1 Integral structures and Euler characteristics

Let be finite-dimensional complex vector spaces, and

$$ \begin{align*}V_* = 0 \to V_0 \to V_1 \to \dots \to V_n \to 0\end{align*} $$

be an exact sequence.

Let $B_i$ be a lattice spanned by a basis for the vector space $V_i$ . Let $\Lambda V$ denote the highest exterior power of V and $\Lambda B$ denote the highest exterior power of B. The alternating tensor product of the $\Lambda V_i's$ is canonically isomorphic to $\mathbb C$ , and the alternating tensor product of the $\Lambda B_i's$ is isomorphic to $\mathbb Z$ . The natural inclusions of $B_i$ in $V_i$ induce a map from $\mathbb Z$ to $\mathbb C$ and the determinant $det (V_*, B_*)$ in $\mathbb C^*/\pm 1$ of the pair $(V_*, B_*)$ is defined to be the image of a generator of $\mathbb Z$ in $\mathbb C$ .

An example of an integral structure on a finite-dimensional complex vector space V comes from a finitely generated abelian group M with a homomorphism from M to V whose image is a lattice $M_0$ in V and whose kernel is the torsion subgroup $M_{tor}$ of M. The integral structure is then $(M_0, |M_{tor}|)$ .

Definition 1.4. Let $(A_*, a_*)$ be an integral structure on the exact complex $V_*$ , $( B_*, b_*)$ be an integral structure on the exact complex $W_*$ and $\phi _*$ be a map of complexes from $V_*$ to $W_*$ such that $\phi _j$ is an isomorphism for all j. Let $det_j$ be the determinant of $\phi _j$ with respect to the lattices $ {A_j}$ and ${B_j}$ , and let

$$\begin{align*}\chi(\phi_j) = \frac{b_j~det_j}{a_j}.\end{align*}$$

Define the Euler characteristic

$$\begin{align*}\chi(\phi):= \prod_{j=0}^n (\chi(\phi_j))^{(-1)^{j+1}}.\end{align*}$$

2 The groups and maps involved in the conjecture

2.1 Weil-étale motivic cohomology

As before, let X be a regular scheme, projective and flat over Spec $\mathbb Z$ . Let $X_0$ be the fiber of X over Spec $\mathbb Q$ , and let K be the algebraic closure of $\mathbb Q$ in the function field of X. Let $O_K$ be the ring of integers in K. We may regard X as a scheme projective and flat over Spec $O_K$ .

We will first define Weil-étale motivic cohomology groups and then discuss their relation to the groups defined by Fontaine and Perrin-Riou [Reference Fontaine8, Reference Flach4].

Let r be an integer and j a nonnegative integer. We would like to define a Weil-étale site and complexes of sheaves $\mathbb Z (r) $ on this site whose cohomology groups $H^j_W(X, \mathbb Z(r))$ would be Weil-étale motivic cohomology, but unfortunately we do not know how to do this. Instead, for $j \leq 2r$ we define $H^j_W(X, \mathbb Z(r))$ to be the hypercohomology groups $H^j_{et}(X, \mathbb Z(r))$ , where $\mathbb Z(r)$ denotes Bloch’s higher Chow group complex sheafified for the étale topology [Reference Bloch1, Reference Levine13]. Sometimes, these groups are referred to as étale motivic cohomology. For $j \geq 2r +1$ , we define $H^j_W( X, \mathbb Z (r))$ to be $h^j(RHom (R \Gamma _{et} (X, \mathbb Z(d-r)) , \mathbb Z[-2d-1])$ , so we have the exact sequence

$$ \begin{align*} 0 \to \mathrm{Ext}^1(H_{et}^{2d +2-j} &(X, \mathbb Z(d-r)), \mathbb Z) \to\\ &\qquad H^j_W(X, \mathbb Z(r))\to \mathrm{Hom} (H_{et}^{2d +1 -j}(X. \mathbb Z(d-r)), \mathbb Z) \to 0. \end{align*} $$

If we had our hypothetical Weil-étale site, with a global sections functor denoted by $\Gamma _W$ , this would follow, up to 2-torsion, from a duality theorem which asserted that $R\Gamma _W(X, \mathbb Z(d-r))$ was isomorphic to $RHom (R\Gamma _W(X, \mathbb Z(r)), \mathbb Z[-2d-1])$ . The analogue of this theorem, assuming the usual conjectures, is true for Weil-étale cohomology in the geometric case, as shown in [Reference Geisser10]. We note here that in [Reference Flach and Morin5], Flach and Morin have constructed such a complex of abelian groups, which satisfies this duality theorem assuming that standard finiteness conjectures hold.

The group $H^{2r}_W(X, \mathbb Z(r))$ is by definition $H^{2r}_{et}(X, \mathbb Z(r))$ , and by standard arguments this agrees with the group $H^{2r}_{Zar}(X, \mathbb Z(r))$ of codimension r cycles on X modulo rational equivalence after tensoring with $\mathbb Q$ . Hence, there is a cycle map $\phi $ from $H^{2r}_W(X, \mathbb Z(r))$ to singular cohomology with rational coefficients. Let $H^{2r}_W(X, \mathbb Z(r))^1$ denote Ker $\phi $ (cycles homologous to zero) and $H^{2r}_W(X, \mathbb Z(r))^2$ denote Image $\phi $ (cycles modulo homological equivalence.).

We have the exact sequence

$$ \begin{align*}0 \to H^{2(d-r)}_W(X, \mathbb Z(d-r))^1 \to H^{2(d-r)}_W(X, \mathbb Z(d-r)) \to H^{2(d-r)}_W(X, \mathbb Z(d-r))^2 \to 0.\end{align*} $$

Conjecture 2.1. The groups $H^j_{et}(X, \mathbb Z(r))$ are finitely generated for $j \leq 2r +1$ , and finite for $j= 2r+1$ .

This implies

Conjecture 2.2. The cohomology groups $H^j_W(X, \mathbb Z(r))$ are finitely generated for all j.

Assuming the validity of Conjecture 2.2, we give the complex vector space $H^j_W(X, \mathbb Z(r))_{\mathbb C}$ the standard integral structure $H^j_W(X, \mathbb Z(r))$ . We also need the following.

Conjecture 2.3. The finite groups $H^{2r+1}_{et}(X, \mathbb Z(r))$ and $H^{2(d-r)+1}_{et}(X, \mathbb Z(d-r))$ are Pontriagin duals.

Flach and Morin showed in [Reference Flach and Morin5, Proposition 3.4] that this follows from Conjecture 2.2 for $d \leq 2$ and, under some restrictions, in the general case.

2.2 Singular and de Rham cohomology

We also will have need of singular cohomology groups. Let $X_{\mathbb C} = X \times _{\mathbb Z} \mathbb C$ . Complex conjugation c acts on $H^j_B(X_{\mathbb C}, \mathbb Z)$ via the natural action of conjugation on $\mathbb C$ . If r is even (resp. odd), let $\tilde H_B^j(X, \mathbb C(r))^+$ and $\tilde H_B^j(X, \mathbb Z(r))^+$ be the set of elements y in $H_B^j(X_{\mathbb C}, \mathbb C)$ and $H_B^j(X_{\mathbb C}, \mathbb Z)$ such that $c(y) = y$ (resp. $c(y) = -y$ ). We define $H_B^j(X, \mathbb C(r)) $ (resp. $ H_B^j(X,\mathbb C(r))^+$ ) to be $H_B^j(X_{\mathbb C}, \mathbb C)$ (resp. $\tilde H_B^j(X_{\mathbb C}, \mathbb C)^+)$ , and its standard integral structure is given by mapping $H_B^j(X_{\mathbb C}, \mathbb Z)$ (resp. $\tilde H_B^j(X_{\mathbb C}, \mathbb Z)^+$ ) to $H_B^j(X_{\mathbb C}, \mathbb C)$ (resp $\tilde H_B^j(X_{\mathbb C}, \mathbb C)^+) $ via the natural map followed by multiplication by $(2\pi i)^r$ .

