(A,s)~\sim~~c~\cdot~(s-1)^{r(A)} \qquad s\to 1,\end{align*} $$

where $r(A)$ is a nonnegative integer and c is nonzero (the special value $L^{*}(A,1)$ ). Birch and Swinnerton–Dyer and Tate have conjectured that $r(A)$ is equal to the rank of the finitely generated group $A(F)$ of points of A over F; they have also given a conjectural formula for $L^{*}(A,1)$ in terms of certain arithmetic invariants of A which we now recall.

Let $\mathcal N$ be the Néron model of A over S. Let $\mathcal N^{0}$ be the largest open subgroup scheme of $\mathcal N$ in which all fibres are connected. For any nonzero prime v of $\mathcal O_{F}$ (i.e., $v \in S$ ), we write

$$ \begin{align*}\mathcal N_{v}, \qquad \mathcal N^{0}_{v}\end{align*} $$

for the fibres over v. Thus, $\mathcal N_{v}$ is a commutative group scheme over $k(v)$ and $\mathcal N^{0}_{v}$ is the connected component of the identity of $\mathcal N_{v}$ . We have an exact sequence of group schemes over $k(v)$ :

$$ \begin{align*}0 \to H_{v} \to \mathcal N_{v}^{0} \to B_{v} \to 0\end{align*} $$

where $H_{v}$ is a commutative linear group scheme and $B_{v}$ is an abelian variety. For almost all v, we have

$$ \begin{align*}\mathcal N_{v} =\mathcal N^{0}_{v} = B_{v};\end{align*} $$

these are the primes where A has good reduction. Let $Y_{v} = \textrm {Hom}(H_{v}, \mathbb G_{m})$ be the character group of $H_{v}$ and $\pi _{v}$ be the Frobenius endomorphism of $B_{v}$ .

We write $\Phi _{v}=\pi _{0}(\mathcal N_{v})$ for the finite group scheme of connected components of the fibre $\mathcal N_{v}$ over v. The Tamagawa number of A at v, denoted $c_{v}$ , is the size of the group $\Phi _{v}(k(v))$ of the group of rational points of $\Phi _{v}$ . The Tamagawa number $c_{v}$ is one for almost all $v \in S$ ; we may define the product

(26) $$ \begin{align} P_{A, fin} = \prod_{v\in S} c_{v}.\end{align} $$

Let $\Gamma _{v}$ be a decomposition group for v in $\Gamma _{F} = \textrm {Gal}(\bar {F}/F)$ and let $I_{v}$ be the inertia subgroup of $\Gamma _{v}$ and $\sigma _{v}$ be an arithmetic Frobenius, a topological generator of the quotient $\Gamma _{v}/{I_{v}}$ . For any prime $\ell $ distinct from the characteristic of $k(v)$ , consider the $\ell $ -adic Tate module $T_{\ell }A$ of A; this is a free $\mathbb Z_{\ell }$ -module of rank $2g$ which admits a continuous $\mathbb Z_{\ell }$ -linear action of $\Gamma _{F}$ . We define the local L-factor of A at v by the formula

(27) $$ \begin{align} L_{v}(A,t) = \textrm{det}(1-\sigma_{v}^{-1}t~|~\textrm{Hom}_{\mathbb Z_{\ell}}( T_{\ell}A, \mathbb Z_{\ell})^{I_{v}}).\end{align} $$

The characteristic polynomial $L_{v}(A,t)$ has integral coefficients which are independent of $\ell $ : this is evident from the formula

$$ \begin{align*}L_{v}(A,t) = \textrm{det}(1-\sigma_{v} t~|~ Y_{v})~\cdot~\textrm{det}(1-\pi_{v} t~|~ T_{\ell} B_{v} )\end{align*} $$

where the first determinant is clearly independent of $\ell $ and the second is the characteristic polynomial of an endomorphism of the abelian variety $B_{v}$ which has integral coefficients independent of $\ell $ by a theorem of A. Weil.

The roots of the first factor have complex absolute value 1, whereas those of the second factor have complex absolute value $q_{v}^{-1/2}$ . Therefore, the global L-series $L(A,s)$ defined by

(28) $$ \begin{align} L(A,s) = L(A/F,s) = \prod_{v\in S} \frac{1}{L_{v}(A, q_{v}^{-s})}\end{align} $$

converges for Re $(s)>\frac {3}{2}$ . We shall assume that it has an analytic continuation to the entire complex plane.

3.2 The archimedean period of A

Let $\omega _{\mathcal N}$ denote the projective $\mathcal O_{F}$ -module of invariant differentials of the Néron model $\mathcal N$ . Since g is the rank of $\omega _{\mathcal N}$ , the module $\Lambda ^{g}\omega _{\mathcal N}$ is a rank 1 $\mathcal O_{F}$ -submodule of $H^{0}(A, \Omega ^{g})$ , a vector space over F of dimension 1. Let $\{w_{1}, \cdots , w_{g}\}$ be an F-basis of $H^{0}(A, \Omega ^{1})$ and put $\eta = w_{1}\wedge \cdots \wedge w_{g}$ . We have

(29) $$ \begin{align} \Lambda^{g}\omega_{\mathcal N} = \eta~\cdot~\mathfrak a_{\eta} \quad \subset H^{0}(A, \Omega^{g}) \end{align} $$

where $\mathfrak a_{\eta }$ is a fractional ideal of F and

(30) $$ \begin{align} \mathfrak a_{\eta} = \mathfrak a_{c\eta} ~\cdot~(c) \qquad c \in F^{\times}.\end{align} $$

For any $\sigma \in S_{\mathbb R}$ (corresponds to a real embedding $\sigma : F \to \mathbb R$ ), let $H^{+}$ denote the submodule of $H_{1}(A_{\sigma }(\mathbb C), \mathbb Z)$ which is fixed by complex conjugation; it is a free $\mathbb Z$ -module of rank g and if $\{ \gamma _{1}, \cdots , \gamma _{g}\}$ denotes a basis, we put

(31) $$ \begin{align} p_{\sigma} (A, \eta) = [(\pi_{0}(A_{\sigma}(\mathbb R)))]~\cdot~\Bigg|~\textrm{det} \bigg( \bigg(\int_{\gamma_{i}}\omega_{j} \bigg)\bigg)~\Bigg|.\end{align} $$

For any complex place v of S (corresponding to $\sigma : F \to \mathbb C$ and its conjugate), let $\{\gamma _{1}, \cdots , \gamma _{2g}\}$ be a basis of the free module $H_{1}(A_{\sigma }(\mathbb C), \mathbb Z)$ of rank $2g$ and define the period

(32) $$ \begin{align} p_{\sigma} (A, \eta) = \big | \textrm{det}( M_{v}(A, \eta)) \big |, \qquad M_{v}(A, \eta) = \bigg(\bigg(\int_{\gamma_{i}}\omega_{j}, {\overline{{\int_{\gamma_{i}}{{\omega}_{j}}}}}~\bigg) \bigg)~;\end{align} $$

this period is nonzero and depends only on the differential form $\eta $ and the place v. The period of A (relative to $\eta $ ) is the real number

(33) $$ \begin{align} P_{A, \infty}(\eta) = \prod_{\sigma \in S_{\infty}} p_{\sigma}(A, \eta). \end{align} $$

Remark 16. We recall the well-known fact that the determinant of $M_{v}(A, \eta )$ is $(\sqrt {-1})^{g}$ times a real number. In the basis $\{\kappa _{1}, \cdots , \kappa _{g}\}$ of $H^{0}(A_{\sigma }(\mathbb C), \Omega ^{1})$ defined by $\int _{\gamma _{j}}\kappa _{i} = \delta _{ij}$ for $1\le i,j\le g$ , the $2g\times 2g$ -period matrix $(\int _{\gamma _{i}}\kappa _{j}, {\overline {{\int _{\gamma _{i}} \kappa _{j}}}})$ is a block matrix of the form

$$ \begin{align*}M^{\prime}= \begin{bmatrix} I_{g} & I_{g}\\ \Omega & {\overline{\Omega}} \end{bmatrix}, \qquad \Omega = \bigg(\int_{\gamma_{g+i}}\kappa_{j}\bigg)_{1\le i,j \le g} \end{align*} $$

whose determinant is equal to the determinant of the $g \times g$ -matrix ${\overline {\Omega }} - \Omega $ (all of its entries have zero real part) and hence $\textrm {det}(M^{\prime })$ is the product of a real number with $\sqrt {-1}^{g}$ . If K is the change of basis matrix from the basis $\{w_{1}, \cdots , w_{g}\}$ to the basis $\{\kappa _{1}, \cdots , \kappa _{g}\}$ , then

$$ \begin{align*} M_{v}(A, \eta) = M^{\prime}~\cdot~ \begin{bmatrix} K & 0\\ 0 & {\overline{K}} \end{bmatrix} \implies \textrm{det} (M_{v}(A, \eta)) = \textrm{det}(M^{\prime})~\cdot~\textrm{det}(K)~\cdot~{\overline{\textrm{det}(K)}}. \end{align*} $$

Lemma 17. The number

$$ \begin{align*}P_{A, \infty}(\eta)~\cdot~ \mathbb N_{F/{\mathbb Q}}(\mathfrak a_{\eta}) = \mathbb N_{F/{\mathbb Q}}(\mathfrak a_{\eta})~\cdot~\prod_{v \in S_{\infty}} p_{v}(A,\eta)\end{align*} $$

is independent of $\eta $ . It will be denoted $P_{A, \infty }$ .

