2. The motivic Euler characteristic and related invariants

In this section, we first recall the definition of the motivic Euler characteristic from [Reference Arcila-Maya, Bethea, Opie, Wickelgren and ZakharevichABO⁺22] that forms the basis for our arithmetic curve counting before showing how it relates to three different invariants, namely, the complex compactly supported Euler characteristic, the real compactly supported Euler characteristic, and the determinant of the $\ell$ -adic cohomology.

Define $\mathbb{H} := \langle 1 \rangle + \langle -1 \rangle$ . The Witt ring, denoted by $\mathrm{W}(k)$ , is the quotient $\widehat{\mathrm{W}}(k)/(\mathbb{H})$ . We say that an element $q \in \widehat{\mathrm{W}}(k)$ is hyperbolic if it lies in the ideal generated by $\mathbb{H}$ , which is the same as the abelian subgroup consisting of elements of the form $n\mathbb{H}$ for $n \in \mathbb{Z}$ .

For a non-degenerate quadratic form q, the rank of q is the dimension of the underlying vector space, denoted by rank(q). It extends uniquely to a ring homomorphism $\mathrm{rank}:\widehat{\mathrm{W}}(k)\to\mathbb Z$ . Write I for the kernel of this homomorphism. The subset $T\subset\widehat{\mathrm{W}}(k)$ of torsion elements is also an ideal, and we let $J=I^2\cap T$ .

Definition 2.2. Let X be a smooth, irreducible projective variety over k of dimension n. Let $H_{\rm dR}^*(X/k)$ denote the de Rham cohomology of X over k. Then the composition of the cup-product square $a \mapsto a \cup a$ and the trace map $\mathrm{Tr}:H_{\rm dR}^*(X/k)\to k$ defines a quadratic form on the vector space $H_{\rm dR}^{2*}(X/k)$ , which we may see is non-degenerate using Poincaré duality. Let $[H_{\rm dR}^{2*}(X/k)]$ denote the class of this quadratic form in $\widehat{\mathrm{W}}(k)$ , and set

\begin{align*}\chi^{\rm dR}(X)=[H_{\rm dR}^{2*}(X/k)]-\frac{1}{2}\bigg(\sum_{i\geq0}\mathop{\mathrm{dim}}\nolimits(H_{\rm dR}^{2i+1}(X/k))\!\bigg)\cdot\mathbb H\in\widehat{\mathrm{W}}(k),\end{align*}

which we will call the de Rham Euler characteristic of X.

In this paper, we will often work with the de Rham Euler characteristic due to its compatibility with results of Saito from [Reference SaitoSai94]. However, we may compare the de Rham Euler characteristic to other Euler characteristics taking values in $\widehat{\mathrm{W}}(k)$ .

Consider the trace map $\mathrm{Tr}: H^d(X, \Omega^d_{X/k}) \to k$ . We can define a cohomologically bounded complex of k vector spaces $\mathrm{Hdg}(X/k)$ such that differentials are all zero and the vector space in degree n is given by $\bigoplus_{j=i+n} H^i(X, \Omega^j_{X/k})$ . The cup product composed with the trace map then furnishes a non-degenerate symmetric bilinear form Tr on $\mathrm{Hdg}(X/k)$ , as with the de Rham Euler characteristic. This therefore defines a class in the $0{\text{th}}$ Grothendieck–Witt group of the exact category of cohomologically bounded cochain complexes of k vector spaces in the sense of [Reference SchlichtingSch10, § 2.2]. Since cochain complexes of k vector spaces are always split, we can identify this group with $\widehat{\mathrm{W}}(k)$ . We will write $\chi^{\rm Hdg}(X)$ to mean the element $(\mathrm{Hdg}(X/k), \mathrm{Tr}) \in \widehat{\mathrm{W}}(k)$ constructed above.

The following elegant proof was suggested to us by one of the referees of this paper. There is an alternative, more complicated, proof of this statement based on an argument by Speyer [Reference SpeyerSpe15], by closely studying the Hodge–de Rham spectral sequence.

Let $i_j:\mathrm{Spec}(k)\to\mathbb A^1_K$ be the inclusion at the points $j=0,1$ . By the proper base-change theorem, we have an equality in the Grothendieck–Witt ring of k,

\begin{align*}\widehat{\mathrm{W}}(\mathrm{Ch}^b\mathrm{Coh}(\mathcal{O}_{\mathrm{Spec}(k)})) \ni i_1^* Rp_{2*}(p_1^*\Omega^*_{X/k}, t\cdot d)=\bigoplus_n H^n_{\rm dR}(X)[-n] = \chi^{\rm dR}(X) \in \widehat{\mathrm{W}}(k),\end{align*}

with cup product and trace form to k. Over $t=0$ , the differential vanishes and so

\begin{align*}\widehat{\mathrm{W}}(\mathrm{Ch}^b\mathrm{Coh}(\mathcal{O}_{\mathrm{Spec}(k)})) \ni i_0^* Rp_{2*}(p_1^*\Omega^*_{X/k}, t\cdot d)= \bigoplus_{p,q}H^p(X,\Omega^q)[-p-q] = \chi^{\rm Hdg}(X) \in \widehat{\mathrm{W}}(k),\end{align*}

again with cup product and trace form to k, where we note that there is a potential even shift in the grading in the second complex, but this does not affect the class in $\widehat{\mathrm{W}}(k)$ . Since the Hermitian K-theory groups are representable by a spectrum in the stable motivic homotopy category of k by [Reference Panin and WalterPW19, Theorem 1.1], they are $\mathbb A^1$ -invariant. Therefore, $\widehat{\mathrm{W}}(k) = KO^{[0]}_0(\mathrm{Spec}(k)) = KO^{[0]}_0(\mathbb{A}^1_k)$ , and $i^*_0, i^*_1$ induce identifications $KO^{[0]}_0(\mathbb{A}^1_k) = \widehat{\mathrm{W}}(k)$ with $i_0^* = i_1^*$ . In particular, there is an equality in $\widehat{\mathrm{W}}(k)$ :

\begin{align*}\chi^{\rm dR}(X) = i_0^*(Rp_{2*}(p_1^*\Omega^*_{X/k}, t\cdot d))=i_1^*(Rp_{2*}(p_1^*\Omega^*_{X/k}, t\cdot d)) = \chi^{\rm Hdg}(X),\end{align*}

as required.

Definition 2.5. In [Reference LevineLev20, Remark 2.1 (2)], the categorical Euler characteristic $\chi^{\rm cat}(X)$ is defined for an arbitrary algebraic variety X over k by using the infinite suspension spectrum $\Sigma_T^\infty X_+$ as an object of $\mathrm{SH}(k)$ , the stable motivic homotopy category over k. Since $\Sigma_T^{\infty} X_+$ is dualisable, it gives us a canonical endomorphism of the unit object $\mathbb S_k$ :

\begin{align*}\chi^{\rm cat}(X)\in\mathrm{End}_{\mathrm{SH}(k)}(\mathbb S_k).\end{align*}

By Morel’s theorem [Reference MorelMor02, Theorem 6.4.1], we can identify $\mathrm{End}_{\mathrm{SH}(k)}(\mathbb S_k)\cong\widehat{\mathrm{W}}(k)$ when k has characteristic 0, and $\mathrm{End}_{\mathrm{SH}(k)}(\mathbb S_k)\cong\widehat{\mathrm{W}}(k)[\frac1p]$ when k has positive odd characteristic.

One of the main theorems of [Reference Levine and RaksitLR20] is the following.

Assume that $\mathrm{char}(k)=0$ for the rest of the paper, unless otherwise stated.

A motivic measure is a ring homomorphism out of $K_0(\mathrm{Var}_k)$ .

The above Euler characteristic does not give us a motivic measure; see Remark 2.10. However, [Reference Arcila-Maya, Bethea, Opie, Wickelgren and ZakharevichABO⁺22, Theorem 2.13] gives the following.