If r is even (resp. odd), let $\tilde H_B^j(X, \mathbb C(r))^-$ and $\tilde H_B^j(X, \mathbb Z(r))^-$ be the set of elements y in $H_B^j(X_{\mathbb C}, \mathbb C)$ and $H_B^j(X_{\mathbb C}, \mathbb Z)$ such that $c(y) = - y$ (resp. $c(y) = y$ ). We define $ H_B^j(X,\mathbb C(r))^-$ to be $\tilde H_B^j(X_{\mathbb C}, \mathbb C)^-$ , and its standard integral structure is given by mapping $\tilde H_B^j(X_{\mathbb C}, \mathbb Z)^-$ to $\tilde H_B^j(X_{\mathbb C}, \mathbb C)^- $ via the natural map followed by multiplication by $(2\pi i)^r$ .

We will also need the following Euler characteristics:

$$\begin{align*}\chi(X_{\mathbb C}) = \sum_j (-1)^j \mathrm{dim}~H^j_B(X_{\mathbb C}), \quad \chi^+ (X_{\mathbb C}) = \sum_j(-1)^j \mathrm{dim}~H^{j,+}_B (X_{\mathbb C}, \mathbb C(0)),\end{align*}$$

$$\begin{align*}\chi^-(X_{\mathbb C}) = \sum_j (-1)^j \mathrm{dim}~H^{j,-}_B(X_{\mathbb C}, \mathbb C(0)).\end{align*}$$

Let $\chi ^+(X_{\mathbb C}, Z(r)) = \chi ^+(X_{\mathbb C})$ if r is even and $\chi ^{-}(X_{\mathbb C})$ if r is odd.

Let $\chi ^-(X_{\mathbb C}, Z(r)) = \chi ^-(X_{\mathbb C})$ if r is even and $\chi ^+(X_{\mathbb C})$ if r is odd.

Let $\Omega = \Omega _{X_{\mathbb C}/\mathbb C}$ . The de Rham cohomology group $H_{DR}^j(X_{\mathbb C}, \mathbb C)$ has the Hodge decomposition

$$ \begin{align*}\prod _{i+k=j}H^i(X_{\mathbb C}, \Lambda^k \Omega )\end{align*} $$

which gives rise to the Hodge filtration $G_m = \prod _{k\geq m} H^{j-k}(X, \Lambda ^k \Omega )$ .

Then, $H_{DR}^j(X_{\mathbb C}, \mathbb C(r))$ is defined to be $H_{DR}^j(X_{\mathbb C}, \mathbb C)$ but with the Hodge filtration F given by $F_m = G_{m+r}$ . If M is $H^j(X_{\mathbb C}, \mathbb Z(r))$ , we define $t_M = t_{j,r}$ to be

$$\begin{align*}H^j_{DR}(X_{\mathbb C} ,\mathbb C(r))/F_0 :=\prod_{k <r}H^{j-k} (X_{\mathbb C}, \Lambda^k \Omega) = \prod_\sigma \prod_{k < r}H^{j-k}(X_\sigma, \Lambda^k \Omega).\end{align*}$$

Here, $\sigma $ runs through all embeddings of the number field K into $\mathbb C$ . and $X_\sigma = X \times _{O_K} \mathbb C$ where the map from $O_K$ to $\mathbb C$ is induced by $\sigma $ . The standard integral structure on $ t_M$ is given by

$$\begin{align*}\prod_\sigma \prod _{k < r} H^{j-k}(X, \lambda^k \Omega _{X/{O_K}}),\end{align*}$$

where $\lambda ^k$ denotes the kth derived exterior power. (See Appendix A for a discussion of derived exterior powers).

2.3 Maps between cohomology groups

Let $M = M_{j,r}$ be the motive $h^j(X_{\mathbb C}, \mathbb Q(r))$ . Let $M^* = H_B^{2d-2-j}(X_{\mathbb C}, \mathbb Z(d-1-r))$ , and $N = M^*(1) = H_B^{2d-2-j}(X_{\mathbb C}, \mathbb Z(d-r))$ . The classical period map $\beta _{j,0}$ maps $H_B(M_{j,0})_{\mathbb C}= H_B^j(X_{\mathbb C}, \mathbb C)$ to $H_{DR}(M_{j,o})_{\mathbb C} = H^j_{DR}(X_{\mathbb C}, \mathbb C)$ . Let $\pi _{j,r}$ be the projection of $H_{DR}(M)_{\mathbb C}$ onto $(t_M)_{\mathbb C}$ , and let $\alpha _{j,r} = \pi _{j,r} \circ (2\pi i)^r\beta _{j,0}$ ) from $H_B(M)_{\mathbb C}^+ = H^j_B(X, \mathbb Z(r))^+_{\mathbb C}$ to $(t_M)_{\mathbb C}$ . We now define a new map $\gamma _M$ (which we call the enhanced period map) as follows: $H_{DR}(M)$ has a decreasing Hodge filtration $F_{q'}(M)$ . Let $H^{q'} = F_{q'}/F_{q'+1}$ . Let $h_{q'}$ be the dimension of $H_{q'}$ . Then $H_{DR}(M)$ has the direct sum Hodge decomposition $\coprod H^{q'}$ . We decompose $\alpha _M= \alpha _{j,r}$ into the direct sum of the maps $\alpha ^{q'}(M)$ , where $\alpha ^{q'}$ is the map $\alpha _M$ followed by the projection onto $H^{q'}$ . Let $\Gamma $ be the usual gamma-function. Recall that the weight $w(M)$ of M is equal to $j-2r$ , Now, let $\gamma ^{q'}(M)$ be $ \Gamma ^*(-w(M) +q')) \alpha ^{q'}(M)$ and let $\gamma _{j,r}$ be the isomorphism $ \tilde \gamma _M = \coprod _{q'} \gamma ^{q'}(M)$ . Since $h_{q'} = h(p,q)$ , where $p+q=j$ and $q'=q-r$ , the determinant of $\gamma _{j,r}$ is equal to the determinant of $\alpha _{j,r}$ multiplied by $\prod _{q'} \Gamma ^*(-w(M) +q')^{h_{q'}}$ which is equal to $\prod _p\Gamma ^*(r-p)^{h(p,q)}$ , where $p+q=j$ and the product is over all p between $0$ and j.

Consider the following diagram of exact sequences:

Let $\gamma _M = q \circ \tilde \gamma _M \circ i $ . Let $\beta _M = p \circ \tilde \gamma _M^{-1} \circ j$ . Diagram-chasing immediately shows that $\tilde \gamma _M$ induces isomorphisms from Ker $\gamma _M$ to Ker $\beta _M$ and from Coker $\gamma _M$ to Coker $\beta _M$ .

Proposition 2.4. The exact sequence of complex vector spaces

(2.4.1(M)) $$\begin{align} 0 \to Ker~\gamma_M \to (H_B(M)^+ )_{\mathbb C} \to (t_M)_{\mathbb C} \to Coker~\gamma_M \to 0 \end{align}$$

is dual to the exact sequence

(2.4.2(N)) $$\begin{align} 0 \to Ker~\beta_N \to( F_0(N))_{\mathbb C} \to (H_B^-(N))_{\mathbb C} \to Coker~\beta_N \to 0. \end{align}$$

Proof. $F_0(N)_{\mathbb C}$ is the Serre dual of $(t_M)_{\mathbb C}$ . $H_B(N)^-$ may be identified with $H_B(M^*)^+$ , which is the Poincarè dual of $H_B(M)^+$ . $H_B^{2d-2}(X, \mathbb C(d-1)) $ may be canonically identified with $H_{DR}^{2d-2}(X, \mathbb C)$ . Poincarè duality is compatible with Serre duality, which implies the proposition.

We from now on choose an arbitrary basis for Ker $\gamma _M$ , the basis for Ker $\beta _M$ induced by the isomorphism between Ker $\gamma _M$ and Ker $\beta _M$ , the basis for Coker $\gamma _N$ induced by the above duality and the basis for Coker $\beta _N$ induced by the isomorphism between Coker $\gamma _N$ and Coker $\beta _N$ We will use these integral structures on the various kernels and cokernels,

If A is a finitely generated abelian group, let $A_{tor}$ denote the torsion subgroup of A and $A_{tf}$ denote $A/A_{tor}$ .