Proof. For any $0 \neq k \in F$ and any $v \in S_{\infty }$ , one has

$$ \begin{align*} p_{v}(A, k\eta)= \begin{cases} p_{v}(A, \eta)~\cdot~\sigma(k)& \mbox{if}~v\in S_{\mathbb R}\\ p_{v}(A, \eta)~\cdot~\overline{\sigma(k)}~\cdot~\sigma(k)&\mbox{if}~v\in S_{\mathbb C} \end{cases}.\end{align*} $$

On the other hand, we see from (30) that $\mathfrak a_{k\eta } = (k)^{-1} \mathfrak a_{\eta }$ . Since

(34) $$ \begin{align} \prod_{\sigma \in \tilde{S}} \sigma(k) = \mathbb N_{F/{\mathbb Q}}(k),\end{align} $$

one has

$$ \begin{align*} \mathbb N_{F/{\mathbb Q}}(\mathfrak a_{k\eta}) & \cdot~\prod_{\sigma \in S_{\infty}} p_{\sigma}(A,k\eta) = \mathbb N_{F/{\mathbb Q}}(k)^{-1}~\cdot~ \mathbb N_{F/{\mathbb Q}}(\mathfrak a_{\eta}) ~\cdot~\prod_{\sigma \in S_{\infty}} p_{\sigma}(A,\eta) \nonumber \\[3pt] &\cdot~(\prod_{\sigma \in \tilde{S}} \sigma(k)) = \mathbb N_{F/{\mathbb Q}}(\mathfrak a_{\eta}) ~\cdot~\prod_{\sigma \in {S}_{\infty} } p_{\sigma}(A,\eta). \\[-40pt] \end{align*} $$

Definition 18. The global volume $P_{A}$ of A is defined by

$$ \begin{align*} P_{A} = \frac{P_{A, fin}~\cdot~P_{A, \infty}}{|d_{F}|^{g/2}} = \frac{P_{A, fin}~\cdot~P_{A, \infty}(\eta)~\cdot~ \mathbb N_{F/{\mathbb Q}}(\mathfrak a_{\eta}) }{|d_{F}|^{g/2}}.\end{align*} $$

Remark 19. It is known that $P_{A}$ is the volume $m_{A}(A(\mathbb A_{F}))$ of the adelic points $A(\mathbb A_{F})$ with respect to the (canonical) Tamagawa measure $m_{A}$ . If $A^{\prime }$ is the Weil restriction of A to $F^{\prime }\subset F$ , then $L(A,s) = L(A^{\prime },s)$ . One can compare the terms in Conjecture 20 for A and $A^{\prime }$ :

see Remark 23 for the the comparison of $P_{A}$ and $P_{A^{\prime }}$ . Conjecture 20 is compatible with restriction of scalars [Reference Milne33] [Reference Conrad7, §6]: Conjecture 20 for A over F is equivalent to $A^{\prime }$ over $F^{\prime }$ ; see [Reference Conrad7, §6] for a beautiful exposition.

Let $A^{t}$ be the dual abelian variety. We write $\Theta _{NT}(A)$ for the determinant of the Néron–Tate height pairing

(35) $$ \begin{align} \frac{A(F)}{A(F)_{tor}} \times \frac{A^{t}(F)}{A^{t}(F)_{tor}} \to \mathbb R. \end{align} $$

Since this height pairing corresponds to the Poincaré divisor on $A \times A^{t}$ which is symmetric, it follows that $\Theta _{NT}(A) = \Theta _{NT}(A^{t})$ . Following Subsection 2.3.5, one has

$$ \begin{align*} \Delta_{NT}(A) = \frac{\Theta_{NT}(A)}{([A(F)_{tor}]~\cdot~[A^{t}(F)_{tor}])}.\end{align*} $$

Finally, let denote the Tate–Shafarevich group of A over F.

The BSD conjecture states that the order of vanishing of $L(A,s)$ at $s=1$ is equal to the rank of the Mordell–Weil group $A(F)$ of A. The strong BSD conjecture [Reference Tate36] (we follow Gross’s formulation [Reference Gross19, Conjecture 2.10] which is shown by him to be equivalent to the one in [Reference Tate36]) concerns the special value $L^{*}(A,1)$ of $L(A,s)$ at $s=1$ .

Conjecture 20. is finite and $L^{*}(A,1)$ satisfies

(36)

3.3 The case of everywhere good reduction

If A has good reduction everywhere, then $c_{v}=1$ for all $v \in S$ and $P_{A,fin} =1$ ; also, $A_{v} = \mathcal N_{v} = \mathcal N^{0}_{v} = B_{v}.$ Conjecture 20 simplifies to the following.

Conjecture 21. If A has good reduction everywhere, then

4.0.1 The case that $N= {\omega }_{ \mathcal J}$ is free

If N is free as an ${\mathcal O}_{F}$ -module (its rank is g), then pick an ${\mathcal O}_{F}$ -basis $\mathcal B = \{v_{1}, \cdots , v_{g}\}$ for it. If ${\mathcal B}^{\prime }$ is a different basis for N, then the determinant $d(\mathcal B, {\mathcal B}^{\prime })$ of the change of basis matrix is a unit in ${\mathcal O}_{F}$ . Note that $\mathcal B$ is also a basis for the F-vector space $\omega _{\mathcal J}\otimes _{{\mathcal O}_{F}}F = H^{0}(J, \Omega ^{1})$ . For any embedding $\sigma : F \to \mathbb C$ , let

$$ \begin{align*} \sigma(\mathcal B)= \{\sigma(v_{1}),\cdots, \sigma(v_{g})\}\end{align*} $$

be the image of $\mathcal B$ under the isomorphism (via $\sigma $ )

$$ \begin{align*}H^{0}(J, \Omega^{1})\otimes_{F}\mathbb C \cong H^{0}(J_{\sigma}(\mathbb C), \Omega^{1}).\end{align*} $$

So $\sigma (\mathcal B)$ is a basis for the complex vector space $H^{0}(J_{\sigma }(\mathbb C), \Omega ^{1})$ . Just as a basis for V and W produces a basis for $V \times W$ , we can combine the bases $\sigma (\mathcal B)$ for $\sigma \in \tilde {S}$ to get a basis $\tilde {\mathcal B}$ for the complex vector space

$$ \begin{align*}H^{0}(J_{\mathbb C}, \Omega^{1}) = H^{0}(J, \Omega^{1})\otimes_{\mathbb Z} \mathbb C= \prod_{\sigma \in \tilde{S}} H^{0}(J_{\sigma}(\mathbb C), \Omega^{1}).\end{align*} $$

The lattice ( $\mathbb Z$ -module) spanned by $\tilde {\mathcal B}$ represents the ${\mathcal O}_{F}$ -integral structure $j_{N}$ on $V =N\otimes _{\mathbb Z}\mathbb C$ (see Definition 8). The determinant $D(\mathcal B)$ of (37) computed with respect to the integral structure provided by $\tilde {\mathcal B}$ is equal to $det_{{\mathcal O}_{F}}(\gamma ^{*}_{\mathcal J})$ . Let us show that $D(\mathcal B)$ is independent of the basis $\tilde {\mathcal B}$ .

Given another basis ${\mathcal B}^{\prime }$ for N, we have

$$ \begin{align*}D({\mathcal B}^{\prime}) = D(\mathcal B) \prod_{\sigma\in \tilde{S} }\sigma(d_{\mathcal B, {\mathcal B}^{\prime}}).\end{align*} $$

The determinant of the change of basis matrix from $\tilde {\mathcal B}$ to $\tilde {{\mathcal B}^{\prime }}$ is

$$ \begin{align*}\prod_{\sigma\in \tilde{S}}\sigma(d_{\mathcal B, {\mathcal B}^{\prime}}) = \mathbb N_{F/{\mathbb Q}}d_{\mathcal B, {\mathcal B}^{\prime}} =\pm 1\end{align*} $$

as $d_{\mathcal B, {\mathcal B}^{\prime }}$ is a unit in ${\mathcal O}_{F}$ . Though the integral structure on $N\otimes _{\mathbb Z} \mathbb C$ provided by $\tilde {\mathcal B}$ depends on the choice of $\mathcal B$ , the number $D(\mathcal B)$ is independent of the basis $\mathcal {B}$ : Any two ${\mathcal O}_{F}$ -bases of N provide Euler-equivalent integral structures representing $j_{N}$ on $V=N\otimes _{\mathbb Z} \mathbb C$ . Thus, in this case, for any basis $\mathcal B$ , one has

$$ \begin{align*} det(\gamma^{*}_{\mathcal J}) = \frac{D(\mathcal B)}{{\sqrt{d_{F}}^{g}}}.\end{align*} $$