Theorem 2.9. There is a unique ring homomorphism $\chi^{\rm mot}:K_0(\mathrm{Var}_k)\to\widehat{\mathrm{W}}(k)$ such that for every smooth, irreducible, projective variety $X/k$ ,

\begin{align*}\chi^{\rm mot}([X])=\chi^{\rm Hdg}(X) = \chi^{\rm dR}(X).\end{align*}

A key ingredient in the proof of the above is Bittner’s presentation of the ring of varieties [Reference BittnerBit04, Theorem 3.1], which tells us that the underlying abelian group of $K_0(\mathrm{Var}_k)$ can be generated by smooth projective varieties, subject to only a smooth blow-up excision relation. In particular, the above theorem requires us to restrict to characteristic 0.

For the rest of the paper, we will write $\chi^{\rm mot}(X)$ for $\chi^{\rm mot}([X])$ for every variety X over k and call this the motivic Euler characteristic of X. This is a slight abuse of notation, since $\chi^{\rm mot}$ depends on k. When the base field is ambiguous, we will write $\chi^{\rm mot}_k$ ; however, for the most part we will omit the subscript for readability.

Definition 2.11. Suppose that the dimension n of X is even. The composition of the cup product and the trace map $\mathrm{Tr}:H_{\rm dR}^{2n}(X/k)\to k$ defines a non-degenerate quadratic form on $H_{\rm dR}^{n}(X/k)$ . Let $[H_{\rm dR}^{n}(X/k)]\in\widehat{\mathrm{W}}(k)$ denote the class of this quadratic form. Define integers $b^+$ and $b^-$ by

\begin{align*}b^{-}&:=\sum_{i\lt n}\mathop{\mathrm{dim}}\nolimits_k H_{\rm dR}^{i}(X/k),\\b^+ &:= \sum_{i \lt n} (-1)^i \mathop{\mathrm{dim}}\nolimits_k H_{\rm dR}^{i}(X/k).\end{align*}

Lemma 2.12. Suppose that X has even dimension n. Then

\begin{align*}\chi^{\rm dR}(X)=b^{+}\cdot\mathbb H+[H_{\rm dR}^{n}(X/k)].\end{align*}

If X has odd dimension, then $\chi^{\rm dR}(X) = (b^+ -\frac{1}{2}\mathrm{dim}_k(H^n_{\rm dR}(X/k)))\cdot\mathbb{H}$ .

We are interested in recovering invariants related to quadratic forms from the Grothendieck–Witt ring. The motivic Euler characteristic encodes classical invariants on varieties, the first being the compactly supported Euler characteristic of the complex points.

Proof of Theorem 2.14. For X a projective variety over $\mathbb{C}$ , this is proven in [Reference LevineLev20, Remark 2.3 (1)]. The general case can be reduced to this case as follows. The assignment $X\mapsto e(X)$ gives rise to a ring homomorphism $ K_0(\mathrm{Var}_k) \to\mathbb Z$ . Since $K_0(\mathrm{Var}_k)$ is generated by smooth projective varieties by [Reference BittnerBit04, Theorem 3.1], it is enough prove the claim for X smooth and projective. Let X be such a variety and let $F\subset k$ be a subfield that is finitely generated over $\mathbb{Q}$ such that X is already defined over F, i.e. there is a smooth projective variety Y over F whose base change to k is X. Fix an embedding of F into $\mathbb C$ , which exists since F is countable. Using the naturality of the categorical Euler characteristic with respect to base change, and the invariance of the rank function on the Grothendieck–Witt ring under base change, we obtain that

\begin{align*}\mathrm{rank}(\chi^{\rm mot}(X))=\mathrm{rank}(\chi^{\rm mot}(Y\times_Fk))=\mathrm{rank}(\chi^{\rm mot}(Y))=\mathrm{rank}(\chi^{\rm mot}(Y\times_F\mathbb C)).\end{align*}

By proper base change for our choice of Weil cohomology theory, we have

\begin{align*}e(X)=e(Y\times_Fk)=e(Y)=e(Y\times_F\mathbb C).\end{align*}

Applying [Reference LevineLev20, Remark 2.3 (1)] gives $\mathrm{rank}(\chi^{\rm mot}(Y\times_F\mathbb C))=e(Y\times_F\mathbb C)$ , as required.

As well as the complex points of our variety, the motivic Euler characteristic allows us to recover information about the real points of our variety and natural generalisations of this.

Definition 2.16. Assume that k is a real closed field. Then, by Sylvester’s law of inertia, every non-degenerate quadratic form q over k is isomorphic to a diagonal form

\begin{align*}\underbrace{\langle1\rangle\oplus\langle1\rangle\oplus\cdots\oplus\langle1\rangle}_m\oplus\underbrace{\langle-1\rangle\oplus\cdots\oplus\langle-1\rangle}_n,\end{align*}

and $m-n$ only depends on the isomorphism class of q. This is called the signature of q and we write $\mathrm{sign}(q)$ . It induces an isomorphism $\mathrm{sign}:\mathrm{W}(k)\to \mathbb Z.$

Now let k again be any field whose characteristic is not 2.

Definition 2.18. For a topological space X and a ring A, let $\mathcal C(X,A)$ denote the ring of continuous A-valued functions, where we give A the discrete topology. For $\lt$ in $\mathrm{Spr}(k)$ , let $k_\lt$ denote the real closure of the ordered field $(k,\lt)$ . For every non-degenerate quadratic form q over k, let $\mathrm{sign}(q):\mathrm{Spr}(k)\to\mathbb Z$ be the function sending each $\lt$ in $\mathrm{Spr}(k)$ to the signature of the base change of q to $k_\lt$ . This is continuous with respect to the Harrison topology, so we obtain a ring homomorphism,

\begin{align*}\mathrm{sign}:\mathrm{W}(k)\to\mathcal C(\mathrm{Spr}(k),\mathbb Z),\vphantom{_{\big(}}\end{align*}

where we note that if $\mathrm{Spr}(k)=\emptyset$ , then $\mathcal C(\mathrm{Spr}(k), \mathbb{Z}) = \{0\}$ , so sign is the trivial homomorphism. By slight abuse of notation, let sign $:\widehat{\mathrm{W}}(k)\to\mathcal C(\mathrm{Spr}(k),\mathbb Z)$ denote the composition of the quotient map $\widehat{\mathrm{W}}(k)\to\mathrm{W}(k)$ with the map sign above.

As we have already remarked, for every quasi-projective variety X over k, the set X(k) has the structure of a semi-algebraic set over k. Let $S_\bullet(X(k))$ denote the simplicial set of semi-algebraic singular simplexes of the semi-algebraic space X(k) of k-valued points of X as defined in [Pál22, Definition 2.5]. The assignment $X\mapsto S_\bullet(X(k))$ induces a functor $S_\bullet$ from the category of smooth quasi-projective varieties $Sm/k$ to $s\mathbf{Set}$ , the category of simplicial sets. Let SH denote the stable homotopy category of simplicial sets. The functor $S_\bullet$ gives rise to a realisation functor $\mathrm{Re}_B:\mathrm{SH}(k)\to\mathrm{SH}$ that for $X\in Sm/k$ sends the suspension spectrum $\Sigma_T^\infty X_+$ to the suspension spectrum $\Sigma^{\infty}(S_\bullet(X(k)))$ . This induces a map $\sigma:\mathrm{End}_{\mathrm{SH}(k)}(\mathbb S_k)\to\mathrm{End}_{\mathrm{SH}}(\mathbb S_k)\cong\mathbb Z$ . We claim that $\sigma$ is the signature map. It is enough to show that $\langle-1\rangle$ goes to $-1$ . Using the naturality of the constructions, we only need to check this for any real closed subfield $L \subseteq k$ . By taking L to be the algebraic closure of $\mathbb{Q}$ in k, assume without loss of generality that L is the real closure $\mathbb{Q}^{\rm real}$ of $\mathbb{Q}$ . Since $\mathbb R$ contains $\mathbb{Q}^{\rm real}$ , we may use naturality again to reduce to the case $k=\mathbb R$ . In this case, [Reference LevineLev20, Remark 2.3 (2)] applies, since the forgetful map from the simplicial set of semi-algebraic singular simplexes to the usual simplicial set of continuous singular simplexes is a homotopy equivalence by [Pál22, Theorem 2.20(ii)].