If $\phi : A \to B$ is a homomorphism of finitely generated abelian groups, let ${\phi }_{tf}$ be the induced homomorphism from $A_{tf}$ to $B_{tf}$ and let $\phi _{tor}$ be the induced homomorphism from $A_{tor}$ to $B_{tor}$ .

Lemma 2.5. Let $ 0 \to A \buildrel f \over \to B \buildrel g \over \to C \to 0$ be an exact sequence of finitely generated abelian groups. There is a natural isomorphism from Ker ${g_{tf}}/ Im f_{tf}$ to Coker $g_{tor}$ , and the determinant of $0 \to A_{tf}\to B_{tf} \to C_{tf} \to 0$ is equal to the Euler characteristic $|B_{tor}|/|A_{tor}||C_{tor}|$ of $0 \to A_{tor} \to B_{tor} \to C_{tor} \to 0$ .

Proof. Exercise

Proposition 2.6. $ \chi (2.4.2(M) ) = \chi (\tilde {\gamma _M}) \chi ( 2.4.1(M))$

Proof. Let $\Lambda $ denote highest exterior power. Let $A_1^* = H_B(M)^+$ , let $A_2^* = H_B(M)$ , and let $A_3^*= H_B(M)^-$ . Let $A_j$ be a generator of $\Lambda ((A_j^*)_{tf})$ . Let $B_1^* = F_0(M)$ , $B_2^* = H_{DR}(M)$ , and $B_3^* = t_M$ . Let $B_j$ be a generator of $\Lambda ((B_j^*)_{tf})$ .

Let $A = A_2/A_1A_3$ and $B = B_2/B_1B_3$ . Let $a_j = |(A_j^*)_{tor}|$ and $b_j = |(B_j^*)_{tor}|$ . Let $\chi (A_{tor}) = a_1a_3/a_2$ and $\chi (B_{tor}) = b_1b_3/b_2 $ . $\chi _{tor} (2.4.1(M)) = b_3/a_1$ , and $\chi _{tor} (2.4.2(M)) = a_3/b_1$ .

We have $\chi _{tor}(2.4.1(M))/\chi _{tor}(2.4.2(M)) = b_1b_3/a_1a_3$ , and

$$ \begin{align*} \frac{\chi(2.4.1(M))}{\chi(2.4.2(M))}&=\frac{det (2.4.1(M))}{det(2.4.2(M))} \frac{\chi_{tor}(2.4.2(M))}{\chi_{tor}(2.4.1(M))}\\ &= \frac{B_1B_3a_1a_3}{A_1A_3b_1b_3}\\ &= \frac{B_2~det(B)}{A_2~det(A)}\frac{\chi(A_{tor})a_2}{\chi(B_{tor})b_2}. \end{align*} $$

Lemma 2.5 implies that this equals $B_2a_2/b_2A_2$ , which is $\chi (\bar {\gamma }_M)^{-1}$ .

Beilinson defines Chern class maps from the algebraic K-theory groups to Deligne cohomology. Let $\gamma _{j,r} =\gamma _M$ , where $M=h^j(X_{\mathbb C},\mathbb Q(r))$ . In our language, Beilinson’s map becomes a map $c_{j,r}$ (for $j\leq 2r-2$ ) from $H^{j+1}_W(X, \mathbb Z(r))_{\mathbb C}$ to Coker $\gamma _{j,r}$ . Let N be the motive $M^*(1) = h^{2d-2-j}(X_{\mathbb C},\mathbb Q(d-r))$ . Since Ker $\gamma _N$ may be identified with the dual of Coker $\gamma _{2d-2-j,d-r}$ , we also have the dual $b_{j,r}$ (for $j\geq 2d-2r)$ of Beilinson’s Chern class map which maps Ker $\gamma _{2d-2-j,d-r}$ to $H^{2d-j}_W(X, \mathbb Z(d-r))_{\mathbb C}$ . We have:

Conjecture 2.7 (Beilinson)

If $j \leq 2r-3$ the maps $c_{j,r}$ and $b_{j,r}$ are isomorphisms.

Recall that by definition $H^{2r+1}_W(X, \mathbb Z(r))_{\mathbb C}$ is dual to $H^{2(d-r)}_{et}(X, \mathbb Z(d-r))_{\mathbb C}$ which is the same as $H^{2(d-r)}_{Zar}(X, \mathbb Z(d-r))_{\mathbb C}$ , that is, codimension (d-r) cycles on X modulo rational equivalence. Recall that $H^{2(d-r)}_{et}(X, \mathbb Z(d-r))^1$ denote cycles homologically equivalent to zero, and $H^{2(d-r)}_{et}(X, \mathbb Z(d-r))^2$ denote cycles modulo homological equivalence. Let $H^{2r+1}_W(X, \mathbb Z(r))^1_{\mathbb C}$ be the dual of $H^{2d-r}_{et}(X, \mathbb Z(d-r))^1_{\mathbb C}$ and $H^{2r+1}_W(X,\mathbb Z(r))^2_{\mathbb C}$ be the dual of $H^{2d-r}_{et}(X,\mathbb Z(d-r))^2_{\mathbb C}$ .

Conjecture 2.8 (Beilinson)

There is an exact sequence

$$ \begin{align*}0 \to H^{2r-1}_W(X, \mathbb Z(r))_{\mathbb C} \buildrel c_{2r-2,r} \over \to Coker (\gamma_M) \to H^{2r+1}_W(X, \mathbb Z(r))_{\mathbb C}^2 \to 0\end{align*} $$

with $M = H^{2r-2}(X, \mathbb Z(r))$ .

This is a slightly different but more natural variant of Beilinson’s original conjecture, and it is implicitly used by Fontaine [Reference Fontaine8].

Conjecture 2.9. There is an exact sequence

$$ \begin{align*}0 \to H^{2r}_W(X, \mathbb Z(r))^2_{\mathbb C} \to Ker (\gamma_N) \to H^{2r+2}_W(X, \mathbb Z(r))_{\mathbb C} \to 0.\end{align*} $$

This is the dual of Conjecture 2.8, with M replaced by

$$\begin{align*}N=M^*(1)= H^{2d-2r}(X, \mathbb Z(d-r)).\end{align*}$$

Conjecture 2.10 (Beilinson)

The Arakelov intersection pairing induces an isomorphism from $(H^{2r+1}_W(X, \mathbb Z(r))^1)_{\mathbb C}$ to $(H^{2r}_W(X, \mathbb Z(r))^1)_{\mathbb C}$ .

This is the nondegeneracy of the Arakelov intersection pairing restricted to finite cycles homologous to zero, where it is independent of metrics.

3 The statement of the conjecture

We would like to first explain the relationship between Weil-étale motivic cohomology groups and the groups which occur in Fontaine’s Deligne–Beilinson conjecture [Reference Fontaine8, Reference Fontaine and Perrin-Riou9]. We look at the motive $M =h^j(X_{\mathbb C},\mathbb Q (r))$ , Recall that $N = M^*(1)$ is $h^{2d-2-j}(X_{\mathbb C}, \mathbb Q(d-r))$ .

Fontaine starts with a projective nonsingular algebraic variety $X_0$ over Spec $\mathbb Q$ . He chooses a regular model X for $X_0$ projective and flat over Spec $\mathbb Z$ . He conjectures that the following six-term sequence is always exact:

$$ \begin{align*}0 \to H^0_f(M)_{\mathbb C} \to Ker (\gamma_M) \to H^1_c(M)_{\mathbb C} \to H^1_f(M)_{\mathbb C} \to Coker (\gamma_M) \to H^2_c(M)_{\mathbb C} \to 0.\end{align*} $$

If $j \leq 2r-2$ , Fontaine’s $H^1_f(M)$ is our $H^{j+1}_W(X, \mathbb Z(r))_{\mathbb Q} $ ; we are using motivic cohomology instead of algebraic K-theory, but these two groups agree after tensoring with $\mathbb Q$ . (Actually, Fontaine’s group is the image of $K(X)$ in $K(X_0)$ , but we conjecture that the natural map is always injective.)

If $j = 2r-1$ , Fontaine’s $H^1_f(M)$ is the group of codimension r cycles on $X_0$ homologically equivalent to zero, tensored with $\mathbb Q$ ,

If $j \geq 2r$ , $H^1_f(M) = 0$ .