4.0.2 The case that $N= \omega _{\mathcal J}$ is not free as a ${\mathcal O}_{F}$ -module

In this case, we proceed as in Subsection 3.2 using the top exterior power $\Lambda ^{g} N$ of the projective ${\mathcal O}_{F}$ -module N. Let $\mathcal B = \{v_{1}, \cdots , v_{g}\}$ be a F-basis of $H^{0}(J, \Omega ^{1})$ and put $\rho = v_{1} \wedge \cdots \wedge v_{g}$ . We have

(38) $$ \begin{align} \Lambda^{g} N = \mathfrak b_{\rho}~\cdot~ \rho \subset \Lambda^{g} H^{0}(J, \Omega^{1}) \end{align} $$

where $\mathfrak b_{\rho }$ is a fractional ideal of ${\mathcal O}_{F}$ . For any $k \neq 0 \in F$ , one has an equality of fractional ideals

(39) $$ \begin{align} \mathfrak b_{\rho} = \mathfrak b_{k\rho}~\cdot~(k). \end{align} $$

As before, let $\tilde {\mathcal B}$ denote the basis of $V=N\otimes _{\mathbb Z} \mathbb C$ obtained from $\mathcal B$ . Let $D(\mathcal B)$ denote the determinant of (37) with respect to the basis $\tilde {\mathcal B}$ and $H^{1}_{B}(J_{\mathbb C}, \mathbb Z)^{+}$ . Using (38), one has

$$ \begin{align*} det_{{\mathcal O}_{F}}(\gamma^{*}_{\mathcal J}) = D(\mathcal B)~\cdot~\mathbb N_{F/{\mathbb Q}} \mathfrak b_{\rho}.\end{align*} $$

Let us directly show that the right-hand side is independent of the basis. Given any basis ${\mathcal B}^{\prime }=\{v_{1}^{\prime },\cdots , v_{g}^{\prime }\}$ ,

$$ \begin{align*} D({\mathcal B}^{\prime}) = D(\mathcal B) \prod_{\sigma \in \tilde{S} } \sigma(k) \qquad k:= d(\mathcal B, {\mathcal B}^{\prime}).\end{align*} $$

Note that $k= d(\mathcal B, {\mathcal B}^{\prime })\in F^{\times }$ need not be a unit.

If $\rho ^{\prime }= v_{1}^{\prime }\wedge \cdots \wedge v_{g}^{\prime }$ , then

$$ \begin{align*}\rho^{\prime} = \rho~\cdot~k, \qquad k= d(\mathcal B, {\mathcal B}^{\prime}),\end{align*} $$

which gives

$$ \begin{align*}b_{\rho^{\prime}} = b_{\rho}~\cdot~(k)^{-1}.\end{align*} $$

Thus, using (34) as in Lemma 17, we find

$$ \begin{align*}D({\mathcal B}^{\prime}) &\cdot~\mathbb N_{F/{\mathbb Q}} \mathfrak b_{\rho^{\prime}} = D(\mathcal B)~\cdot~( \prod_{\sigma \in \tilde{S} } \sigma(k))~\cdot~\mathbb N_{F/{\mathbb Q}} \mathfrak b_{\rho^{\prime}} = D(\mathcal B) \nonumber \\ & \cdot~(\prod_{\sigma \in \tilde{S}} \sigma(k))~\cdot~\mathbb N_{F/{\mathbb Q}} \mathfrak b_{\rho}~\cdot~(\mathbb N_{F/{\mathbb Q}}(k))^{-1} = D(\mathcal B)~\cdot~\mathbb N_{F/{\mathbb Q}} \mathfrak b_{\rho}.\end{align*} $$

Thus, $D(\mathcal B)~\cdot ~\mathbb N_{F/{\mathbb Q}} \mathfrak b_{\rho }$ is independent of $\mathcal B$ .

4.0.3 Betti lattices

It should be clear that the above computation of $det_{{\mathcal O}_{F}}(\gamma ^{*}_{\mathcal J})$ is exactly the computation of $P_{J,\infty }$ in Subsection 3.2. In fact, for any g-dimensional abelian variety A over F, one has (up to a power of 2)

(40) $$ \begin{align} det_{{\mathcal O}_{F}}(\gamma^{*}_{\mathcal N}) = \pm ({\sqrt{-1}})^{g~\cdot~r_{2}}~\cdot~P_{A,\infty}.\end{align} $$

Here, as in Subsection 3.2, $\mathcal N$ is the Néron model of A and $\gamma ^{*}_{\mathcal N}$ is the period isomorphism

$$ \begin{align*}\gamma^{*}_{\mathcal N}:~H^{1}_{B}(A_{\mathbb C}, \mathbb Z)^{+}_{\mathbb C} \xleftarrow{\sim} \omega_{\mathcal N} \otimes_{\mathbb Z} \mathbb C.\end{align*} $$

Let $B= \{w_{1}, \cdots , w_{g}\}$ be an F-basis of $H^{0}(A, \Omega ^{1})$ and put $\eta = w_{1}\wedge \cdots \wedge w_{g}$ . The discrepancy in (40) arises from the fact that the Betti lattices are different.

For a real place of S corresponding to $\sigma : F \to \mathbb R$ , consider the period isomorphism

$$ \begin{align*} \gamma_{\sigma}= H^{1}_{B}(A_{\sigma}(\mathbb C) , \mathbb Z)^{+}_{\mathbb C} \xleftarrow{\sim} H^{0}(A_{\sigma}, \Omega^{1}).\end{align*} $$

For a complex place v of S (corresponding to $\sigma : F \to \mathbb C$ and its conjugate $c\sigma $ ), consider the complex abelian variety

$$ \begin{align*}A_{v} = A_{\sigma} \times A_{c\sigma}\end{align*} $$

and the period isomorphism

$$ \begin{align*}\gamma_{v}: H^{1}_{B}(A_{v}, \mathbb Z)^{+}_{\mathbb C} \xleftarrow{\sim} H^{0}(A_{v}, \Omega^{1}).\end{align*} $$

For real places, the groups in $\gamma _{\sigma }$ are the same as the one used in (31); in this case, $p_{\sigma }(A, \eta )$ is the determinantFootnote ² of $\gamma _{\sigma }$ relative to the basis $H^{1}_{B}(A_{\sigma }(\mathbb C) , \mathbb Z)^{+}$ and $B= \{w_{1}, \cdots , w_{g}\}$ .

For complex places v, (32) uses $H^{1}(A_{\sigma }, \mathbb Z)$ instead of $H^{1}_{B}(A_{v}, \mathbb Z)^{+}$ . In this case, $(\sqrt {-1})^{g}~\cdot ~p_{v}(A, \eta )$ is the determinant of $\gamma _{v}$ relative to the basis $H^{1}_{B}(A_{v}, \mathbb Z)^{+}$ and B. This is what leads to the discrepancy between the two invariants. What follows is presumably well-known, but we include it for sake of completeness.

Complex conjugation

$$ \begin{align*}c: A_{\sigma} \to A_{c\sigma}\end{align*} $$

simply permutes the factors of ${A}_{v}$ . One has the decomposition

$$ \begin{align*}H_{1}({A_{v}}(\mathbb C), \mathbb Z) &= H_{1}(A_{\sigma}(\mathbb C), \mathbb Z) \oplus H_{1}(A_{c\sigma} (\mathbb C), \mathbb Z), \nonumber \\ H^{0}({A_{v}}, \Omega^{1}) &= H^{0}(A_{\sigma}, \Omega^{1}) \oplus H^{0}(A_{c\sigma}, \Omega^{1}).\end{align*} $$

Concretely, a basis for $H_{1}({A_{v}}, \mathbb Z)$ is given by

$$ \begin{align*}\{(\gamma_{1}, 0), (\gamma_{2}, 0), \cdots, (\gamma_{2g}, 0), (0, c\gamma_{1}), (0,c\gamma_{2}), \cdots, (0,c\gamma_{2g})\};\end{align*} $$

a basis for $H^{0}(A_{v}, \Omega ^{1})$ is given by (here the image of $w_{j}$ is denoted as $w_{j}$ on $A_{\sigma }$ and as $cw_{j}$ on $A_{c\sigma }$ )

$$ \begin{align*}\{(w_{1}, 0), (w_{2}, 0), \cdots, (w_{g}, 0), (0, cw_{1}), (0,cw_{2}), \cdots, (0, cw_{g})\}.\end{align*} $$

So a basis for $H_{1}(A_{v}, \mathbb Z)^{+}$ is given by

$$ \begin{align*}\{(\gamma_{1}, c\gamma_{1}), \cdots, (\gamma_{2g}, c\gamma_{2g})\}.\end{align*} $$

For the complex place v and $\eta $ , the determinant of $\gamma _{v}$ relative to the basis $H^{1}_{B}(A_{v}, \mathbb Z)^{+}$ and B is the determinant of the $2g \times 2g$ -matrix M

Observe that (in a product $V \times W$ of manifolds, a differential form from W does not pair with a cycle from V)

$$ \begin{align*}\int_{(\gamma_{i}, c\gamma_{i})} (w_{j}, 0) = \int_{\gamma_{i}} w_{j}, \qquad \int_{(\gamma_{i}, c\gamma_{i})} (0, cw_{j}) = \int_{c\gamma_{i}} c w_{j}.\end{align*} $$