Remark 2.21. By Pfister’s local–global principle (see [Reference LamLam05, § VIII, Theorems 3.2, 4.9, and 6.9]), the induced map

\begin{align*}\mathrm{sign}\otimes\mathrm{id}_{\mathbb Z[\frac{1}{2}]}:\mathrm{W}(k)\otimes\mathbb Z[\tfrac{1}{2}]\to\mathcal C(\mathrm{Spr}(k),\mathbb Z[\tfrac{1}{2}])\end{align*}

is an isomorphism. Therefore, the invariants rank and sign compute the Grothendieck–Witt ring up to torsion, so the motivic Euler characteristic recovers both the algebraically closed Euler characteristic in the form of the rank as well as the real closed Euler characteristic in the form of the signature. Conversely, the motivic Euler characteristic, up to torsion, can be computed from these invariants. The full motivic Euler characteristic contains interesting torsion information that cannot be recovered from these invariants. We turn our attention to some torsion information about $\chi^{\rm mot}$ that also relates to a more classical invariant.

Definition 2.22. Assume that k is a field whose characteristic is not 2. Then every non-degenerate quadratic form q is isomorphic to a diagonal form

\begin{align*}\langle a_1\rangle\oplus\langle a_2\rangle\oplus\cdots\oplus\langle a_n\rangle.\end{align*}

The discriminant disc(q) of the form q above is defined as the class of the product

\begin{align*}a_1\cdot a_2\cdots a_n\;\mathrm{in}\;k^\times/k^{\times2}.\end{align*}

This is independent of the diagonalisation of q, since it is the determinant of the symmetric matrix defining the bilinear form associated to q. It induces a homomorphism of the underlying abelian groups

\begin{align*}\mathrm{disc}:\widehat{\mathrm{W}}(k)\to k^\times/k^{\times2}.\end{align*}

We will freely use the identification $k^\times/k^{\times2} = \mathop{\mathrm{Hom}}\nolimits(\mathop{\mathrm{Gal}}\nolimits_k, \mathbb{Z}/2\mathbb{Z})$ .

For X a variety over k, we would like to relate $\mathrm{disc}(\chi^{\rm mot}(X))$ to other invariants on X.

Definition 2.23. Let $\ell$ be a prime number different from the characteristic of k, let U be a smooth scheme over k, and let $\mathcal{F}$ be a smooth $\ell$ -adic sheaf on U. Following Saito in [Reference SaitoSai94], we will consider the one-dimensional $\ell$ -adic representation of $\mathop{\mathrm{Gal}}\nolimits_k$ given by the determinant

\begin{align*}\mathop{\mathrm{det}}\nolimits R\Gamma_c(U_{\overline{k}}, \mathcal{F}):=\bigotimes_i \mathop{\mathrm{det}}\nolimits H^i_c(U_{\overline{k}}, \mathcal{F})^{\otimes(-1)^i}.\end{align*}

The above representation is called the determinant of cohomology of $\mathcal F$ , and for $\mathcal{F}=\mathbb{Q}_{\ell}$ , we will call it the $\ell$ -adic determinant of cohomology of X. This is a one-dimensional $\ell$ -adic representation, so we will view it as an element of the abelian group $\mathop{\mathrm{Hom}}\nolimits(\mathop{\mathrm{Gal}}\nolimits_k, \mathbb{Q}_{\ell}^\times)$ . For $\mathcal{F}=\mathbb{Q}_{\ell}$ , we will write $\mathrm{det}_\ell(U)$ to mean $\mathop{\mathrm{det}}\nolimits R\Gamma_c(U_{\overline{k}}, \mathbb{Q}_\ell)$ .

Let X be a smooth proper variety over k of dimension n. As shown in [Reference SaitoSai94, Lemma 5], Poincaré duality allows us to obtain the identity

\begin{align*}\mathop{\mathrm{det}}\nolimits R\Gamma(X_{\overline{k}},\mathbb{Q}_{\ell})=\mathbb{Q}_{\ell}({-}\tfrac{1}{2}ne(X))\otimes \kappa(X),\end{align*}

where $\kappa(X)$ is a character of order at most 2 of $\mathop{\mathrm{Gal}}\nolimits_k$ , which is independent of $\ell$ . Suppose that n is odd. As in [Reference SaitoSai94, Lemma 5], the existence of a non-degenerate alternating form on $H^n_{\acute{e}t}(X_{\overline{k}},\mathbb{Q}_{\ell})$ given by the cup product allows us to obtain that $\mathrm{det}( H^n_{\acute{e}t}(X_{\overline{k}}, \mathbb{Q}_{\ell}))=\mathbb{Q}_{\ell}( -j)$ where $j=\frac12 \mathrm{dim}_{\mathbb{Q}_{\ell}} (H^n_{\acute{e}t}(X_{\overline{k}}, \mathbb{Q}_{\ell}))$ . In particular, $\kappa(X)=1$ if $\mathrm{dim}(X)$ is odd.

Theorem 2.24. For every smooth projective variety X over k of dimension n, we have

\begin{align*}\mathrm{disc}(\chi^{\rm cat}(X))=\mathrm{disc}(\chi^{\rm dR}(X))=(-1)^{(1/2) n \cdot e(X)}\cdot\kappa(X).\end{align*}

If n is odd, then e(X) is even, so $\frac12 n \cdot e(X)$ is always a positive integer. For X a K3 surface, this formula gives $\mathrm{disc}(\chi^{\rm mot}(X))=\kappa(X)$ . This result is essentially equivalent to a theorem of Saito from [Reference SaitoSai94]. First, we are going to formulate this as Saito did. The main result of Saito’s paper [Reference SaitoSai94] linking the determinant of cohomology with de Rham cohomology is the following.

Theorem 2.25 (Saito’s theorem [Reference SaitoSai94]). Assume that X has even dimension n. Then

\begin{align*}\kappa(X)=(-1)^{(1/2)e(X)+b^{-}}\cdot\mathrm{disc}([H_{\rm dR}^{n}(X/k)]).\end{align*}

Proof of Theorem 2.24. By Lemma 2.12, the result is trivial if $\mathrm{dim}(X)$ is odd, since we have $\chi^{\rm dR}(X) = \frac12 e(X) \cdot \mathbb{H}$ . Therefore, suppose that $\mathrm{dim}(X)$ is even.

We can work with $\chi^{\rm dR}$ instead of $\chi^{\rm cat}$ , by Theorem 2.6. For all $q,q' \in \widehat{\mathrm{W}}(k)$ , we have that the discriminant $\mathrm{disc}(q+q')=\mathrm{disc}(q)\cdot\mathrm{disc}(q')$ . Applying the map disc to both sides of the equation in Lemma 2.12 therefore gives

\begin{align*}\mathrm{disc}(\chi^{\rm cat}(X))=\mathrm{disc}(\chi^{\rm dR}(X))&=\mathrm{disc}(\mathbb H)^{b^{+}}\cdot\mathrm{disc}([H_{\rm dR}^{n}(X/k)])\\&=(-1)^{b^{-}}\cdot\mathrm{disc}([H_{\rm dR}^{n}(X/k)]),\end{align*}

where we use that $b^{-}\equiv b^{+}\!\!\mod2$ . The claim follows by Theorem 2.25.

We want to extend Theorem 2.24 to all varieties by replacing $\chi^{\rm dR}$ with $\chi^{\rm mot}$ , and we do this by showing that the terms in Theorem 2.24 extend to motivic measures. Assume that $\mathrm{char}(k)=0$ .