Fontaine’s $H^0_f(M)$ is zero unless $j = 2r$ , in which case it equals the group of codimension r cycles on $X_0$ modulo homological equivalence, tensored with $\mathbb Q$ .

Fontaine’s $H^i_c(M)$ is the $\mathbb Q$ -dual of $H^{2-i}_f(N)$ for $i=1,2$ .

For each $ j $ and r with $j \leq min (2d-1,2r-3)$ , we will define a sequence of integral structures $A(j,r)$ . For each j and r with $j\geq max(0, 2r+1)$ , we will define a sequence of integral structures $A'(j,r)$ . $A(j,r)$ is given by:

$$ \begin{align*}(j \leq 2r-3) \quad \quad c_{j,r}: H^{j+1}_W(X, \mathbb Z(r))_{\mathbb C} \to Coker (\gamma_{j,r})\end{align*} $$

while $A'(j,r)$ is given by:

$$ \begin{align*}(j \geq 2r +1) \quad \quad b_{j,r}: Ker (\gamma_{j,r}) \to H^{j+2}_W(X, \mathbb Z(r))_{\mathbb C}.\end{align*} $$

If $1<r<d$ , we define an integral structure $C(r)$ given by: $C(r)=$

$$ \begin{align*} &0 \to H^{2r-1}_W(X, \mathbb Z(r))_{\mathbb C} \buildrel c_{2r-2,r} \over \to Coker (\gamma_{2r-2,r}) \to H^{2r+1}_W(X, \mathbb Z(r)) _{\mathbb C} \buildrel e_r \over \to\\ &\qquad\qquad\qquad\qquad\to H^{2r}_W(X, \mathbb Z(r))_{\mathbb C} \to Ker (\gamma_{2r,r}) \buildrel b_{2r,r} \over \to H^{2r+2}_W(X. \mathbb Z(r))_{\mathbb C} \to 0.\end{align*} $$

Here, $e_r$ is induced by the Arakelov intersection pairing. We give these vector spaces the standard integral structures previously defined in §2.2.

Conjectures 2.7, 2.8, 2.9 and 2.10 imply that these sequences are exact.

We give degrees to the terms of these complexes by requiring that $Ker (\gamma _M)$ has even degree and $Coker (\gamma _M) $ has odd degree. (These sequences are all truncations of modified versions of Fontaine’s six-term sequence in [Reference Fontaine8], and this convention makes the degrees agree)

Finally, we define exact sequences $B(j, r)_{\mathbb C}$ given for all j and r by

$$ \begin{align*}0 \to Ker(\gamma_M) \to (H^j_B(X_{\mathbb C}, \mathbb Z(r))^+)_{\mathbb C} \buildrel \gamma_M \over \to (t_{j,r})_{\mathbb C} \to Coker(\gamma_M) \to 0.\end{align*} $$

We put $Ker(\gamma _M)$ in degree zero.

The integral structures on the cohomology groups here are induced by the standard integral structures defined in §2.2. Let

$$\begin{align*}\chi_{A,C} (X, r) = \chi(C(r)) \prod _{j=0}^{min(2d-1,2r-3)} (\chi(A(j,r)) ^{(-1)^{j}} \prod_{j=max (0,2r+1)}^{2d-1} \chi(A'(j,r))^{(-1)^{j}},\end{align*}$$

$$\begin{align*}\chi _B(X,r) = \prod_{j=0}^{\infty}\chi(B(j,r))^{(-1)^{j}},\end{align*}$$

and let

$$\begin{align*}\chi(X,r) = \frac{\chi_{A,C}(X,r)}{\chi_B(X,r)}.\end{align*}$$

Conjecture 3.1. Give all groups in the above exact sequences their standard integral structures. Then

$$ \begin{align*}\zeta^*(X,r)= \chi (X,r)\end{align*} $$

up to sign and powers of 2.

(Note that $B(j,r)$ is torsion for $j\geq 2d-1$ and zero for j large).

If $j \neq 2r-1$ , each of the terms Ker $(\gamma _M)$ and Coker $(\gamma _M)$ occurs exactly twice in the conjecture with degrees of opposite parity, so the conjecture is independent of the choice of integral structure. If $j = 2r-1$ , Ker $(\gamma _M)$ and Coker $(\gamma _M)$ are both zero.

Proof. $\chi (C(r)) = det (C(r))/ |(C(r)_{tor}|$ . $C(r)_{tf}$ is dual to $C(d-r)_{tf}$ , so $det (C(r)) = det (C(d-r))$ . $\chi (A(j,r) = det(A(j,r))/ |A(j,r)_{tor}|$ . $\chi (A'(j,r)) = det (A'(j,r))/ | (A'(j,r)_{tor}|$ . $A(j,r)_{tf}$ is dual to $A'(2d-2-j, d-r)_{tf}$ so $det(A(j,r)) = det (A'(2d-2-j, d-r))$ and $det(A'(j,r) = det (A(2d-2-j, d-r))$ .

On the other hand,

$$\begin{align*}|C(r)_{tor}|\prod_{j=0}^{2r-3}|A(j,r)_{tor}|^{(-1)^j} \prod_{j=2r+1}^{2d-1} |A'(j,r)_{tor}|^{(-1)^j} = \prod_{j=1}^{2d+1} |H^j(X,\mathbb Z(r))_{tor}|^{(-1)^j} \end{align*}$$

which is equal by duality to

$$\begin{align*}\prod_{j=1}^{2d+1}|H^j(X, \mathbb Z(d-r))_{tor}|^{(-1)^j}\end{align*}$$

which is equal to

$$\begin{align*}|C(d-r)_{tor}|\prod _{j=0}^{2d-2r-3} |A(j, d-r)_{tor}|^{(-1)^j} \prod _{j=2d-2r+1}^{2d-1} |A'(j, d-r)_{tor}|^{(-1)^j}.\\[-48pt]\end{align*}$$

4 The Euler characteristic of the period map

Let K be the integral closure of $\mathbb Q$ in the function field of X. Then we can view X as a scheme over $S = $ Spec $O_K$ , and $X_{\mathbb C}$ is canonically isomorphic to $\prod _\sigma X \times _S$ Spec $\mathbb C$ , where the product is taken over all embeddings $\sigma $ of K in $\mathbb C$ . Let $\kappa (v)$ be the residue field of the closed point v, and let $X_v= X\times _S $ Spec $\kappa (v)$

Recall that A is the positive rational number which appears in the conjectured functional equation $A^{s/2} \Gamma (X, s) \zeta (X,s) = \pm A^{(d-s)/2}\Gamma (X, d-s) \zeta (X, d-s)$ for $\zeta (X,s)$ . Let $\omega _{X/S}$ be the relative canonical class of X over S.

(This definition is taken from [Reference Kato and Saito12]. We will not use it directly in what follows. What we need is stated in Theorems 4.2 and 4.3.)

Here, the product is taken over all closed points v of S such that X is not smooth over S at v. Note that the Chern class ${c_d}^X_{X_v}(\Omega )$ is equal to 1 if X is smooth over S at v.

Let $O_v = O_K$ localized at v. Let $X_v = X \times _{O_K} O_v$ .

Let $\lambda ^k$ denote the k-th derived exterior power.

This theorem will follow from the following propositions.

Proof. Let $\Omega = \Omega _{X_{\mathbb C}/\mathbb C}$ and $\Omega _K = \Omega _{X/{O_K}}$ , By a spectral sequence argument, $H^{j-k}(X_{\mathbb C}, \Lambda ^k \Omega ) $ is canonically isomorphic to $Ext^{j-k}_{X_{\mathbb C}} (\Lambda ^{d-1-k}\Omega , \omega _{\mathbb C})$ . So $ A_{j,k} = H^{j-k}(X, \lambda ^k\Omega _K)$ and $B_{j,k} = Ext_X^{j-k}(\lambda ^{d-1-k}\Omega _K, \omega )$ give two different integral structures on $H^{j-k}(X_{\mathbb C}, \Lambda ^k\Omega )$ . By Serre duality, $A_{j,k}$ is dual to $B_{2d-2-j,d-1-k}$ . Let $(x_{i.j})$ be a basis for $\coprod _k A_{j.k}$ , and let $(y_{i,j})$ be a basis for $\coprod _k B_{j,k}$ .