So the matrix M can be rewritten as

$$ \begin{align*} M=\begin{bmatrix} \int_{\gamma_{1}} w_{1} & \int_{\gamma_{1}} w_{2} & \cdots & \int_{\gamma_{1}} w_{g} & \int_{c\gamma_{1}} cw_{1} & \cdots & \int_{ c\gamma_{1}} cw_{g}\\ \int_{\gamma_{2}} w_{1} & \int_{\gamma_{2}} w_{2} & \cdots & \int_{\gamma_{2}} w_{g} &\int_{c\gamma_{2}} cw_{1} & \cdots & \int_{c\gamma_{2}} cw_{g}\\ \cdots &\cdots & \cdots &\cdots &\cdots &\cdots & \cdots\\ \cdots &\cdots & \cdots &\cdots &\cdots &\cdots & \cdots\\ \cdots &\cdots & \cdots &\cdots &\cdots &\cdots & \cdots\\ \int_{\gamma_{2g}} w_{1} & \int_{\gamma_{2g}} w_{2} & \cdots & \int_{\gamma_{2g}} w_{g} & \int_{ c\gamma_{2g} } cw_{1} & \cdots & \int_{ c\gamma_{2g}} cw_{g} \end{bmatrix}. \end{align*} $$

As $A_{\sigma }$ and $A_{c\sigma }$ are complex conjugate abelian varieties, one has

$$ \begin{align*}\int_{ c\gamma_{i} } cw_{j} = \overline{\int_{\gamma_{i} } w_{j}};\end{align*} $$

it follows that M and $M_{v}(A, \eta )$ of (32) satisfy

$$ \begin{align*} \textrm{det} (M) = \textrm{det}(M_{v}(A, \eta)).\end{align*} $$

Remark 16 shows that

$$ \begin{align*}\prod_{v\in S_{\infty}} \textrm{det}(M_{v} (A, \eta))\end{align*} $$

is a real number multiplied by $({\sqrt {-1}})^{g~\cdot ~r_{2}}$ . This proves the identity (40); combining it with Proposition 9 provides (up to a power of 2)

$$ \begin{align*} det(\gamma^{*}_{\mathcal J}) = \frac{ det_{{\mathcal O}_{F}}(\gamma^{*}_{\mathcal J})}{\sqrt{d_{F}}^{g}} = \pm ({\sqrt{-1}})^{g~\cdot~r_{2}} \frac{P_{J,\infty}}{\sqrt{d_{F}}^{g}}.\end{align*} $$

Theorem 23 follows using the well-known result that the sign of $d_{F}$ is $(-1)^{r_{2}}$ .

Proposition 25. (i) $det_{{\mathcal O}_{F}}(\gamma _{\mathcal J}^{*}) $ is equal to the determinant $det_{{\mathcal O}_{F}}(\gamma _{\mathcal J})$ of

$$ \begin{align*}\gamma_{\mathcal J}:~H^{1}_{B}(J_{\mathbb C}, \mathbb Z(1))^{+}_{\mathbb C} \xrightarrow{\sim} \textrm{Lie}~(\mathcal J) \otimes_{\mathbb Z} \mathbb C\end{align*} $$

calculated using the ${\mathcal O}_{F}$ -integral structure $j_{\textrm {Lie}~(\mathcal J)}$ (Definition 8) and the lattice $H^{1}_{B}(J_{\mathbb C}, \mathbb Z(1))^{+}$ .

(ii) $det(\gamma _{\mathcal J}^{*})= det(\gamma _{\mathcal J})$ .

Proof. (i) For any complex abelian variety A, the dual abelian variety $A^{t}$ satisfies

$$ \begin{align*} H_{1}(A^{t}, \mathbb Z) = H^{1}_{B}(A, \mathbb Z(1))& & \textrm{Lie}~A = H^{1}(A^{t}, \mathcal O) \\ \textrm{Hom}(H^{1}_{B}(A^{t}, \mathbb Z), \mathbb Z) = H^{1}_{B}(A, \mathbb Z(1)) && \textrm{Hom}_{\mathbb C}(H^{1}(A, \mathcal O), \mathbb C) = H^{0}(A^{t}, \Omega^{1}). \end{align*} $$

This shows that the period isomorphism

$$ \begin{align*}\gamma_{A}: H^{1}_{B}(A, \mathbb Z(1))^{+}_{\mathbb C} \xrightarrow{\sim} H^{1}(A, \mathcal O)\end{align*} $$

is dual to the map

$$ \begin{align*}\gamma^{*}_{A^{t}}: H^{1}_{B}(A^{t}, \mathbb Z)^{+}_{\mathbb C} \xleftarrow{\sim} H^{0}(A^{t}, \Omega^{1}).\end{align*} $$

Applying these to the self-dual Jacobian $J_{\mathbb C}$ gives that the lattice $H^{1}_{B}(J_{\mathbb C}, \mathbb Z(1))^{+}$ is dual to the lattice $H^{1}_{B}(J_{\mathbb C}, \mathbb Z)^{+}$ . The natural duality of the projective ${\mathcal O}_{F}$ -modules $\textrm {Lie}~(\mathcal J)$ and $\omega _{\mathcal J}$ shows, using Proposition 9, that the ${\mathcal O}_{F}$ -integral structures $j_{\textrm {Lie}~(\mathcal J)}$ and $j_{ \omega _{\mathcal J} }$ are dual. As the determinants of dual maps computed with respect to dual lattices are equal, the result follows.

(ii) Proposition 9 and (i) imply $det(\gamma _{\mathcal J}^{*})~\cdot ~(\sqrt {d_{F}})^{g} = det_{{\mathcal O}_{F}}(\gamma _{\mathcal J}^{*}) =det_{{\mathcal O}_{F}}(\gamma _{\mathcal J}) =det(\gamma _{\mathcal J}) ~\cdot ~(\sqrt {d_{F}})^{g}$ .

5.1 The groups $H^{*}_{W}(X, \mathbb Z(1))$

By Subsection 2.8.1, we need to understand $\textrm {Pic}(X)$ and $\textrm {Br}(X)$ .

Any $x\in X_{0}(F)$ provides $ \text {Pic}(X_{0}) = J(F) \times \mathbb Z$ .

Proposition 26. One has (neglecting 2-torsion)

Proof. As $\pi :X\to S$ is smooth proper, any $x\in X_{0}(F)$ provides a splitting of the map $\textrm {Pic}(S) \to \textrm {Pic}(X)$ . The identity component $\textrm {Pic}^{0}_{X/S}$ of the relative Picard scheme $\textrm {Pic}_{X/S}$ is the Néron model $\mathcal J$ of J by [Reference Bosch, Lütkebohmert and Raynaud4, Theorem 1, page 264] and $\textrm {Pic}_{X/S}(S) = \textrm {Pic}^{0}_{X/S}(S) \times \mathbb Z$ by [Reference Bosch, Lütkebohmert and Raynaud4, Theorem 1, page 252]. As $\pi _{*}{\mathcal O}_{X} ={\mathcal O}_{S}$ , [Reference Bosch, Lütkebohmert and Raynaud4, Proposition 4, page 204] says $\textrm {Pic}_{X/S}(S) = \text {Pic}(X)/{\text {Pic}(S)}$ and $\textrm {Pic}_{X/S}(F) = \text {Pic}(X_{0})$ . This proves (i). For (ii), we note that [Reference Tate36, Theorem 3.1] provides an exact (modulo 2-torsion) sequence

by class field theory, $\textrm {Br}(S)$ is a finite $2$ -group.

5.2 The Euler characteristic $\chi _{B}(X,1)$

We now compute the Euler characteristic of (21),

$$ \begin{align*}\chi_{B}(X,1) = \frac{\chi(B(0,1)) \chi(B(2,1))}{\chi(B(1,1))}.\end{align*} $$

5.2.1 The Euler characteristic of $B(1,1)$

We write $det(\gamma _{X})$ for the determinant of

(41) $$ \begin{align} \gamma_{X}: H^{1}_{B}(X_{\mathbb C}, \mathbb Z(1))^{+}_{\mathbb C} \xrightarrow{\sim} H^{1}(X, \mathcal O)\otimes_{\mathbb Z} \mathbb C \end{align} $$

with respect to the lattices $H^{1}_{B}(X_{\mathbb C}, \mathbb Z(1))^{+}$ and the abelian group underlying $H^{1}(X, \mathcal O)$ .

Lemma 28. The projective ${\mathcal O}_{F}$ -modules $H^{1}(\mathcal J, \mathcal O)$ and $\omega _{\mathcal J}$ are dual.