Proof. Let X be a variety over k, let $j:U\to X$ be an open immersion, and let $i:Z\to X$ be the closed immersion, which is the complement of j. We claim that $\mathrm{det}_{\ell}(X)=\mathrm{det}_{\ell}(U)\cdot\mathrm{det}_{\ell}(Z)$ . Consider the short exact sequence of sheaves on X:

\begin{align*}0 \to j_!(\mathbb{Q}_{\ell}) \to\mathbb{Q}_{\ell} \to i_*(\mathbb{Q}_{\ell}) \to 0.\end{align*}

This short exact sequence gives rise to a distinguished triangle:

\begin{align*}R_c\Gamma(X_{\overline{k}},j_!(\mathbb{Q}_{\ell}))\to R_c\Gamma(X_{\overline{k}}, \mathbb{Q}_{\ell})\to R_c\Gamma(X_{\overline{k}},i_*(\mathbb{Q}_{\ell}))\to R_c\Gamma(X_{\overline{k}},j_!(\mathbb{Q}_{\ell}))[1],\end{align*}

so taking the determinant gives

\begin{align*}\mathop{\mathrm{det}}\nolimits R_c\Gamma(X_{\overline{k}}, \mathbb{Q}_{\ell})=\mathop{\mathrm{det}}\nolimits R_c\Gamma(X_{\overline{k}},j_!(\mathbb{Q}_{\ell}))\cdot\mathop{\mathrm{det}}\nolimits R_c\Gamma(X_{\overline{k}},i_*(\mathbb{Q}_{\ell})).\end{align*}

Since $j_!$ is exact, we have $\mathop{\mathrm{det}}\nolimits R_c\Gamma(U_{\overline{k}},\mathbb{Q}_{\ell})=\mathop{\mathrm{det}}\nolimits R_c\Gamma(X_{\overline{k}},j_!(\mathbb{Q}_{\ell}))$ , and the proper base change theorem gives $\mathop{\mathrm{det}}\nolimits R_c\Gamma(Z_{\overline{k}}, \mathbb{Q}_{\ell})= \mathop{\mathrm{det}}\nolimits R_c\Gamma(X_{\overline{k}},i_*(\mathbb{Q}_{\ell}))$ , as required.

Definition 2.28. Since $b_i(X)=0$ for $i\gt2n$ for every smooth and projective of dimension n, every formal linear combination $\sum_{i\in\mathbb N}c_ib_i$ where $c_i \in \mathbb{R}$ is also well defined and gives rise to a unique homomorphism of abelian groups $K_0(\mathrm{Var}_k) \to\mathbb R$ . When X is smooth and projective of dimension n, then

\begin{align*}\sum_{i\in\mathbb N}\frac{(-1)^ii}{2}b_i&=\frac{(-1)^nn}{2}b_n+\sum_{i\lt n}\bigg(\frac{(-1)^ii}{2}b_i+\frac{(-1)^{2n-i}(2n-i)}{2}b_{2n-i}\bigg)\\&=\frac{(-1)^nn}{2}b_n+\sum_{i\lt n}\bigg(\frac{(-1)^ii}{2}b_i+\frac{(-1)^{i}(2n-i)}{2}b_i\bigg)\\&=\frac{n}{2}\bigg((-1)^nb_n+\sum_{i\lt n}(-1)^i2b_i\bigg)=\frac{e(X)n}{2},\end{align*}

using Poincaré duality. This gives a unique homomorphism of abelian groups $w: K_0(\mathrm{Var}_k)\to\mathbb R$ extending the function $w(X) = [e(X)n]/2 $ on smooth projective varieties over k, which we call the total weight. Since $[{e(X)n}]/{2}$ is an integer for X smooth and projective, [Reference BittnerBit04, Theorem 3.1] tells us that w(X) is always an integer.

Theorem 2.29. For every algebraic variety X over k, we have

\begin{align*}\mathrm{disc}(\chi^{\rm mot}(X))(-1)^{w(X)}=\mathrm{det}_{\ell}(X)\cdot\mathbb{Q}_{\ell}(w(X)) \in\mathop{\mathrm{Hom}}\nolimits(\mathop{\mathrm{Gal}}\nolimits_k,\mathbb{Q}_{\ell}^\times)._{\vphantom{\big(}}\end{align*}

Definition 2.37. Let $\chi_{\ell}$ and $\chi_w$ denote the maps

\begin{align*}\chi_{\ell}: K_0(\mathrm{Var}_k)\to E(\mathrm{Hom}(\mathop{\mathrm{Gal}}\nolimits_k,\mathbb{Q}_{\ell}^\times))\quad\mathrm{and}\quad\chi_w: K_0(\mathrm{Var}_k) \to E(\mathbb Z)\end{align*}

given by

\begin{align*}[X]\mapsto(e(X),\mathrm{det}_{\ell}(X))\quad\mathrm{and}\quad[X]\mapsto(e(X),w(X)),\end{align*}

respectively. We will call $\chi_{\ell}$ and $\chi_w$ the augmented determinant of cohomology and the augmented total weight, respectively.

Proposition 2.38. The maps

\begin{align*}\chi_{\ell}: K_0(\mathrm{Var}_k)\to E(\mathrm{Hom}(\mathop{\mathrm{Gal}}\nolimits_k,\mathbb{Q}_{\ell}^\times))\quad\textrm{and}\quad\chi_w: K_0(\mathrm{Var}_k)\to E(\mathbb Z)\end{align*}

are ring homomorphisms.

Proof. We may assume without loss of generality that k is finitely generated over $\mathbb{Q}$ . Define $s:E(\mathrm{Hom}(\mathop{\mathrm{Gal}}\nolimits_k,\mathbb{Q}_{\ell}^\times))\to E(\mathrm{Hom}(\mathop{\mathrm{Gal}}\nolimits_k,\mathbb{Q}_{\ell}^\times))$ to be the ring homomorphism induced by the group homomorphism $\mathrm{Hom}(\mathop{\mathrm{Gal}}\nolimits_k,\mathbb{Q}_{\ell}^\times)\to\mathrm{Hom}(\mathop{\mathrm{Gal}}\nolimits_k,\mathbb{Q}_{\ell}^\times)$ induced by the squaring map $\mathbb{Q}^\times_{\ell}\to\mathbb{Q}^\times_{\ell}$ . Let $c:E(\mathbb Z)\to E(\mathrm{Hom}(\mathop{\mathrm{Gal}}\nolimits_k,\mathbb{Q}_{\ell}^\times))$ be the ring homomorphism induced by the unique group homomorphism $\mathbb Z\to\mathrm{Hom}(\mathop{\mathrm{Gal}}\nolimits_k,\mathbb{Q}_{\ell}^\times)$ such that the image of $-1$ is the cyclotomic character. Then $c\circ\chi_w=s\circ\chi_{\ell}$ . Since c is injective, we obtain that $\chi_w$ is a ring homomorphism if $\chi_{\ell}$ is. Since $\chi_{\ell}$ is a group homomorphism, we only need that for any X,Y, we have

\begin{align*}\mathrm{det}_{\ell}(X\times Y)=\mathrm{det}_{\ell}(X)^{\otimes e(Y)}\otimes\mathrm{det}_{\ell}(Y)^{\otimes e(Y)}.\end{align*}

Using the Künneth formula,

\begin{align*}\mathrm{det}_{\ell}(X\times Y)&=\bigotimes_n\mathop{\mathrm{det}}\nolimits H^n_c(X_{\overline{k}}\times Y_{\overline{k}},\mathbb{Q}_{\ell})^{\otimes(-1)^n}\\&=\bigotimes_n\bigotimes_{i+j=n}(\mathop{\mathrm{det}}\nolimits H^i_c(X_{\overline{k}},\mathbb{Q}_{\ell})\otimes H^j_c(Y_{\overline{k}},\mathbb{Q}_{\ell}))^{\otimes(-1)^n}\\&=\bigotimes_{i,j}\mathop{\mathrm{det}}\nolimits H^i_c(X_{\overline{k}},\mathbb{Q}_{\ell})^{\otimes(-1)^{i+j}b_j(Y)}\otimes\mathop{\mathrm{det}}\nolimits H^j_c(Y_{\overline{k}},\mathbb{Q}_{\ell})^{\otimes(-1)^{j+i}b_i(X)}\\&=\bigotimes_i\mathop{\mathrm{det}}\nolimits H^i_c(X_{\overline{k}},\mathbb{Q}_{\ell})^{\otimes(-1)^ie(Y)}\otimes\bigotimes_j\mathop{\mathrm{det}}\nolimits H^j_c(Y_{\overline{k}},\mathbb{Q}_{\ell})^{\otimes(-1)^je(X)},\end{align*}

so the claim is now clear.