Let $d_{j,k}$ be the determinant of the identity map of $H^{j-k}(X_{\mathbb C}, \Lambda ^k\Omega )$ with respect to these two integral structures. (This is independent of the choice of basis for the integral structures, up to sign.) Then Theorem 4.3 asserts that

$$\begin{align*}\prod_j (d_{j,k})^{(-1)^{j-k}}\chi((A_{j,k})_{tor})^{-1} \chi ((B_{j,k})_{tor}) = (A')\end{align*}$$

if k is even and is equal to $(A')^{-1}$ if k is odd. By Serre duality, $\chi ((A_{j,k})_{tor})= (\chi ((B_{j,k})_{tor})^{-1}$ .

We conclude easily from this that $\prod _k \prod _j (d_{j,k})^{(-1)^j} (\chi (\lambda ^k(\Omega ))_{tor}))^{-2} = (A')^d$ , and the product on the left is the Euler characteristic of the identity map.

Let $Q_{DR}$ denote the de Rham cup product and $Q_B$ denote the cup product in singular cohomology. Let $\alpha ^j: H^j_{DR}(X_{\mathbb C}, \mathbb C) \rightarrow H^j_B(X_{\mathbb C}, \mathbb C)$ be the classical period map, We know that $Q_{DR}$ and $Q_B$ are compatible, that is, $Q_B(\alpha ^j(a) , \alpha ^k(b)) = \alpha ^{j+k}(Q_{DR}(a, b))$ .

Proposition 4.6. Let $ [v_{i,j} ]$ be a basis of $H^j_{DR}(X_{\mathbb C}, \mathbb C)$ . Let $[u_{i,j}]$ be a basis of $H^j_B(X_{\mathbb C}, \mathbb C)$ coming from a basis of $H^j_B(X_{\mathbb C}, \mathbb Z )$ modulo torsion. Let $E^j$ be the matrix of $\alpha ^j$ with respect to the bases $[ v_{i,j}]$ and $[ u_{i,j}]$ . Then

$$\begin{align*}(\prod_j (det E^j)^{(-1)^j})^2 = \prod_j det (Q_{DR}( v_{i,j}, v_{k, 2d-2-j}) )^{(-1)^j} (2\pi i)^{\chi(X_{\mathbb C})(d-1)}.\end{align*}$$

Proof. Fix j. On the top-dimensional cohomology group $H_{DR}^{2d-2}(X_{\mathbb C}, \mathbb C)$ , $\alpha (1_{DR})= (2\pi i)^{d-1} 1_B$ . Therefore, $\alpha (Q_{DR}(v_{i,j} , v_{k,2d-2-j} )) = Q_B(\alpha (v_{i,j}), \alpha (v_{k, 2d-2-j}))$ implies

$$ \begin{align*} &(2\pi i)^{(d-1) B_j} det (Q_{DR}(v_{i,j}, v_{k, 2d-2-j})) = det Q_B(\alpha(v_{i,j}) , \alpha(v_{k, 2d-2-j})) = \\ &\quad= det (E^j) det (E^{2d-2-j}) det(Q_B(u_{i,j}, u_{k, 2d-2-j}))= \pm det(E^j) det(E^{2d-2-j}). \end{align*} $$

Raising to the $(-1)^j$ power and taking products over j, we obtain the proposition.

Proof of Theorem 4.4

Recall that by definition and Serre duality $\chi (\alpha ) = \prod _j (det E^j)^{(-1)^j} \chi (H^*_{DR}(X)_{tor})^{-1}$

and $\chi _{I} = \prod d_j^{(-1)^j} \chi (H^*_{DR}(X)_{tor})^{-2}$ . Substituting x for v in Proposition 4.6 and using Proposition 4.5, we obtain

$$\begin{align*}(A')^d (\chi(H^*_{DR}(X)_{tor}))^2 = \prod_j det (Q_{DR}([x_{i,j}], [x_{k, 2d-2-j}])^{(-1)^j}.\end{align*}$$

Proposition 4.6 implies

$$\begin{align*}(A')^d (\chi(H^*_{DR}(X)_{tor})^2 = (2\pi i)^{-\chi(X_{\mathbb C})(d-1)}(\prod det(\alpha^j)^{(-1)^j})^{-2} \end{align*}$$

which implies

$$\begin{align*}(2\pi i)^{\chi(X_{\mathbb C}) (d-1)/2} (A')^{d/2} = \det (\alpha^*) \chi ((H^*_{DR}(X))_{tor})^{(-1)}.\end{align*}$$

By Poincarè duality, $\chi (H^*_B(X_{\mathbb C}, \mathbb Z)_{tor} ) = 1$ , so

$$\begin{align*}\det (\alpha^*) \chi((H^*_{DR}(X))_{tor})^{(-1)} \chi((H^*_B(X_{\mathbb C}, \mathbb Z)_{tor}) = (2\pi i)^{\chi (X_{\mathbb C})(d-1)/2 }(A')^{d/2} \end{align*}$$

$$\begin{align*}\chi(\alpha) = (2 \pi i)^{\chi(X_{\mathbb C})(d-1)/2 } (A'){d/2},\end{align*}$$

which is Theorem 4.4.

This follows from the definition of twisting by r.

Recall that $\Gamma _{r,j} = \prod _{p+q=j}\Gamma ^*(r-p)^{h(p,q)}$ . Let $\Gamma _r = \prod \Gamma _{r,j}^{(-1)^{j+1}}$ .

Corollary 4.9. The Euler characteristic $\chi (\gamma _r)$ of the enhanced period map $\gamma _r$ is

$$\begin{align*}\Gamma_r( A')^{d/2} (2\pi i)^{\chi(X_{\mathbb C})((d-1)/2 -r)}.\end{align*}$$

5 Serre’s functional equation and $\Gamma $ -function identities

Let $X_0$ be a smooth projective algebraic variety of dimension $d -1$ over the number field K. Let j be a nonnegative integer, and let $L^j(X_0,s)$ be the L-function attached by Serre in [Reference Serre19] to the j-th cohomology group of $X_0$ ,

Let $\sigma $ be an embedding of K into $\mathbb C$ , and let v be the place of K induced by $\sigma $ . Let $K_v$ be the completion of K at v. Let $X_v = X_0 \times _K \mathbb C$ , where $\sigma $ maps K into $\mathbb C$ , and let $\Omega =\Omega _{X_v/\mathbb C}$ . Recall that Hodge theory gives us a decomposition $H_{DR}^j(X_v)= \coprod H_v^{p.q}$ , where the sum is taken over pairs $(p.q)$ such that $p+q =j$ and $ H_v^{p.q} = H^q(X_v, \Omega ^p)$ . Let c be the automorphism of $X_v$ induced by complex conjugation acting on $\mathbb C/K_v$ . Then if j is even and equal to $2n$ , $c $ acts as an involution on $H_v^{n,n}.$ Let $h_v(p,q)$ be the dimension of $H_v^{p,q}$ .

Then $H_v^{n,n} = H_v^{(n,+)} \oplus H_v^{(n,-)}$ , where

$$\begin{align*}H_v^{(n,+)} = \{x \in H_v^{n,n} , c (x) = (-1)^n x\} \end{align*}$$

$$\begin{align*}H_v^{(n,-)} = \{ x \in H_v^{n,n}, c (x) = (-1)^{(n+1)} x\} \end{align*}$$

Let $h_v(n, +) = \textrm {dim}~ H_v^{(n,+)}$ and $h_v(n ,-) = \textrm {dim}~ H_v^{(n,-)}$ .

Let $B^j_v$ be the rank of $H^j(X_v, \mathbb Z)$ , and let $(B_v^j)^+$ be the rank of the subgroup of $H^j(X_v, \mathbb Z)$ left fixed by c. Let $(B_v^j)^- = B^j_v -(B^j_v)^+$ . Note that if $j = 2n$ , $(B_v^j)^+$ is equal to $\Sigma h_v(p,q) + h_v(n,+)$ if n is even and is equal to $\Sigma h_v(p,q) + h_v(n,-)$ if n is odd, where the sum is taken over all pairs $(p,q)$ where $p<q$ and $p+q =j$ . Let $(B_v^{j,r})^-$ be $(B_v^j)^-$ if r is even and $(B^j_v)^+$ if r is odd.