Proof. As $\mathcal J \to S$ is an abelian scheme, its relative Picard scheme $\textrm {Pic}_{\mathcal J/S}$ exists [Reference Bosch, Lütkebohmert and Raynaud4, Theorem 5, p. 234] and its identity component $\textrm {Pic}^{0}_{\mathcal J/S}$ is the dual abelian scheme $\mathcal J^{t} \to S$ . The S-points $\textrm {Lie}~\textrm {Pic}_{\mathcal J}$ of its Lie algebraFootnote ³ ${Lie}~\textrm {Pic}_{\mathcal J}$ satisfy [Reference Bosch, Lütkebohmert and Raynaud4, Theorem 1, p. 231], [Reference Liu, Lorenzini and Raynaud30, Proposition 1.1 (d)]

(42) $$ \begin{align} \textrm{Lie}~(\mathcal J^{t})= \textrm{Lie~(Pic}_{\mathcal J/S}) \xrightarrow{\sim} H^{1}(\mathcal J, \mathcal O). \end{align} $$

Since $J^{t}$ is the generic fibre of $\mathcal J^{t}$ , the self-duality $J \cong J^{t}$ shows $\mathcal J \cong \mathcal J^{t}$ and $\textrm {Lie}~(\mathcal J^{t}) \cong \textrm {Lie}~(\mathcal J)$ . Combining this with the natural duality between $ \textrm {Lie}~(\mathcal J)$ and $\omega _{\mathcal J}$ [Reference Liu, Lorenzini and Raynaud30, Proposition 1.1 (c)] proves the lemma.

Proposition 29. One has

$$ \begin{align*}\chi(B(1,1)) = \frac{P_{J, fin}}{P_{J}}= \frac{{\sqrt{|d_{F}|}}^{g}}{P_{J,\infty}(\eta)~ \mathbb N_{F/{\mathbb Q}}(\mathfrak a_{\eta})}.\end{align*} $$

Proof. Any $x\in X_{0}(F)$ gives a map

$$ \begin{align*} \beta_{x}: X \to \mathcal J, \qquad X_{0} \to J;\end{align*} $$

the induced map $\beta $ on cohomology is independent of the choice of x; for example, it provides the following isomorphism:

$$ \begin{align*}\beta: H^{1}(\mathcal J, \mathcal O) \xrightarrow{\sim} H^{1}(X, \mathcal O)\end{align*} $$

of ${\mathcal O}_{F}$ -modules, which fits into a commutative diagram (using Lemma 28)

(43)

as the vertical maps $\beta $ are isomorphisms, the integral structure corresponding to the map $\gamma _{\mathcal J}$ (top row) is isomorphic to the one corresponding to $\gamma _{X}$ (bottom row). As they are torsion-free, Theorem 23 and Proposition 25 show that

$$ \begin{align*}\chi(\gamma_{X}) = \chi(\gamma_{\mathcal J}) = det(\gamma_{\mathcal J}) = \frac{P_{J, \infty}}{|d_{F}|^{g/2}}.\end{align*} $$

Our convention for $B(1,1)$ is that $\textrm {Ker}(\gamma _{M})$ is in even degree, say, degree 0. We have

$$ \begin{align*} &B(1,1):~\underset{\textrm{degree 1}}{H^{1}_{B}(X_{\mathbb C}, \mathbb Z(1))^{+}_{\mathbb C}} \to \underset{\textrm{degree 2}}{H^{1}(X, \mathcal O)\otimes_{\mathbb Z} \mathbb C}, \nonumber \\[3pt] & det (B(1,1)) = \frac{{\sqrt{|d_{F}|}}^{g}}{P_{J,\infty}(\eta)~ \mathbb N_{F/{\mathbb Q}}(\mathfrak a_{\eta})} = \chi(B(1,1)). \\[-38pt] \end{align*} $$

Combining Proposition 29 with (24) and (25) yields the following.

Proposition 30. We have

$$ \begin{align*}\frac{1}{\chi_{B}(X,1)} = \frac{\chi(B(1,1))}{\chi(B(0,1)) \chi(B(2,1))} =\frac{(2\pi i)^{r_{2}}}{\sqrt{d_{F}}}~\cdot~ \frac{{\sqrt{|d_{F}|}}^{g}}{P_{J,\infty}(\eta)~ \mathbb N_{F/{\mathbb Q}}(\mathfrak a_{\eta})} .\end{align*} $$

5.3 The determinant of the sequence $C(1)$

Using (16), (17), (18) and Proposition 26, the torsion in (22) satisfies

$$ \begin{align*}\chi(C(1)_{tor}) = \frac{w~\cdot~[Br(X)]}{[\textrm{Pic}(X)_{tor}]^{2} } = \frac{w~\cdot~[Br(X)]} {h~\cdot~[J(F)_{tor}]~\cdot~h~\cdot~[J(F)_{tor}]}.\end{align*} $$

As $\textrm {Hom}(\textrm {Pic}(X), \mathbb Z)\otimes \mathbb C \cong \mathbb C \times \textrm {Hom}(J(F), \mathbb Z)\otimes \mathbb C$ by Proposition 26, the sequence $C(1)_{\mathbb C}$ of (22)

(44) $$ \begin{align} 0 &\to \underset{degree~0}{{\mathcal O}_{F}^{\times}\otimes\mathbb C} \to \underset{degree~1}{\textrm{Coker}(\gamma_{M_{0}})} \to \underset{degree~2}{\textrm{Hom}(\textrm{Pic}(X), \mathbb C)} \xrightarrow{h} \underset{degree~3}{{\textrm{Pic}(X)}\otimes\mathbb C} \to \underset{degree~4}{\textrm{Ker}(\gamma_{M_{2}})} \nonumber \\[3pt] &\to \underset{degree~5}{\textrm{Hom}({\mathcal O}_{F}^{\times}, \mathbb C)} \to 0. \end{align} $$

breaks up into exact sequences

$$ \begin{align*} 0 \to \underset{degree~0}{{\mathcal O}_{F}^{\times}\otimes\mathbb C} \to \underset{degree~1}{\textrm{Coker}(\gamma_{M_{0}})} \to \mathbb C \to 0,\\ \underset{degree~2}{\textrm{Hom}(\textrm{Pic}^{0}(X), \mathbb C)} \xrightarrow{h} \underset{degree~3}{{\textrm{Pic}^{0}(X)}\otimes\mathbb C},\\ 0 \to \mathbb C \to \underset{degree~4}{\textrm{Ker}(\gamma_{M_{2}})} \to \underset{degree~5}{\textrm{Hom}({\mathcal O}_{F}^{\times}, \mathbb Z)\otimes\mathbb C} \to 0. \end{align*} $$

We compute $det(C(1))$ in three steps:

• • By (23), the first sequence is (14) and so has determinant R.

• • By [Reference Lichtenbaum27, Conjecture 2.3.7] or (11), the map h is the Arakelov intersection pairing

  $$ \begin{align*}\textrm{Hom}(J(F), \mathbb Z)\otimes{\mathbb C} \to J(F)\otimes{\mathbb C}. \end{align*} $$
  By Faltings–Hriljac [Reference Faltings8, Reference Hriljac21], $-h$ is the Néron–Tate height pairing (35) and so $det(h) = \pm \Theta _{NT}(J)$ .

• • As $t_{M_{2}} = H^{2}(X_{0}, \mathcal O) =0$ , one has $\textrm {Ker}(\gamma _{M_{2}}) = H^{2}(X_{\mathbb C}, \mathbb C(1))^{+} \cong \mathbb C^{r_{1}+r_{2}}$ . Using (18), the third sequence is (12) and so has determinant R.

Proposition 31. We have

$$ \begin{align*} det(C(1)) = \frac{R \times R}{\Theta_{NT}} \qquad \chi (C(1)) = \frac{det(C(1))}{\chi(C(1)_{tor})}= \frac{h \times h [J(F)_{tor}] [J(F)_{tor}]}{[Br(X)] w } \frac{R \times R}{\Theta_{NT}} .\end{align*} $$

5.4 Completion of the proof of the main theorem in a special case

In this case, J has good reduction everywhere as $\textrm {Pic}^{0}_{X/S}$ is the Néron model of J; the group scheme $\Phi _{v}$ is trivial and the Tamagawa numbers $c_{v} =1$ for all v and $P_{J,fin} =1$ . Conjecture 20 for $L(J,s)$ becomes Conjecture 21:

We recall that

$$ \begin{align*} \zeta(X,s) = \prod_{v\in S}\zeta(X_{v}, s), \qquad \zeta(X_{v}, s) = \frac{P_{1}(X_{v},s)}{P_{0}(X_{v}, s) P_{2}(X_{v}, s)}. \end{align*} $$

As $\pi :X \to S$ is smooth and proper, one has

$$ \begin{align*}P_{0}(X_{v},s) = (1-q_{v}^{-s}), \quad P_{1}(C_{v}, t) = P_{1}(J_{v}, t),\quad P_{2}(X_{v},s) = (1-q_{v}^{1-s})\end{align*} $$

and

(45) $$ \begin{align} \prod_{v\in S}\frac{1}{P_{0}(X_{v},s)} = \zeta(S,s), \qquad \prod_{v\in S}\frac{1}{P_{2}(X_{v},s)} = \zeta(S,s-1), \quad L(J, s) = \prod_{v\in S} \frac{1}{P_{1}(X_{v},s)}.\end{align} $$

Using (13), (15), we see that Conjecture 21 for $J/F$ is equivalent to the equality

Conjecture 1 for $\zeta ^{*}(X,1)$ says

$$ \begin{align*} \zeta^{*}(X,1) = \chi(X,1) = \frac{\chi_{A,C}(X,1)}{ \chi_{B}(X,1)} = \frac{{\chi(C(1))}}{\chi_{B}(X,1)~\cdot~ \chi(A^{\prime}(3,1))}.\end{align*} $$

Propositions 30 and 31 show

$$ \begin{align*}\chi(X,1) = {\frac{1}{w}}~\cdot~{\frac{[\textrm{Pic}(X)_{tor}]^{2}}{w ~[\textrm{Br}(X)]}~\frac{R \times R}{\Theta_{NT}}}~\cdot~ {\frac{(2\pi i )^{r_{2}}}{\sqrt{d_{F}}} \frac{{\sqrt{|d_{F}|}}^{g}}{P_{J,\infty}(\eta)~ \mathbb N_{F/{\mathbb Q}}(\mathfrak a_{\eta}) }}. \end{align*} $$

Proposition 26 shows that Conjecture 1 is equivalent to Conjecture 21 for J over F, up to powers of 2. This proves Theorem 2 in the special case $\pi :X\to S$ is smooth and $X_{0}(F)$ is nonempty.