Proof. Since $\chi_{\ell}(X) = \chi_{\ell}(Y) \chi_{\ell}(Z)$ , we have $e(X) = e(Y)e(Z)$ . This allows us to say the following:

\begin{align*}\chi_{\ell}(X)=\chi_{\ell}(Y)\cdot\chi_{\ell}(Z)&\Longleftrightarrow\mathrm{det}_{\ell}(X)=\mathrm{det}_{\ell}(Y)^{\otimes e(Z)}\otimes\mathrm{det}_{\ell}(Z)^{\otimes e(Y)},\\ \chi_w(X)=\chi_w(Y)\cdot\chi_w(Z)&\Longleftrightarrow w(X)=e(Z)w(Y)+e(Y)w(Z),\\ \chi^{\rm mot}(X)=\chi^{\rm mot}(Y)\cdot\chi^{\rm mot}(Z)\!\!\!\!\mod I^2&\Longleftrightarrow\\ \mathrm{disc}(\chi^{\rm mot}&(X))=\mathrm{disc}(\chi^{\rm mot}(Y))^{e(Z)}\cdot\mathrm{disc}(\chi^{\rm mot}(Z))^{e(Y)},\end{align*}

using that all three augmented rings $E(\mathrm{Hom}(\mathop{\mathrm{Gal}}\nolimits_k,\mathbb{Q}_{\ell}^\times)),E(\mathbb Z)$ , and $\mathbf W(k)$ are elementary. By Theorem 2.29, it is enough to show that $\chi_{\ell}(X)=\chi_{\ell}(Y)\cdot\chi_{\ell}(Z)$ implies $\chi_w(X)=\chi_w(Y)\cdot\chi_w(Z)$ . Let c and s be the homomorphisms from the proof of Proposition 2.38. Since $c\circ\chi_w=s\circ\chi_{\ell}$ and c is injective, the latter is clear.

The main reason why it is easier to work with the $\ell$ -adic determinant of cohomology rather than the determinant is that we may relate the $\ell$ -adic determinant of cohomology to the zeta functions of reductions of the variety over finite fields. For the purposes of this paper, it is often helpful to have a formulation of the $\ell$ -adic determinant of cohomology of a variety in terms of $\zeta$ -functions of the reduction of the variety to finite fields.

Since U is of finite type over $\mathrm{Spec}(\mathbb{Z})$ , if s is a closed point of U, then the residue field $\mathbb{F}_s$ of s is finite. Write $\mathbb F_{s^n}$ to be the unique extension of $\mathbb F_s$ of degree n. Consider the variety $\mathcal{X}_s/\mathbb{F}_s$ . Let $\zeta(\mathcal X_s, t) :=\mathrm{exp}\big( \sum_{n \geq 1} \# \mathcal X_s(\mathbb F_{s^n})t^n \big)$ .

By the proof of the Weil conjectures (see, for example, [Reference PoonenPoo17, § 7.5.7]), we have

\begin{align*}\zeta(\mathcal X_s, t) =\prod_i\mathop{\mathrm{det}}\nolimits(\mathrm{Id}-t\cdot\mathrm{Frob}_s|_{H^i_c(\overline{\mathcal{X}_s}, \mathbb{Q}_{\ell})})^{(-1)^i},\end{align*}

where $\mathrm{Frob}_s$ denotes the geometric Frobenius at s acting on the compactly supported $\ell$ -adic cohomology, and $\ell$ is any prime different from the characteristic of $\mathbb F_s$ .

We claim that the $\ell$ -adic determinant of cohomology of X is determined by the $\zeta$ -functions of the reductions $\mathcal{X}_s$ . In order to prove this, we first need some linear algebra.

Notation 2.41. Let $V=\bigoplus_iV_i$ be a finite-dimensional graded vector space over a field L of characteristic 0, let $\phi:V\to V$ be an invertible L-linear map respecting the grading, and define

\begin{align*}\mathrm{Tr}(\phi) :=\sum_i(-1)^i\mathrm{Tr}(\phi|_{V_i}),\quad\mathop{\mathrm{det}}\nolimits(\phi):=\prod_i\mathop{\mathrm{det}}\nolimits(\phi|_{V_i})^{(-1)^i}.\end{align*}

Define $e(V) := \mathrm{Tr}(\mathrm{Id}_V)$ . Finally, define

\begin{align*}\zeta(\phi,t)=\prod_i\mathop{\mathrm{det}}\nolimits(\mathrm{Id}_{V_i}-t\cdot\phi|_{V_i})^{(-1)^i}.\end{align*}

Lemma 2.42. We have

\begin{align*}\zeta(\phi,t)=\mathrm{det}(\phi)(-t)^{e(V)}\cdot u\quad \text{where }u\in1+L[[t^{-1}]].\end{align*}

Proof. Since $\phi$ is the direct sum of the restrictions $\phi|_{V_i}$ , and the terms $\zeta(\phi,t),\mathrm{det}(\phi)$ and $(-t)^{e(V)}$ are all multiplicative with respect to direct sums, we may assume without loss of generality that only one graded piece of V is non-zero. Shifting the grading by j changes these terms by raising them to the $(-1)^j$ th power, and the operation of raising to the $(-1)^j$ th power preserves power series in $L[[t^{-1}]]$ , so we may also assume that $V=V_0$ . Note that

\begin{align*}\mathop{\mathrm{det}}\nolimits(\mathrm{Id}-t\cdot\phi) = (-t)^{\mathrm{dim}(V)} \cdot \mathop{\mathrm{det}}\nolimits(\phi - t^{-1}\mathrm{Id}) = (-t)^{\mathrm{dim}(V)} c_\phi(t^{-1}),\end{align*}

where $c_{\phi}$ denotes the characteristic polynomial of $\phi$ , and so $c_{\phi}(t^{-1}) = \mathrm{det}(\phi) + t^{-1}u'$ for some polynomial $u' \in L[t^{-1}]\subseteq L[[t^{-1}]]$ . Since $\phi$ is invertible, we may relabel $u'' = ({1}/{\mathrm{det}{\phi}})u'$ and obtain

\begin{align*}\mathop{\mathrm{det}}\nolimits(\mathrm{Id}-t\cdot\phi) = (-t)^{\mathrm{dim}(V)} \cdot \mathrm{det}(\phi) \cdot (1 + t^{-1}u''),\end{align*}

which gives the result.

Proposition 2.43. Let X be a quasi-projective variety over a finite field $\mathbb F$ and let $\ell$ be a prime different from $\mathrm{char}(\mathbb F)$ . Then

\begin{align*}\zeta(X,t)=\mathrm{det}_{\ell}(\mathrm{Frob}_{\mathbb{F}}) \cdot (-t)^{\chi(X)}\cdot u,\quad u\in1+\mathbb{Q}_{\ell}[[t^{-1}]],\end{align*}

where $\chi(X)$ is the compactly supported Euler characteristic of X, $\mathrm{Frob}_{\mathbb{F}} \in\mathrm{Gal}(\overline{ \mathbb F}/\mathbb F)$ is the geometric Frobenius, and $\mathrm{det}_{\ell}(X)$ is the one-dimensional representation given by the determinant of cohomology of X as in Definition 2.23.

We will often prove facts about the motivic Euler characteristic by proving results separately for the rank, signature, and discriminant, then combining them to obtain an element in $\widehat{\mathrm{W}}(k)/J$ . This is made possible by the following lemma.