Let $\Gamma _{\mathbb C}(s) = (2\pi )^{-s}\Gamma (s)$ . Let $\Gamma _{\mathbb R}(s) = \pi ^{-s/2}\Gamma (s/2)$ .

Serre [Reference Serre19] gives the functional equation $\phi ^j(s) =\pm \phi (j+1-s)$ , where $\phi (s) = L^j(s) A_j^{s/2} \Gamma ^j(s)$ , $A_j$ is a certain positive integer, and $\Gamma ^j(s)$ is described as follows:

$\Gamma ^j(s) = \prod _v \Gamma _v^j(s)$ , where $\Gamma ^j_v(s) = \prod _{p+q=j}(\Gamma _{\mathbb C}(s-\textrm {inf}(p,q))^{h_v(p,q)}$ if v is a complex place of K,

$\Gamma ^j_v(s)= \Gamma _{\mathbb R}(s-n)^{h_v(n,+)} \Gamma _{\mathbb R}(s-n+1)^{h_v(n,-)}\prod _{p <j-p}\Gamma _{\mathbb C}(s-p)^{h_v(p,j-p)}$ , if v is a real place of K.

We observe that it is an easy computation that $\Gamma _v^j(s) = \Gamma _v^{2d-2-j}(s+d-j-1)$ so that with our earlier observation that at least in the smooth case $L^j(s) = L^{2d-2-j}(s+d-j-1)$ we obtain that Serre’s functional equation is equivalent to the functional equation $\phi ^j(s) = \pm \phi ^{2d-2-j}(d-s)$ .

Proof. We consider the case when j is even. (The case when j is odd is similar but simpler.) Let $j=2n$ . Fix v, and let $q = j-p$ . First look at terms where $p\neq q$ . Let $p' = d-1-p$ and $q' = d-1 - q$ , so $p' + q' = 2d-2-j$ . We have

$$\begin{align*}\frac{ \prod_{p<q}\Gamma_{\mathbb C}^*(r-p)^{h_v(p,q)}}{\prod_{p'<q'} \Gamma_{\mathbb C}^*(d -r -p')^{h_v(p',q')}} = \frac{\prod_{p<q}\Gamma_{\mathbb C}^*(r-p)^{h_v(p,q)}}{\prod _{p>q} \Gamma_{\mathbb C}^*(1-r+p)^{h_v(p,q)}} \end{align*}$$

since $h_v(p,q) = h_v(p', q')$ by Serre duality. By definition of $\Gamma _{\mathbb C}$ , this is equal to

$$\begin{align*}\frac{ \prod_{p<q} \Gamma^*(r-p)^{h_v(p,q)}}{\prod_{p>q} \Gamma^*(1-r+p)^{h_v(p,q)} }\end{align*}$$

multiplied by

$$\begin{align*}(2\pi) ^{-(\Sigma_{p<q}(h_v(p,q) (r-p)) -\Sigma_{p>q}(h_v(p,q)(1-r+p) ))) }.\end{align*}$$

This product is then equal to

$$\begin{align*}\pm \prod_{p\neq n} \Gamma^*(r-p)^{h_v(p,q)} (2\pi)^{-(\Sigma_{p<q} (h_v(p,q)( (r-p) - (1-r+j-p))))} \end{align*}$$

because of the relation $\Gamma ^*(r) = \pm \Gamma ^*(1-r)^{-1} $ for integral r which follows immediately from the functional equation for the Gamma function. We then obtain:

(5.1) $$ \begin{align}\pm \prod_{p\neq n} \Gamma^*(r-p)^{h_v(p,q)} (2\pi)^{-(B_v^j-h_v(n,n)) (r - (j+1)/2)} \end{align} $$

We now look at the terms involving n with v still fixed. We first observe that the functional equation for the gamma function implies that

$\Gamma ^*(a/2) \Gamma ^*((2-a)/2)$ equals $\pm \pi $ if a is an odd integer and equals $\pm 1$ if a is an even integer.

We compute:

$$\begin{align*}\Gamma_{\mathbb R}^*((r-n))^{h_v(n,+)} \Gamma_{\mathbb R}^*((r-n+1))^{h_v(n,-)}\end{align*}$$

multiplied by

$$\begin{align*}\Gamma_{\mathbb R}^*((d-r-(d-1-n))^{-h_v(n,+)} \Gamma_{\mathbb R}^*(d+1-r-(d-1-n))^{-h_v(n,-)}\end{align*}$$

which we rewrite as

(5.2) $$ \begin{align} (\Gamma^*((r-n)/2) (\Gamma^*(1-r+n)/2)^{-1})^{h_v(n,+)} \end{align} $$

multiplied by

(5.3) $$ \begin{align} \Gamma^*((r-n+1)/2) (\Gamma^*((n+2-r)/2)^{-1} )^{h_v(n,-)} \end{align} $$

multiplied by

(5.4) $$ \begin{align} \pi^{-( h_v(n,+)(2r-2n-1)/2 +h_v(n,-)( 2r-2n-1)/2 )}. \end{align} $$

First, assume that $r-n$ is odd. By the functional equation,

(5.5) $$ \begin{align} \Gamma^*((1-r+n)/2)^{-1} = \pm \Gamma^*(r-n+1)/2 \end{align} $$

(5.6) $$ \begin{align} \Gamma^*((n+2-r)/2)^{-1} = \pm \pi^{-1} \Gamma^*((r-n)/2). \end{align} $$

Now, recall the duplication formula for the gamma function:

(5.7) $$ \begin{align} \Gamma(2s) = 2^{2s-1}\Gamma(s) \Gamma(s + 1/2) /\sqrt \pi. \end{align} $$

Then equation (5.2) becomes (using equations (5.5) and (5.7))

$$\begin{align*}(\pm 2^{n-r+1} \Gamma^*(r-n) \sqrt \pi)^{h_v(n,+)} \end{align*}$$

and equation (5.3) becomes

$$\begin{align*}(\pm 2^{n-r} \Gamma^*(r-n) \sqrt \pi)^{-h_v(n,-)} \end{align*}$$

while equation (5.4) is

(5.8) $$ \begin{align} \pi^{-h_v(n,n)(r-(j+1)/2)}. \end{align} $$

So, up to sign and powers of 2, our product has become

(5.9) $$ \begin{align} \Gamma^*(r-n)^{h_v(n,n)} \pi^{(h_v(n,n)(r- (j+1)/2) +h_v(n,+)/2 -h_v(n,-)/2} \end{align} $$

Multiplying equation (5.1) by equation (5.9), we get

(5.10) $$ \begin{align} \prod_p \Gamma^*(r-p)^{h_v(p,q)} \pi^{-(B^j_v(r -(j+1/2))) +h_v(n,+)/2 -h_v(n,-)/2) } \end{align} $$

which equals

(5.11) $$ \begin{align} \prod_p\Gamma^*(r-p)^{h_v(p,q)} \pi^{-(B^j_v(r-j/2) +(B^j_v)^+} \end{align} $$

if n is even (so r odd) and equals

(5.12) $$ \begin{align} \prod_p\Gamma^*(r-p)^{h_v(p,q)} \pi^{-B^j_v(r-j/2) +(B^j_v)^-} \end{align} $$

if n is odd (so r even) since $(B_j -h_v(n,n)/2) +h_v(n,+)$ is equal to $(B^j_v)^+$ if n is even and $(B^j_v)^-$ if n is odd.

The proof for $r-n$ even is identical, except for switching $h_v(n,+)$ and $h_v(n, -)$ .