6.1 Index and Period

Let C be a smooth proper curve over a field K and $T=$ Spec K.

6.2 Curves over local and global fields [Reference Bosch and Liu3, Reference Liu, Lorenzini and Raynaud30]

Let K be a p-adic field with ring of integers ${\mathcal O}_{K}$ and residue field $k(v) = \mathbb F_{q}$ . Write $T = \text {Spec}~{\mathcal O}_{K}$ and $v: \text {Spec}~\mathbb F_{q} \to T$ for the closed point of T. Let $f: C \to T$ be a flat projective morphism with C regular, $f_{*}{\mathcal O}_{C} = {\mathcal O}_{T}$ and $C_{K} \to \text {Spec}~K$ a geometrically connected smooth curve of genus $g>0$ . Let $\mathcal N$ be the Néron model of the Jacobian J of $C_{K}$ . We write $\Phi _{v}=\pi _{0}(\mathcal N_{v})$ for the finite group scheme of connected components of the special fibre $\mathcal N_{v}$ over v and $c_{v} $ is the order of $\Phi _{v}(k(v))$ .

Consider the map $Lie~(\phi ): R^{1}f_{*} {\mathcal O}_{C} \to Lie~\mathcal N$ of coherent sheaves on T and the induced map on T-points

$$ \begin{align*}\textrm{Lie}(\phi): H^{1}(C, {\mathcal O}_{C}) \to \textrm{Lie}~(\mathcal N).\end{align*} $$

We will need the following result in Section 7.

Let $\Gamma _{i}$ ( $i\in I$ ) be the irreducible components of the special fibre $C_{v}$ .

• • $d_{i}$ is the multiplicity of $\Gamma _{i}$ in $C_{v}$ .

• • $e_{i}$ is the geometric multiplicity of $\Gamma _{i}$ in $C_{v}$ .

• • $r_{i}$ is the number of irreducible components of $\Gamma _{i} \times \mathrm {Spec}~\overline {\mathbb F_{q}}$ .

There are canonical maps [Reference Bosch and Liu3, Proposition 1.9] $\alpha _{C}: \mathbb Z^{I} \to \mathbb Z^{I}$ (defined using intersection on C) and $\beta _{C}: \mathbb Z^{I} \to \mathbb Z$ . Let d be the gcd of the set $\{d_{i}, i\in I\}$ and $d^{\prime }$ be the gcd of the set $\{r_{i}d_{i}, i\in I\}$ .

It is known that [Reference Liu and Erné29, §9.1, Theorem 1.23] that the intersection pairing on $R_{v}$

$$ \begin{align*}R_{v}:= \frac{\mathbb Z^{I}}{\mathbb Z} = \textrm{Coker}~(\mathbb Z \to \mathbb Z^{I}) \qquad 1\mapsto \sum_{i\in I} d_{i} \Gamma_{i}\end{align*} $$

is negative definite. The following corollary of Theorem 34 and [Reference Bosch and Liu3, Theorem 1.11] on $\Delta (R_{v})$ (defined as in Subsection 2.3.5) is due to Flach–Siebel.

Corollary 35 (Flach–Siebel [Reference Flach and Siebel14, Lemma 17])

If $\delta _{v}$ is the index of $C_{K}$ and $\delta ^{\prime }_{v}$ is the period of $C_{K}$ , then

$$ \begin{align*}\Delta(R_{v}) = \frac{c_{v}}{\delta_{v}~\cdot~\delta^{\prime}_{v}} \prod_{i\in I} r_{i}.\end{align*} $$

In terms of the Arakelov intersection pairing on $R_{v}$ (see [Reference Faltings8, p. 390] or [Reference Gross20, (3.7)]), one has

$$ \begin{align*}\Delta_{ar}(R_{v}) = \Delta(R_{v}) (\log q_{v})^{\#G_{v}-1}.\end{align*} $$

Lemma 36. Let $L_{v}(J, t)$ be the local L-factor of J as in (27). The zeta function

$$ \begin{align*}Z(C_{v},t) = \frac{P_{1}(C_{v},t)}{P_{0}(C_{v},t)~\cdot~P_{2}(C_{v},t)}\end{align*} $$

of $C_{v}$ satisfies

$$ \begin{align*}Z(C_{v},t) = \frac{L_{v}(J,t)}{(1-t)~\cdot~\prod_{i\in I}(1-{(qt)}^{r_{i}})}.\end{align*} $$

Proof. As $C_{K}$ is smooth projective and geometrically connected, $C_{v}$ is geometrically connected. So $P_{0}(C_{v},t) = (1-t)$ .

We next show that $P_{1}(C_{v},t)$ equals $L_{v}(J,t)$ of (27). For any prime $\ell $ coprime to q, the Kummer sequence and the perfect pairing (Poincaré duality)

$$ \begin{align*}H^{1}_{et}(C_{K} \times \bar{K}, \mathbb Q_{\ell}) \times H^{1}_{et}(C_{K} \times \bar{K}, \mathbb Q_{\ell}(1) ) \xrightarrow{\cup} H^{2}_{et}(C_{K} \times \bar{K}, \mathbb Q_{\ell}(1) ) \cong \mathbb Q_{\ell}\end{align*} $$

provide isomorphisms of $\textrm {Gal}(\bar {K}/K)$ -representations

$$ \begin{align*}H^{1}_{et}(C_{K} \times \bar{K}, \mathbb Q_{\ell}(1) ) \xrightarrow{\sim} T_{\ell}J_{K}\otimes \mathbb Q_{\ell}, \quad H^{1}_{et}(C_{K} \times \bar{K}, \mathbb Q_{\ell}) \xrightarrow{\sim} \textrm{Hom}(H^{1}_{et}(C_{K} \times \bar{K}, \mathbb Q_{\ell}(1)), \mathbb Q_{\ell}).\end{align*} $$

So we obtain an isomorphism

(46) $$ \begin{align} H^{1}_{et}(C_{K} \times \bar{K}, \mathbb Q_{\ell}) \xrightarrow{\sim} \textrm{Hom}(T_{\ell}J_{K}\otimes \mathbb Q_{\ell}, \mathbb Q_{\ell}) \end{align} $$

of $\textrm {Gal}(\bar {K}/K) $ representations. Since $H^{1}_{et}(C_{v} \times \bar {\mathbb F}_{q}, \mathbb Q_{\ell })$ isomorphic to the subspace of $H^{1}_{et}(C_{K} \times \bar {K}, \mathbb Q_{\ell })$ of invariants under the inertia subgroup [Reference Bloch1, Lemma 1.2], it follows from (46) and (27) that $P_{1}(C_{v}, t)$ is $L_{v}(J, t)$ .

Finally, one has the elementary identity Footnote ⁴ [Reference Gordon18, Proposition 3.3]; see also [Reference Liu, Lorenzini and Raynaud30, p.484]

$$ \begin{align*}P_{2}(C_{v},t) = \prod_{i\in I} (1-{(qt)}^{r_{i}}).\end{align*} $$

This completes the proof.

6.3 Relating and $\textrm {Br}(X)$

For any arithmetic surface $X \to S$ , let $\delta $ be the index of $X_{0}$ over F and $\alpha $ be the order of the (finite) cokernel of the natural map $\textrm {Pic}^{0}(X_{0}) \hookrightarrow J(F)$ . For any finite place v of S, we put $\delta _{v}$ and $\delta ^{\prime }_{v}$ for the (local) index and period of $X\times F_{v}$ over the local field $F_{v}$ . The following result is due to Geisser [Reference Geisser17, Theorem 1.1]; there is also a recent proof by Flach-Siebel [Reference Flach and Siebel14].

Theorem 37. Assume that $\text {Br}(X)$ is finite. The following equality holds (up to powers of $2$ ):

(47)

7.1 Preliminary steps

We are in a position to prove Theorem 2 in the general case. Let $\Sigma =\{v\in S~|~X_{v}~\textrm {is~not~smooth}\}$ ; let $G_{v}$ denote the set of irreducible components of $X_{v}$ .