Corollary 2.46. There is an injective ring homomorphism

\begin{align*}\mathrm{rank} \times \mathrm{sign} \times E_{\rm disc}:\widehat{\mathrm{W}}(k)/J \to \mathbb Z \times\mathcal{C}(\mathrm{Spr}(k), \mathbb{Z}) \times E(\mathrm{Hom}(\mathop{\mathrm{Gal}}\nolimits_k, \mathbb Z/2\mathbb Z)).\end{align*}

3. Galois descent for the motivic Euler characteristic

In this section, we give a detailed account of a Galois-descent-type result for $\chi^{\rm mot}$ . Fix $K/k$ a finite Galois extension with Galois group $\mathop{\mathrm{Gal}}\nolimits(K/k)$ . If k has positive characteristic, assume further that $[K:k]$ is coprime to $\mathrm{char}(k)$ .

where the vertical arrows denote the structure morphism $X \to\mathrm{Spec}(K)$ . A morphism in $\mathrm{Var}_{K, \rm desc}$ , $f: (X, \varphi) \to (X', \varphi')$ is a morphism of K-varieties $f: X \to X'$ that is compatible with the descent data in the sense that the obvious diagram commutes. We say that a morphism is an open (respectively, closed) immersion if the underlying morphism of varieties is an open (respectively, closed) immersion. This category also has a unit object with respect to the monoidal structure induced by the product: namely, the variety $\mathrm{Spec}(K)$ equipped with the identity as the structure morphism, with the only possible descent data. We will write $(\mathrm{Spec}(K), \mathrm{Id})$ for this unit object. This allows us to form the Grothendieck ring of K varieties with descent data, $K_0(\mathrm{Var}_{K, \rm desc})$ , which is the ring generated by symbols $[(X, \varphi)]$ , where $(X,\varphi)$ is a K variety with descent data, subject to the relations:

1. (1) $[(X, \varphi)] = [(X', \varphi')]$ if $(X,\varphi)$ and $(X', \varphi')$ are isomorphic;

2. (2) $[(X, \varphi)] = [(U, \varphi|_U)] + [( X \setminus U, \varphi_{X\setminus U})]$ for $(U, \varphi_U) \hookrightarrow (X, \varphi)$ an open immersion, where the descent data on $X \setminus U$ are given by the restriction of the descent data on X; and

3. (3) $[(X, \varphi)] [(X', \varphi')] = [( X \times_K X', \varphi\times \varphi')]$ for any pair of K varieties with descent data.

By Galois descent (see, for example, [Reference PoonenPoo17, Corollary 4.4.6]), there is an equivalence of categories $\mathrm{Var}_{K, \rm desc} \cong \mathrm{Var}_k$ . This equivalence of categories preserves the unit object in the category since it sends $(\mathrm{Spec}(K), \mathrm{Id})$ to $\mathrm{Spec}(k)$ , where $(\mathrm{Spec}(K), \mathrm{Id})$ . If $(X, \varphi)$ is a K variety with descent data, write $\mathrm{des}_{K/k}(X,\varphi)$ to mean the associated k variety, and similarly if $Y/k$ is a variety, write $(Y_K, \phi_Y)$ to mean the associated K variety with descent data.

This equivalence of categories preserves open and closed immersions (see [Reference PoonenPoo17, Theorem 4.3.7]), so it induces an isomorphism $K_0(\mathrm{Var}_{K, \rm desc}) \cong K_0(\mathrm{Var}_k)$ given on the level of quasi-projective varieties by sending $[(X, \varphi)] \mapsto [\mathrm{desc}_{K/k}(X, \varphi)]$ , with inverse $[X] \mapsto [(X_K, \phi_X)]$ .

We would like to apply a Galois-descent-style result for the motivic Euler characteristic. In order to do this, we need a Galois-descent result for quadratic forms.

Write $\varphi: \mathop{\mathrm{Gal}}\nolimits(K/k) \to\mathrm{Aut}_k(V)$ for the group homomorphism defining this semilinear action. We will write $(q, \varphi)$ to denote the quadratic form on V along with the semilinear $\mathop{\mathrm{Gal}}\nolimits(K/k)$ action on V induced by $\varphi$ . Let $\widehat{\mathrm{W}}(K)^{\rm desc}$ denote the ring generated by the symbols $[(q, \varphi)]$ , where $[(q, \varphi)]$ is a Galois-compatible quadratic form over K, subject to the relations:

1. (1) $[(q,\varphi)] = [(q',\varphi')]$ if there is an isomorphism between the underlying vector spaces of q,q’ that is compatible with the quadratic forms and the semilinear $\mathop{\mathrm{Gal}}\nolimits(K/k)$ actions;

2. (2) $[(q, \varphi)] + [(q', \varphi')] = [(q \oplus q', \varphi \oplus \varphi')]$ , where $q \oplus q'$ (respectively, $\varphi \oplus \varphi'$ ) represents the natural quadratic form (respectively, semilinear $\mathop{\mathrm{Gal}}\nolimits(K/k)$ action) on $V \oplus V'$ ; and

3. (3) $[(q, \varphi)]\cdot[(q',\varphi')] = [(q \otimes_K q', \varphi\otimes_K \varphi')]$ , where $q \otimes_K q'$ (respectively, $\varphi\otimes_K \varphi'$ ) represents the natural quadratic form (respectively, semilinear $\mathop{\mathrm{Gal}}\nolimits(K/k)$ action) on $V \otimes_K V'$ .

For $a \in k^\times$ , write $\langle a \rangle$ to mean the Galois-compatible quadratic form on K given by $x \mapsto ax^2$ , and write $\mathbb{H} := \langle 1 \rangle + \langle -1 \rangle \in\widehat{\mathrm{W}}(K)^{\rm desc}$ .

Proof. This follows by Galois descent for vector spaces. If $q: V \rightarrow k$ is a quadratic form over k, we obtain a Galois-compatible quadratic form over K,

\begin{align*}q \otimes Id_K: V \otimes_k K \rightarrow K,\end{align*}

where $\varphi: \mathop{\mathrm{Gal}}\nolimits(K/k) \to \mathrm{Aut}_k(V \otimes_k K)$ is given by $\tau \mapsto \mathrm{Id} \otimes \tau$ .

If $(q', \varphi) : V' \rightarrow K$ is a Galois-compatible quadratic form over K, and $v' \in V'^{\mathop{\mathrm{Gal}}\nolimits(K/k)}$ , then for all $\tau \in \mathop{\mathrm{Gal}}\nolimits(K/k)$ , we see that $\tau(q'(v')) = q'(\tau(v')) = q'(v')$ , so $q'(v') \in k$ . Restricting to the Galois-invariant subspace gives us a quadratic form over k:

\begin{align*}q'|_{V'^{\mathop{\mathrm{Gal}}\nolimits(K/k)}}: V'^{\mathop{\mathrm{Gal}}\nolimits(K/k)} \rightarrow k.\end{align*}

We claim that $q'|_{V'^{\mathop{\mathrm{Gal}}\nolimits(K/k)}}$ is non-degenerate. Let $B: V' \otimes_K V' \to K$ be the underlying bilinear form, and let $v \in V'^{\mathop{\mathrm{Gal}}\nolimits(K/k)}$ . Since B’ is non-degenerate, pick $w \in V'$ such that $B(v,w)=1$ . Let $\hat{w} = \frac{1}{[K:k]} \sum_{\tau \in \mathop{\mathrm{Gal}}\nolimits(K/k)} \tau(w)$ . Then $\hat{w} \in V'^{\mathop{\mathrm{Gal}}\nolimits(K/k)}$ , and $B'(v,\hat{w}) = 1$ , so $B'|_{V'^{\mathop{\mathrm{Gal}}\nolimits(K/k)}}$ is non-degenerate. Finally, note that the functors $q \mapsto q \otimes Id_K$ and $q' \mapsto q'|_{V'^{\mathop{\mathrm{Gal}}\nolimits(K/k)}}$ are quasi-inverse, which completes the proof.

We claim that Galois descent for varieties is compatible with a Galois-descent result for the Galois-compatible quadratic forms on their de Rham cohomology. This is made exact in the following.

Lemma 3.5. Let X be a smooth projective k variety. Then there is an isomorphism of graded vector spaces with a semilinear $\mathrm{Gal}(K/k)$ action:

\begin{align*}H^{*}_{\rm dR}(X/k) \otimes_k K \cong H^{*}_{\rm dR}(X_K/K).\end{align*}

Let $\cup_k$ denote the cup product on $H^{*}_{\rm dR}(X/k)$ , and let $\cup_K$ denote the cup product on $H^{*}_{\rm dR}(X_K/K)$ . Let $\alpha, \alpha' \in H^*_{\rm dR}(X/k)$ and $\lambda, \lambda'$ in K. Then, under this isomorphism,

\begin{align*}(\alpha \otimes \lambda) \cup_K (\alpha' \otimes \lambda') = (\alpha \cup_k \alpha') \otimes \lambda \lambda'\end{align*}

and $Tr_{X_K}(\alpha \otimes \lambda) = \lambda Tr_X(\alpha)$ .