Proof. Let $q =j-p$ , $p'= d-1-p$ and $q'= d-1-q$ . Note that $0 \leq p, q, p', q' \leq d-1$

We write $(\Gamma ^j)^*_v(r)/(\Gamma ^{2d-2-j})^*_v(d-r) $ =

$$\begin{align*}\frac{ \prod_p \Gamma_{\mathbb C}^*(r-\textrm{inf}(p,q))^{h_v(p,q)}}{\prod_{p'} \Gamma_{\mathbb C}(d-r-\textrm{inf}(p',q'))^{h_v(p',q')}}\end{align*}$$

which is equal to

(5.13) $$ \begin{align} \frac{\prod_p\Gamma^*(r-\textrm{inf}(p,q))^{(h_v(p,q)} }{\prod_{p'}\Gamma^*(d-r -\textrm{inf} (p',q'))^{h_v(p',q')}} \end{align} $$

multiplied by

(5.14) $$ \begin{align} (2\pi)^{-\Sigma_p h_v(p.q) (r- \textrm{inf}(p,q)) -\Sigma_{p'}h_v(p',q')(d-r-\textrm{inf}(p',q'))}. \end{align} $$

Since $h_v(p',q') =h_v(p,q)$ , the functional equation for the gamma function transforms equation (5.13) into (up to sign)

(5.15) $$ \begin{align} \prod_p \Gamma^*(r-\textrm{inf}(p,q))^{h_v(p,q)} \prod_p \Gamma^*(r-sup(p,q))^{h_v(p,q)},\end{align} $$

which in turn equals

$$\begin{align*}(\prod_p\Gamma^*(r-p)^{h_v(p,q)})^2. \end{align*}$$

Now, equation (5.14) easily transforms into $ (\pi ^{2r-j-1})^{-\Sigma _p h_v(p,q)}$ , which becomes $\pi ^{-(B^j_v(2r - (j+1))}$ .

6 Compatibility of the conjecture with the functional equation

Starting with Serre’s conjectured functional equation for cohomological L-functions described in the previous section, Bloch, Kato and T. Saito conclude that the following functional equation holds for the zeta-function of X;

Here, the constant $A = A(X)$ is obtained by taking the alternating product of the constants $A_j$ which occur in the conjectured functional equation for Serre’s L-function $L_j$ and modifying it by terms coming from degenerate fibers. It is still a positive rational number.

The generic fiber $X_0$ of X is a projective algebraic variety smooth over a number field $K = H^0(X_0, O_{X_0})$ . Let $\Gamma (X, s) = \prod _j \prod _{\sigma } \Gamma (X_{v(\sigma )}^j, s)^{(-1)^j}$ , where $\Gamma _v^j$ was defined in the previous section. If we rewrite our conjecture as $\zeta ^*(X, r) = \chi (X,r)$ , then what we want to show is that $\zeta ^*(X, r)/\zeta ^*(X, d-r) = \chi (X, r)/\chi (X, d-r)$ , up to sign and powers of 2, where the left-hand side is computed by the functional equation.

Proposition 6.2. Conjecture 6.1 implies

$$\begin{align*}\frac{\zeta^*(X, r)}{\zeta^*(X, d-r)} = A^{d/2 -r}\prod_j (\prod_{p+q=j} {{\Gamma^*(r-p)}^{h(p,q)})}^{(-1)^j} \pi^{-\chi(X_{\mathbb C}) (r-(d-1)/2) +\chi^-(X_{\mathbb C}, \mathbb Z(r))}.\end{align*}$$

We now wish to compute $\chi (X, r)/\chi (X, d-r)$ and show that it agrees with the expression in Proposition 6.2. We first recall that $\chi (X, r) = \chi _{A,C}(X,r) \chi _B(X, r)$ and that $\chi _{A,C}(X,r)/\chi _{A,C}(X,d-r) = 1$ , by Proposition 3.3.

We now have to look at $\chi _B(X, r)/\chi _B(X, d-r)$ .

Lemma 6.4. Let $\theta _{j,r} $ be the Euler characteristic of the identity map from the complex vector space $t_M = \coprod _{0\leq k < r} H^{j-k}(X_{\mathbb C}, \Lambda ^k \Omega _{X_{\mathbb C}})$ with the integral structure $\coprod _{0\leq k < r} H^{j-k}(X, \lambda ^k \Omega _X )$ to the same vector space with the integral structure given by

$$\begin{align*}\underline {RHom} (H^{d-1-j+k}(X, \lambda^{d-1-k} \Omega_X), \omega).\end{align*}$$

Then $\prod _j \theta _{j,r}^{(-1)^j} = (A')^r $ .

Let $\gamma (j,r)$ denote the integral structure

$$\begin{align*}0 \to Ker~\gamma_* \to H^j_B(X, \mathbb C(r))^+ \to t(j,r)_{\mathbb C} \to Coker~\gamma_* \to 0. \end{align*}$$

Let $\beta (j,r)$ denote the integral structure

$$\begin{align*}0 \to Ker~\beta_* \to {F_0(j,r)}_{\mathbb C} \to H_B^j(X, \mathbb C(r))^- \to Coker~\beta_* \to 0.\end{align*}$$

Proposition 6.6. a) The integral structure $\beta (j,r)$ is dual to the integral structure $\delta (2d-2-j,d-r)$ given as follows:

$$\begin{align*}0 \to Ker\ \gamma_* \to (H_B^{2d-2-j}(X, \mathbb Z(d-1-r))_{\mathbb C})^- \to t(2d-2-j, d-r)_{\mathbb C} \to Coker\ \gamma_* \to 0,\end{align*}$$

where the singular cohomology group has its standard structure and t(2d-2-j, d-r) has the integral structure dual to the standard one on $F_0(j,r)$ .

b) det $(\beta (j,r)) = det (\delta (2d-2-j, d-r))$ .

Let

$$\begin{align*}\chi(\gamma(r)) = \prod_{j=0}^{2d-2} \chi(\gamma(j,r))^{(-1)^j},\quad \chi (\beta(r)) = \prod _{j=0}^{2d-2} \chi(\beta(j,r))^{(-1)^j},\end{align*}$$

and

$$\begin{align*}\chi(\epsilon (d-r)) = \prod_{j=0}^{2d-2} \chi( \epsilon (j, d-r))^{(-1)^j}.\end{align*}$$

Note that $\gamma (2d-2-j, d-r)$ is the integral structure $\delta (2d-2-j, d-r)$ except that the singular cohomology group now has the integral structure given by $H_B^{2d-2-j}(X, \mathbb Z (d-r)_{\mathbb C})^+$ and the tangent space has the standard integral structure rather than the one coming from duality.

Proposition 6.9. One has

$$\begin{align*}\frac{\chi(\gamma(r))}{\chi(\beta(r))} = \Gamma_r(A')^{d/2} (2\pi i) ^{\chi(X_{\mathbb C}) ((d-1)/2 -r) }.\end{align*}$$

Proposition 6.10. One has

$$\begin{align*}\frac{\chi(\gamma(d-r) )}{\chi(\gamma(r) )} = \Gamma_r(A')^{d/2-r}(2\pi i)^{\chi(X_{\mathbb C})((d-1)/2 -r) + \chi^-(X_{\mathbb C}, \mathbb Z(r))}.\end{align*}$$

Theorem 6.11. One has

$$\begin{align*}\frac{\chi_B(X, d-r)}{\chi_B(X, r)} = \Gamma_r(A')^{d/2-r}(2\pi i)^{\chi(X_{\mathbb C})(r-(d-1)/2 ) + \chi^-(X_{\mathbb C}, \mathbb Z(r))}.\end{align*}$$

Proposition 6.2 and Theorem 6.11 immediately imply the compatibility of our conjecture with the functional equation if we replace A by $A'$ .

7 The case of number rings

Let F be a number field with ring of integers $O_F$ , class number h, number of roots of unity w and discriminant $d_F$ . Let $X = $ Spec $O_F$ . We will explain how our conjecture for X and r reduces to standard theorems if $r=0$ or $r=1$ and well-known conjectures if $r <0$ or $r>1$ .

We begin with $r=0$ .

We know that $H^j_{et}(X, \mathbb Z(1)) = 0$ if $j < 1$ , $H^1_{et}(X, \mathbb Z(1)) = O_F^*$ , $H^2_{et}(X, \mathbb Z(1))= Pic(X)$ , $H^3(X, \mathbb Z(1)) = 0$ (up to 2-torsion) and $H^0_{et}(X, \mathbb Z(0)) = \mathbb Z$ .

It immediately follows from the definitions that $H^0_W(X, \mathbb Z(0)) = \mathbb Z$ , $H^1_W(X, \mathbb Z(0))= 0$ , and $H^3(X, \mathbb Z(0)) = \mu _F^\vee $ , the dual of the roots of unity in F.