Lemma 38. If

$$ \begin{align*} Q_{2}(s) = \prod_{v \in \Sigma} \frac{(1-q_{v}^{1-s})}{\prod_{i\in G_{v}} (1-q_{v}^{{r_{i}} (1-s)})},\end{align*} $$

then

(48) $$ \begin{align} Q_{2}^{*}(1) &= \frac{1}{\prod_{v\in \Sigma} ( (\log q_{v})^{\#G_{v}-1}~\cdot~\prod_{i\in G_{v}} r_{i})} = \prod_{v\in \Sigma}\frac{c_{v}}{\Delta_{ar}(R_{v})~.\delta_{v}~\cdot~\delta^{\prime}_{v}} = P_{J,fin} \nonumber \\[3pt] & \quad \cdot~\prod_{v\in \Sigma}\frac{1}{\Delta_{ar}(R_{v})~.\delta_{v}~\cdot~\delta^{\prime}_{v}}.\end{align} $$

Proposition 39. One has

$$ \begin{align*} \chi(B(1,1)) = \frac{P_{J, fin}}{P_{J}} = \frac{1}{P_{J,\infty}}= \frac{|d_{F}|^{g/2}}{P_{J, \infty}(\eta)~\cdot~ \mathbb N_{F/{\mathbb Q}}(\mathfrak a_{\eta})}.\end{align*} $$

Proof. One has the isomorphism of integral structures (its Euler characteristic is 1)

$$ \begin{align*}H^{1}_{B}(J_{\mathbb C}, \mathbb Z(1))^{+}_{\mathbb C} \xrightarrow{\sim} H^{1}_{B}(X_{\mathbb C}, \mathbb Z(1))^{+}_{\mathbb C}.\end{align*} $$

As the notion of Néron model is local on the base [Reference Bosch, Lütkebohmert and Raynaud4, Proposition 4, page 13], Theorem 33 shows that the kernel and cokernel of $\text {Lie}(\phi ): H^{1}(X, \mathcal O) \to {\textrm {Lie}}(\mathcal J)$ are torsion ${\mathcal O}_{F}$ -modules of the same length and the kernel is exactly the torsion of $H^{1}(X,{\mathcal O}_{X})$ . This implies that the Euler characteristic of

$$ \begin{align*}H^{1}(X, \mathcal O)\otimes_{\mathbb Z} \mathbb C\xrightarrow{{\textrm{Lie}}{(\phi})} {\textrm{Lie}}(\mathcal J)\otimes_{\mathbb Z}\mathbb C\end{align*} $$

is 1: $\chi (\text {Lie}(\phi )) =1$ . Here the integral structures are $(H^{1}(X, \mathcal O), [H^{1}(X, \mathcal O)_{tor}])$ and $({\textrm {Lie}}(\mathcal J), 1)$ . We obtain that the Euler characteristic of

$$ \begin{align*} H^{1}_{B}(X_{\mathbb C}, \mathbb Z(1))^{+}_{\mathbb C} \xrightarrow{\gamma_{X}} H^{1}(X, \mathcal O)\otimes_{\mathbb Z} \mathbb C\end{align*} $$

is equal to that of

$$ \begin{align*}\gamma_{\mathcal J}: H^{1}_{B}(J_{\mathbb C}, \mathbb Z(1))^{+}_{\mathbb C} \xrightarrow{\sim}H^{1}_{B}(X_{\mathbb C}, \mathbb Z(1))^{+}_{\mathbb C} \xrightarrow{\gamma_{X}} H^{1}(X, \mathcal O)\otimes_{\mathbb Z} \mathbb C\xrightarrow{{\textrm{Lie}}{(\phi})} {\textrm{Lie}}(\mathcal J)\otimes_{\mathbb Z}\mathbb C.\end{align*} $$

Proposition 25 and Theorem 23 show

(49) $$ \begin{align} \chi(\gamma_{X}) = \chi(\gamma_{\mathcal J}) = \frac{P_{J,\infty}(\eta)~\cdot~\mathbb N_{F/{\mathbb Q}}(\mathfrak a_{\eta})}{{\sqrt{|d_{F}|}}^{g}}.\end{align} $$

Our convention for $B(1,1)$ is that $\textrm {Ker}(\gamma _{M})$ is in even degree, say, degree 0. Thus,

$$ \begin{align*}\underset{\textrm{degree 1}}{H^{1}_{B}(X_{\mathbb C}, \mathbb Z(1))^{+}_{\mathbb C}} \to \underset{\textrm{degree 2}}{H^{1}(X, \mathcal O)\otimes \mathbb C}, \qquad det (B(1,1)) = \frac{{\sqrt{|d_{F}|}}^{g}}{P_{J,\infty}(\eta)~\cdot~\mathbb N_{F/{\mathbb Q}}(\mathfrak a_{\eta})} = \chi(B(1,1)). \end{align*} $$

7.2 Global index

Since X is regular, the natural map $ \textrm {Pic}(X) \to \textrm {Pic}(X_{0})$ is surjective: a natural section (as sets) is provided by sending a divisor on $X_{0}$ to its Zariski closure in X. So the index $\delta $ of $X_{0}$ over F is the order of the cokernel of the composite map $\textrm {Pic}(X) \to \textrm {Pic}(X_{0}) \xrightarrow {d} \mathbb Z$ .

7.2.1 Calculation of $\chi (C(1))$

The torsion in $C(1)$ of (22) satisfies

(50) $$ \begin{align} \chi(C(1)_{tor}) = \frac{w~\cdot~[Br(X)]} {[\textrm{Pic}(X)_{tor}]~\cdot~[\textrm{Pic}(X)_{tor}]}.\end{align} $$

We can rewrite $C(1)_{\mathbb C}$ as

(51) $$ \begin{align} 0 &\to \underset{degree~0}{{\mathcal O}_{F}^{\times}\otimes\mathbb C} \to \underset{degree~1}{\textrm{Coker}(\gamma_{M_{0}})} \to \underset{degree~2}{\textrm{Hom}(\textrm{Pic}(X), \mathbb C)} \to \underset{degree~3}{{\textrm{Pic}(X)}\otimes\mathbb C} \to \underset{degree~4}{\textrm{Ker}(\gamma_{M_{2}})} \nonumber\\[3pt] &\to \underset{degree~5}{\textrm{Hom}({\mathcal O}_{F}^{\times}, \mathbb C)} \to 0. \end{align} $$

Using the exact sequence

(52) $$ \begin{align} 0 \to \textrm{Pic}^{0}(X) \to \textrm{Pic}(X) \xrightarrow{d} \mathbb Z \to \frac{\mathbb Z}{\delta\mathbb Z} \to 0,\end{align} $$

the sequence $C(1)$ breaks up into

• • the sequence (14) with determinant R

  $$ \begin{align*}0 \to \underset{degree~0}{{\mathcal O}_{F}^{\times}\otimes\mathbb C} \to \underset{degree~1}{\textrm{Coker}(\gamma_{M_{0}})} \to \underset{degree~2}{\textrm{Hom}(\mathbb Z, \mathbb C)} \to 0;\end{align*} $$

• • the sequence with determinant $\delta $

  $$ \begin{align*}0 \to \textrm{Hom}(\mathbb Z, \mathbb C) \to \textrm{Hom}(\textrm{Pic}(X), \mathbb C) \to \textrm{Hom}(\textrm{Pic}^{0}(X), \mathbb C) \to 0; \end{align*} $$

• • and the sequence with determinant $det(\psi )$

  $$ \begin{align*}h: \underset{degree~2}{\textrm{Hom}(\textrm{Pic}^{0}(X), \mathbb C)} \to \underset{degree~3}{{\textrm{Pic}^{0}(X)}\otimes\mathbb C} \end{align*} $$
  where h is the Arakelov intersection pairing [Reference Lichtenbaum27, Conjecture 2.3.7] as in (11);

• • the sequence with determinant $\delta $

  $$ \begin{align*}0 \to \textrm{Pic}^{0}(X)\otimes \mathbb C \to \textrm{Pic}(X)\otimes \mathbb C \xrightarrow{d} \mathbb C \to 0;\end{align*} $$

• • the sequence (12) with determinant R

  $$ \begin{align*} 0 \to \mathbb C \to \underset{degree~4}{\textrm{Ker}(\gamma_{M_{2}})} \to \underset{degree~5}{\textrm{Hom}({\mathcal O}_{F}^{\times}, \mathbb Z)\otimes \mathbb C} \to 0.\end{align*} $$

It follows from (52) that, in (51), the image of the lattice in degree 3 has index $\delta $ in the lattice in degree 4; dually, the image of the lattice in degree 1 has index $\delta $ in the lattice in degree 3. Thus,

$$ \begin{align*}det(C(1)) = \frac{R~\cdot~R}{\delta~\cdot~\delta~\cdot~det(h)}.\end{align*} $$

So $ \chi (C(1)) $ is given as

(53) $$ \begin{align} &= \frac{det(C(1))}{\chi(C(1)_{tor})} = \frac{R^{2}}{\delta^{2}~\cdot~\Delta_{ar}(\textrm{Pic}^{0}(X))~\cdot~[\textrm{Pic}^{0}(X)_{tor}]^{2}}~\cdot~\frac{[\textrm{Pic}^{0}(X)_{tor}]^{2}}{w~\cdot~[\textrm{Br}(X)]} \nonumber \\ &= \frac{R^{2}}{\delta^{2}~\cdot~\Delta_{ar}(\textrm{Pic}^{0}(X))~\cdot~w~\cdot~[\textrm{Br}(X)]},\end{align} $$

where, as in Subsection 2.3.5,

(54) $$ \begin{align} \Delta_{ar}(\textrm{Pic}^{0}(X)) = \frac{det(h)}{[\textrm{Pic}^{0}(X)_{tor}]^{2}}.\end{align} $$

Our next task is to calculate $\Delta _{ar}(\textrm {Pic}^{0}(X))$ and relate it to the Néron–Tate pairing (56) on $J(F)$ .