Proof. By the Künneth formula, there is an isomorphism of K vector spaces,

\begin{align*}H^*_{\rm dR}(X/k) \otimes_k H^*_{\rm dR}(\mathrm{Spec}(K)/k) \cong H^*_{\rm dR}(X_K/K),\end{align*}

which respects the cup product and the trace map. Moreover, we see that we can identify $H^*_{\rm dR}(\mathrm{Spec}(K)/k) \cong K$ . The left-hand side of the above isomorphism has a canonical semilinear action of $\mathrm{Gal}(K/k)$ on it, given by $\tau(x \otimes \alpha) = x \otimes \tau(\alpha)$ . The right-hand side has a left $\mathrm{Gal}(K/k)$ action on it, induced by the right action on $X_K$ . We claim that these two actions are the same, which is enough to prove the theorem.

First, consider $\mathrm{Spec}(K)$ as a k variety. The $\mathrm{Gal}(K/k)$ action on $\mathrm{Spec}(K)$ gives us a left action of $\mathrm{Gal}(K/k)$ on $H^*_{\rm dR}(K/k)\cong K$ , which corresponds to the standard action of $\mathrm{Gal}(K/k)$ on K. For $\tau \in \mathop{\mathrm{Gal}}\nolimits(K/k)$ , write $\hat{\tau}$ for the automorphism of $X_K$ given by the commutative diagram

where $\hat{\tau}$ is the morphism induced by the universal property of the fibre product. Write $p^*$ , $q^*$ , $\tau^*$ , and $\hat{\tau}^*$ for the images of the morphisms p, q, $\tau$ and $\hat{\tau}$ after applying the contravariant functor, $H^*_{\rm dR}(-/k)$ . For $x \in H^*_{\rm dR}(X)$ and $y \in K$ , we have that

\begin{align*}p^*(x) &= x \otimes 1 , \\q^*(y) &= 1 \otimes y , \\ \tau^*(y) &= \tau(y).\end{align*}

Therefore, the map $\hat{\tau}^*: H^*_{\rm dR}(X_K/k) \rightarrow H^*_{\rm dR}(X_K/k) \otimes_k K$ is $x \otimes y \mapsto x \otimes \tau(y)$ .

Note that $H^*_{\rm dR}(X_K/k)$ is the image of $H^*_{\rm dR}(X_K/K)$ under the forgetful functor from K vector spaces to k vector spaces, and the map $\hat{\tau}^*$ is the map on $H^*_{\rm dR}(X_K/K)$ induced by the semilinear $\mathop{\mathrm{Gal}}\nolimits(K/k)$ action. This gives that $H^*_{\rm dR}(X_K/k) \cong H^*_{\rm dR}(X) \otimes_k K$ , as required.

Theorem 3.8. There is a unique ring homomorphism $\chi^{\rm mot, desc}_k: K_0(\mathrm{Var}_{K, \rm desc}) \to \widehat{\mathrm{W}}(K)^{\rm desc}$ such that if $(X, \varphi)$ is a smooth projective variety, then

\begin{align*}\chi^{\rm mot, desc}_k( [(X,\varphi)]) = [H^{2*}_{\rm dR}(X/K)] - [H^{2*+1}_{\rm dR}(X/K)].\end{align*}

Proof. Define $\chi^{\rm mot, desc}$ to be the composition $K_0(\mathrm{Var}_{K, \rm desc}) \cong K_0(\mathrm{Var}_k) \xrightarrow{\chi^{\rm mot}} \widehat{\mathrm{W}}(k) \to \widehat{\mathrm{W}}(K)^{\rm desc}$ . The above corollary guarantees that this satisfies the desired property. Uniqueness is also immediate by the above corollary, since if $\varphi$ is any other morphism as above, then the composition

\begin{align*}K_0(\mathrm{Var}_k) \to K_0(\mathrm{Var}_{K, \rm desc}) \xrightarrow{\varphi} \widehat{\mathrm{W}}(K)^{\rm desc} \cong \widehat{\mathrm{W}}(k)\end{align*}

is such that for a smooth projective variety X,

\begin{align*}[X] \mapsto [H^{2*}_{\rm dR}(X/k)] - \tfrac12 \mathrm{dim}_k H^{2*+1}_{\rm dR}(X/k) \cdot \mathbb{H} = \chi^{\rm dR}(X),\end{align*}

so the composition is equal to $\chi^{\rm mot}$ . As noted in Theorem 2.9, the morphism $\chi^{\rm mot}$ is unique with respect to this property, so $\chi^{\rm mot, desc}$ will also be unique.

4. The motivic Euler characteristic and transfer maps

Let $L/k$ be a finite separable field extension. We can define various transfer maps between the rings $K_0(\mathrm{Var}_L)$ and $K_0(\mathrm{Var}_k)$ , given by base change, forgetting the base, and Weil restriction, and analogously for $\widehat{\mathrm{W}}(L)$ and $\widehat{\mathrm{W}}(k)$ by base change, the trace transfer, and Rost’s norm transfer from [Reference RostRos03]. In this section, we show that the motivic Euler characteristic is compatible with these transfer maps.

Lemma 4.2. The following diagram is commutative.

Since $X/k$ is smooth and projective, there is a canonical isomorphism $H^{2n}_{\rm dR}(X/k) \cong H^n(X, \Omega^n_{X/k})$ via the cycle class map. The trace map $H^n(X, \Omega^n_{X/k}) \to k$ is defined to be the map given by the image of the identity under the isomorphism

\begin{align*}Rp_{X/k,*} R\mathcal{H}\mathrm{om}_{\mathcal{O}_X}(\Omega_{X/k}, \Omega_{X/k})^0 \cong H^n(X, \Omega^n_{X/k})^\vee\end{align*}

induced by Grothendieck–Verdier duality (see [Reference HartshorneHar66, Chapter III, Corollary 11.2(f)]). In particular, we see that the trace map commutes with arbitrary base extension by [Reference HartshorneHar66, Corollary 11.2(c)], so the trace map $\mathrm{Tr}_{X/L}$ is given by $\mathrm{Tr}_{X/k} \otimes_k \mathrm{Id}_L$ , which gives us the result.

We now focus on two other natural transfer maps, which are maps $\widehat{\mathrm{W}}(L) \to \widehat{\mathrm{W}}(k)$ for quadratic forms and maps $K_0(\mathrm{Var}_L) \to K_0(\mathrm{Var}_k)$ for varieties.

Write $\mathrm{disc}(L/k) := \mathrm{disc}(\mathrm{Tr}_{L/k}(\langle 1 \rangle))$ . This is not misleading: for $k=\mathbb{Q}$ , this agrees with the discriminant of the number field L. For $w \in \widehat{\mathrm{W}}(k)$ , we have $\mathrm{rank} (\mathrm{Tr}_{L/k}(w)) = [L:k] \mathrm{rank}(w)$ , so $\mathrm{Tr}_{L/k}$ is not a ring homomorphism.

Analogously to the above, if we have a variety $X \xrightarrow{p} \mathrm{Spec}(L)$ , we can consider X as a variety over k with structure map $X \xrightarrow{p} \mathrm{Spec}(L) \to \mathrm{Spec}(k)$ . This induces a homomorphism of abelian groups $K_0(\mathrm{Var}_L)\to K_0(\mathrm{Var}_k)$ , though not a ring homomorphism.