We also have the exact sequence $0 \to Pic(X)^\vee \to H^2_W(X, \mathbb Z(0)) \to Hom(O_F^*, \mathbb Z) \to 0 $ .

We also have $t_{j,0} = 0$ for all j. It follows that $A(j,0)$ is always equal to $0$ .

We also see that $A'(j,0) = 0$ unless $j = 1$ , when $A(1,0)$ reduces to $(\mu _F)^\vee $ in degree 2, so $\chi (A'(1,0)) = w$ .

$C(0)$ becomes $0 \to \mathbb C \to \mathbb C^{r_1+r_2} \to Hom (O_F^*, \mathbb C) \to 0 $ , with the integral structure on the last term being $H^2_W(X, \mathbb Z(2))$ and the second map being the dual of the classical regulator. So $\chi (C(0))= hR$ , and $\chi (X, 0) = hR/w$ . As is well known, $\zeta ^*(X,0) = -hR/w$ .

We now consider the case when $r =1$ .

$A(j,1)$ is easily seen to be zero for $j < -1$ .

The complex part of $\mathbb C(1)$ is given by $0 \to O_F^*\otimes \mathbb C \to \mathbb C^{r_1+r_2} \to \mathbb C \to 0 $ , with the last two terms getting the standard bases and the first term a basis coming from $O_F^*$ , so $det (\mathbb C(1)$ is the classical regulator R. The Euler characteristic of $\mathbb C(1)_{tor}$ is

$$\begin{align*}|H^1_W(X, \mathbb Z(1))_{tor}|/|H^2(X,\mathbb Z(1))_{tor}| =w/h.\end{align*}$$

It follows that the Euler characteristic of $\mathbb C(1)$ is $hR/w$ .

Since $t_{j,1} = 0$ for $j \geq 0$ and $H^j_W(X, \mathbb Z(1)) = 0$ for $j \geq 5$ , $A'(j,1) = 0$ for $j \geq 3$ .

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

(7.1) $$ \begin{align} 0 \to \mathbb C^{r_2} \to O_F \otimes \mathbb C \to \mathbb C^{r_1 +r_2} \to 0.\end{align} $$

Note that we have the usual map $\theta $ mapping $O_F \otimes \mathbb C$ to $\mathbb C^{r_1+2r_2}$ by sending $x \otimes 1$ to the collection of $\sigma (x) $ as $\sigma $ runs through the embeddings of F in $\mathbb C$ . The map from $\mathbb C^{r_2}$ to $O_F \otimes \mathbb C$ is given by the natural inclusion of $\mathbb C^{r_2}$ in $\mathbb C^{r_1 +2r_2}$ multiplied by $2\pi i$ , followed by the inverse of $\theta $ . Since the determinant of $\theta $ with respect to the usual bases is $\sqrt d_F$ , we see that the determinant of equation (7.1) is equal to $(2\pi i)^{r_2}/\sqrt d_F$ . Hence, the Euler characteristic $\chi (X, 1)$ is equal to $hR (2\pi i)^{r_2}/w\sqrt d_F$ which is equal to the usual formula $hR(2\pi )^{r_2}2^{r_1}/w\sqrt |d_F|$ for $\zeta ^*(X, 1)$ up to a power of 2.

Now, let $r <0$ .

The only nonzero groups $H^k_W(X, \mathbb Z(r))$ occur when $k =2$ or $k =3$ , so $A(j,r)$ is equal to zero for all j in the appropriate range, $C(r) =0$ , and $A'(j,r) = 0$ unless $j=0$ or $j =1$ .

$B(j,r) = 0$ unless $j =0$ . $B(0,r)$ reduces to the isomorphism $Ker( b_{0,r}) \to H^2_W(X, \mathbb Z(r))^+$ so we give $Ker (b_{0,r})$ the integral structure induced from $H^2_W(X, \mathbb Z(r))^+$ , and $\chi (B(0,r)) =1$ .

$A'(0,r)_{\mathbb C}$ is given by $Ker (b_{0,r}) \to H^2_W(X,\mathbb Z(r))_{\mathbb C}$ . $H^2_W(X, \mathbb Z(r))_{\mathbb C}$ is dual to $H^1_W(X, \mathbb Z (1-r))_{\mathbb C}$ which is canonically isomorphic to $K_{1-2r}(O_F)_{\mathbb C}$ , and $Ker (b_{0,r})$ is canonically dual to $Coker (c_{0,1-r})$ , where $c_{0,1-r}$ is the Beilinson regulator map. We conclude that $det(A'(0,r))$ is dual to the determinant $D_{1-r}$ of the Beilinson regulator map with respect to the natural bases coming from singular cohomology and K-theory.

Then $\chi (X,r)$ is equal to $D_{1-r} |H^2_W(X, \mathbb Z(r))_{tor}|/|H^3_W(X, \mathbb Z(r))|$ , which is equal to $D_{1-r} |H^2_W(X, \mathbb Z(1-r))|/|H^1_W(X, \mathbb Z(1-r))_{tor}|$ which in turn is equal to

$$\begin{align*}\frac{D_{1-r} |K_{-2r}(O_F)|}{|K_{1-2r}(O_F)_{tor}|},\end{align*}$$

up to 2-torsion. This is essentially what was conjectured in [Reference Lichtenbaum16] to be $\zeta ^*(X, r)$ .

Finally, let $r> 1$ . By definition, since j has to be between $2r+1$ and $2d-1$ , there is no contribution from $A'(j,r)$

$A(j,r)_{\mathbb C}$ is the complex $H^{j+1}_W(X, \mathbb Z(r)) _{\mathbb C}\to Coker (\gamma _{j,r})$ , which is only nonzero when $j =0$ , in which case the map is the Beilinson regulator $c_{0,r}$ , Since j has to be either $0$ or $1$ , the torsion Euler characteristic is $|H^2(X, \mathbb Z(r))|/|H^1(X, \mathbb Z(r))_{tor}$ which up to $2$ -torsion is $K_{2r-2}(O_F)|/|K_{2r-1}(O_F)_{tor}|$ . So letting $R_r$ be the determinant of the Beilinson regulator map, the Euler characteristic

$$\begin{align*}\chi(A,r)= \frac{|K_{2r-2}(O_F)}{|K_{2r-1}(O_F)_{tor}|} R_r,\end{align*}$$

up to 2-torsion.

Now, $B(j,r)_{\mathbb C}$ obviously is zero if $j \neq 0$ . Let $s(r) = r_2$ If r is even and $s(r )= r_1+r_2$ if r is odd. Then by the same arguments as in the case when $r =1$ we have $det(B(0,r)_{\mathbb C} )= (2\pi i)^{rs(r)}\sqrt {d_F}$ .

The integral structure on $t_{j,r} $ is given by $\coprod _{0\leq k < r}H^{j-k}(X, \lambda ^k (\Omega ))$ In the appendix, we compute that $\lambda ^k(\Omega )$ is equal to $\Omega [1-k]$ . So $t_{j,r}= \coprod _{k<r} H^{j-k}(X,\Omega [1-k])$ . Since $\Omega $ only has cohomology in dimension $0$ , we see that $t_{j,r} =0$ unless $j =1$ , and the order of $H^0(X, \Omega )$ is $d_F$ so $\chi _{tor}(t_{j,r}) = d_F^{-r}$ and $\chi (B(j,r)) = (2\pi i)^{rs(r)} d_F^{(r-1/2)}$ .

Putting everything together, we get that our conjecture says that up to 2-torsion $\zeta (X,r) = \chi (X,r)$ , where

$$\begin{align*}\chi(X,r) = \frac{|K_{2r-2}(O_F)|}{|K_{2r-1}(O_F)_{tor}|} (2\pi i)^{rs(r)} d_F^{r-1/2}R_r \end{align*}$$

since) $H^{k-1}(X, \lambda ^k \Omega ) = H^{k-1}(X,\Omega [1-k]) = d_F$ , and $H^j(X, \lambda ^k \Omega ) = 0$ if $j \neq k-1$ . This is compatible with our conjecture for $s=1-r$ via the functional equation, which of course here is a well-known theorem.