7.2.2 Calculation of $\Delta _{ar}(\textrm {Pic}^{0}(X))$

This is based on localisation sequences on X and S.

Let $U = S - \Sigma $ . So the map $X_{U} = \pi ^{-1}(U) \to U$ is smooth. For any finite $\Sigma ^{\prime } \subset S$ containing $\Sigma $ , we put $U^{\prime } = S - \Sigma ^{\prime }$ and $X_{U^{\prime }} = X - \pi ^{-1}(U^{\prime })$ .

Lemma 40.

1. (i) The maps

   $$ \begin{align*}\textrm{Pic}(S) \to \textrm{Pic}(X), \qquad \textrm{Pic}(U^{\prime}) \to \textrm{Pic}(X_{U^{\prime}})\end{align*} $$
   are injective.

2. (ii) There is an exact sequence

   (55) $$ \begin{align} 0 \to \oplus_{v\in \Sigma} \frac{ \mathbb Z^{G_{v}}}{\mathbb Z} \to \frac{\textrm{Pic}^{0} (X)}{\textrm{Pic}(S)} \to \textrm{Pic}^{0} ( X_{0}) \to 0.\end{align} $$

Proof.

1. (i) From the Leray spectral sequence for $\pi : X\to S$ and the étale sheaf $\mathbb G_{m}$ on X, we get the exact sequence

   $$ \begin{align*} 0 \to H^{1}(S, \pi_{*} \mathbb G_{m}) \to H^{1}(X, \mathbb G_{m}) \to H^{0}(S, R^{1}\pi_{*}\mathbb G {}_{m}) \to \textrm{Br}(S).\end{align*} $$
   Now use that $\pi _{*}\mathbb G_{m}$ is the sheaf $\mathbb G_{m}$ on S. This provides the injectivity of the first map. A similar argument provides the injectivity of the second.

2. (ii) We can compare the localisation sequences for $X_{U^{\prime }} \subset X$ and $U^{\prime } \subset S$

Note that S and X are regular and for any regular scheme Y, one has an isomorphism $\textrm {Cl}(Y) = \textrm {Pic}(Y)$ between the class group and the Picard group. The natural map between the localisation sequences is injective on all terms and, by assumption, is an isomorphism on the first and second terms. This provides the exact sequence

$$ \begin{align*}0 \to \oplus_{v\in \Sigma^{\prime}} \frac{ \mathbb Z^{G_{v}}}{\mathbb Z} \to \frac{\textrm{Pic}(X)}{\textrm{Pic}(S)} \to \frac{\textrm{Pic}(X_{U^{\prime}})}{\textrm{Pic}(U^{\prime})} \to 0.\end{align*} $$

In particular, we get this sequence for $\Sigma $ and U. Using (52), we obtain the exact sequence

$$ \begin{align*}0 \to \oplus_{v\in \Sigma} \frac{ \mathbb Z^{G_{v}}}{\mathbb Z} \to \frac{\textrm{Pic}^{0}(X)}{\textrm{Pic}(S)} \to \frac{\textrm{Pic}^{0}(X_{U})}{\textrm{Pic}(U)} \to 0.\end{align*} $$

By assumption, $X_{v}$ is geometrically irreducible for any $v \notin \Sigma $ . So for any $U^{\prime } = S - \Sigma ^{\prime }$ with $U^{\prime } \subset U$ , the induced maps

$$ \begin{align*}\frac{\textrm{Pic}^{0}(X_{U})}{\textrm{Pic}(U)}\to \frac{\textrm{Pic}^{0}(X_{U^{\prime}})}{\textrm{Pic}(U^{\prime})}\end{align*} $$

are isomorphisms. Taking the limit over $\Sigma ^{\prime }$ gives us an exact sequence

$$ \begin{align*}0 \to \oplus_{v\in \Sigma} \frac{ \mathbb Z^{G_{v}}}{\mathbb Z} \to \frac{\textrm{Pic}^{0} (X)}{\textrm{Pic}(S)} \to \textrm{Pic}^{0}(X_{0}) \to 0.\end{align*} $$

This proves the lemma.

For any $v \in S$ , recall $\Delta _{ar}(R_{v})$ from Corollary 35, where

$$ \begin{align*}R_{v} =\frac{\mathbb Z^{G_{v}}}{\mathbb Z}\end{align*} $$

and, as before, $G_{v}$ is the set of irreducible components of $X_{v}$ . Let us define (see Subsection 2.3.5)

(56) $$ \begin{align} \Delta_{NT}(J(F)) = \frac{\Theta_{NT}(J)}{ [ J(F)_{tor}]~\cdot~[J(F)_{tor}]} \end{align} $$

using the Néron–Tate pairing (35) on $J(F)$ , analogous to $\Delta _{ar}(\textrm {Pic}^{0}(X))$ from (54).

Proposition 41. If $\alpha $ is the order of the cokernel of the natural map $\textrm {Pic}^{0}(X_{0}) \hookrightarrow J(F)$ , then one has

(57) $$ \begin{align} \Delta_{ar}(\textrm{Pic}^{0}(X))= \pm \frac{\alpha^{2}}{h^{2}}~\cdot~\Delta_{NT}(J(F))~\cdot~\prod_{v\in\Sigma} \Delta_{ar}(R_{v}).\end{align} $$

Proof. By [Reference Hriljac21, Proposition 3.3], the Arakelov intersection pairing h is negative-definite on $\textrm {Pic}^{0}(X)\otimes \mathbb Q$ and there exists a map $\kappa : \textrm {Pic}^{0}(X_{0}) \to \textrm {Pic}^{0}(X)$ such that

$$ \begin{align*} (y,y^{\prime}) \mapsto h(\kappa(y), \kappa(y^{\prime}) )\end{align*} $$

gives the intersection pairing on $\textrm {Pic}^{0}(X_{0})$ which, by Faltings–Hriljac [Reference Faltings8], [Reference Hriljac21, Theorem 3.1], is the negative of the Néron–Tate pairing (35) on $J(F)$ . Note that this means

(58) $$ \begin{align} \Delta_{NT}(J(F)) = \pm \Delta_{ar}(J(F)). \end{align} $$

So the sequence (55) splits over $\mathbb Q$ as an orthogonal direct sum with respect to the Arakelov intersection pairing

(59) $$ \begin{align} (\textrm{Pic}^{0} ( X_{0})\otimes\mathbb Q)~\oplus~(\underset{v\in \Sigma}{\oplus} \frac{ \mathbb Q^{G_{v}}}{\mathbb Q}) \cong \textrm{Pic}^{0} (X)\otimes\mathbb Q.\end{align} $$

The map $\kappa $ is defined as follows: given any element y of $\textrm {Pic}^{0} ( X_{0})$ , consider its Zariski closure $\bar {y}$ in X. As the intersection pairing is negative-definite [Reference Liu and Erné29, §9.1, Theorem 1.23] on $R_{v}$ , the linear mapping $R_{v} \to \mathbb Z$ defined by $z \mapsto z.\bar {y}$ is represented by a unique element $\kappa _{v}(y) \in R_{v}\otimes \mathbb Q$ . Clearly, the element

$$ \begin{align*}\kappa(y) = \bar{y} - \sum_{v\in \Sigma} \kappa_{v}(y) \end{align*} $$

is orthogonal to $\oplus _{v\in \Sigma }R_{v} \subset \textrm {Pic}^{0}(X)$ ; so the assignment $y \mapsto \kappa (y)$ provides (59).

Now (7) shows

$$ \begin{align*} \Delta_{ar}(\frac{\textrm{Pic}^{0}(X)}{\textrm{Pic}(S)})= \Delta_{ar}(\textrm{Pic}^{0}(X_{0} ))~\cdot~\prod_{v\in\Sigma} \Delta_{ar}(R_{v}).\end{align*} $$

Using $h = [\textrm {Pic}(S)]$ , this becomes

(60) $$ \begin{align}\Delta_{ar}(\textrm{Pic}^{0}(X))~\cdot~ {h^{2}} = \Delta_{ar}(\frac{\textrm{Pic}^{0}(X)}{\textrm{Pic}(S)})= \Delta_{ar}(\textrm{Pic}^{0}(X_{0} ))~\cdot~\prod_{v\in\Sigma} \Delta_{ar}(R_{v}).\end{align} $$

As $\textrm {Pic}^{0}(X_{0}) \hookrightarrow J(F)$ is a subgroup of index $\alpha $ , we see $ \Delta _{ar}(\textrm {Pic}^{0}(X_{0} )) = \alpha ^{2} \Delta _{ar}(J(F)) = \pm \alpha ^{2}\cdot \Delta _{NT}(J(F))$ .