As in Remark 2.36, identify $\widehat{\mathrm{W}}(k)/I^2 = \mathbf{W}(k)$ with $E(\mathop{\mathrm{Hom}}\nolimits(\mathop{\mathrm{Gal}}\nolimits_k, \mathbb{Z}/2\mathbb{Z}))$ . Let $E_{\rm disc}$ denote the quotient map

\begin{align*}E_{\rm disc}:\widehat{\mathrm{W}}(k) &\to E(\mathop{\mathrm{Hom}}\nolimits(\mathop{\mathrm{Gal}}\nolimits_k, \mathbb{Z}/2\mathbb{Z})),\\w &\mapsto (\mathrm{rank}(w), \mathrm{disc}(w)).\end{align*}

Lemma 4.4. Let $w \in \widehat{\mathrm{W}}(L)$ , such that $E_{\rm disc}(w) = (\alpha, \beta)$ . Then $\mathrm{Tr}_{L/k}(w)$ satisfies

\begin{align*}E_{\rm disc}(\mathrm{Tr}_{L/k}(w)) = ([L:k]\alpha,\mathrm{disc}(L/k)^{\alpha}N_{L/k}(\beta))._{\vphantom{\big(}}\end{align*}

Proof. The statement about the rank is immediate, so we only need to show this for the discriminant. Since both disc and $\mathrm{Tr}_{L/k}$ are abelian group homomorphisms, it is enough to show the statement on generators of $\widehat{\mathrm{W}}(L)$ , so assume that $w=\langle \beta \rangle$ . Let $m_{\beta}$ be the matrix defining the k linear map on L given by multiplication by $\beta$ , so that $\mathrm{det}(m_\beta) = N_{L/k}(\beta)$ . Let A be a symmetric matrix representing the symmetric k-bilinear form on L given by $(x,y) \mapsto \mathrm{Tr}_{L/k}(xy)$ for all $x, y \in L$ . Then A is a symmetric matrix such that $\mathrm{det}(A) = \mathrm{disc}(L/k)$ . We obtain a matrix representing the quadratic form $\mathrm{Tr}_{L/k}(\langle \beta \rangle)$ , given by $Am_\beta$ . The result follows since

\begin{eqnarray*}\mathrm{disc}(\mathrm{Tr}_{L/k}(\langle \beta \rangle)) =\mathrm{det}(Am_\beta) = \mathrm{det}(A)\mathrm{det}(m_\beta) =\mathrm{disc}(L/k)N_{L/k}(\beta).\\[-36pt]\end{eqnarray*}

Lemma 4.5. The following diagram is commutative.

After applying the cycle class isomorphism $H^{2n}_{\rm dR}(X/k)\cong H^n(X, \Omega^n_{X/k})$ , we see that the trace map $H^n(X, \Omega^n_{X/k}) \to k$ is the image of the identity under the isomorphism

\begin{align*}Rp_{X/k,*}\mathcal{H}\mathrm{om}_{\mathcal{O}_X}(\Omega_{X/k}, \Omega_{X/k})^0 \cong H^n(X, \Omega^n_{X/k})^\vee\end{align*}

induced by Grothendieck–Verdier duality, and similarly if we replace k by L as above. By [Reference HartshorneHar66, Chapter 3, Corollary 11.2(e)], it is implied that the image of the identity under the isomorphism

\begin{align*}Rp_{L/k,*}\mathcal{H}\mathrm{om}_{L}(\Omega_{L/k}, \Omega_{L/k})^0 \cong H^n(\mathrm{Spec}(L), \Omega^n_{L/k})^\vee\end{align*}

is given by the standard trace map $\mathrm{Tr}_{L/k}$ . Since $p_{L/k}$ is étale, $p^!_{L/k}=p^*_{L/k}$ is the left adjoint of $Rp_{L/k,*}$ , and since it is finite, $Rp_{L/k,*}=p_{L/k,*}$ . Moreover, since $p^!$ is right adjoint to $p_*$ , we have canonical isomorphisms $p^!_{X/k}(\mathcal{O}_k) = \Omega^n_{X/k}[-n]$ , and similarly when replacing k with L. Similarly, we see that

\begin{align*}p^!_{X/k}(k) = p^!_{X/L}p^!_{L/k}(k) = p^!_{X/L}( p^*_{L/k}(k)) = p^!_{X/L}(L).\end{align*}

Using this, we have the following commutative diagram, in which all of the arrows are isomorphisms.

Again by [Reference HartshorneHar66, Chapter 3, Corollary 11.2(e) and (f)], the isomorphism $\Phi$ takes a morphism $f \in \mathop{\mathrm{Hom}}\nolimits_L( H^n(X, \Omega^n_{X/L}), L)$ to $\mathrm{Tr}_{L/k} \circ f \in \mathop{\mathrm{Hom}}\nolimits_k(H^n(X, \Omega^n_{X/L}), k)$ . Consider the identity element $\mathrm{Id} \in Rp_{X/k,*} {R}\mathcal{H}\mathrm{om}_{\mathcal{O}_X}(p^!_{X/k}(k), p^!_{X/k}(k))^0$ . Going clockwise around the above diagram maps the element Id to $\mathrm{Tr}_{L/k} \circ \mathrm{Tr}_{X/L}$ . However, going counterclockwise maps Id to $\mathrm{Tr}_{X/k}$ , giving the desired equality.

We now turn our attention to the norm and Weil restriction transfer maps. This first requires us to use the machinery of polynomial maps: while the transfer maps defined by the trace and forgetful functor are group homomorphisms, the transfer maps that we obtain by looking at Weil restrictions and norms have a different structure.

Definition 4.6. Let B be an abelian group, let A an abelian monoid, and let $P:A\to B$ be a function on the underlying sets. We say that P is a polynomial map of degree $\leq -1$ if $P=0$ . For $n\geq0$ , we say that P is a polynomial map of degree $\leq n$ if, for each $x\in A$ , the map

\begin{align*}\Delta_xP:A&\to B , \\ \Delta_xP(y)&=P(y+x)-P(y)\end{align*}

is a polynomial map of degree $\leq n-1$ . We say that P is a polynomial map if it is a polynomial map of degree $\leq n$ for some n. Note that P is a polynomial map of degree $\leq 0$ if and only if P is constant.

Now suppose that A and B are commutative rings. A polynomial map is called multiplicative if $P(1)=1$ and if for every $x,y \in A$ , we have $P(xy)=P(x)P(y)$ . Note that a multiplicative polynomial map of degree $\leq 1$ is either identically 1 or a ring homomorphism.

Polynomial maps can be extended from monoids to rings by the following.

The statement about the degree of the polynomial map is not in the statement of [Reference JoukhovitskiJou00, Proposition 1.6]; however, it follows from the proof.

For every finite separable extension $K/F$ of fields, let $\mathrm{N}_{K/F}:K\to F$ denote the norm map. The following proposition also serves as a definition of the norm map.

This rest of this section is dedicated to proving Theorem 4.22, which says that this norm polynomial map computes the motivic Euler characteristic of the Weil restriction of a variety. In order to prove this, we first need some smaller lemmas.

For the second part of the lemma, we proceed by induction on the degree of the polynomial map, with the case that f has degree $\leq 0$ being trivial, since f is then identically 1. We compute

\begin{align*}\Delta_x(fg)(y) &= g(x+y)f(x+y) - g(y)f(y)\\&= g(x)f(x+y) + g(y)f(x+y) - g(y)f(y) \\&= g(x)f(x+y) + g(y) (\Delta_xf(y)).\end{align*}

By induction, we have that $g \cdot (\Delta_xf)$ is a polynomial map of degree $\leq n$ . Note that we have a polynomial map $y \mapsto g(x) f(x+y)$ of degree $\leq n$ , since g(x) is constant, and $f(x+y) = \Delta_xf + f$ is a polynomial map of degree $\leq n$ by the first part of the lemma. Therefore, $\Delta_x(fg)$ is polynomial of degree $\leq n$ , so fg is polynomial of degree $\leq n+1$ . The fact that fg is multiplicative is immediate.

The next few results are dedicated to showing that the Weil restriction functor from L to k gives rise to a polynomial map $K_0(\mathrm{Var}_L)\to K_0(\mathrm{Var}_k)$ .

If $K/k$ is a finite separable field extension with $L \subseteq K$ , we have a $\mathop{\mathrm{Gal}}\nolimits_k$ equivariant map $I_K \to I_L$ given by restriction, and the fibres of this map have cardinality $[K:L]$ . From here, fix $L/k$ a finite field extension, let K be the Galois closure of L in $\overline{k}$ , and let $G=\mathop{\mathrm{Gal}}\nolimits(K/k)$ . If H is the subgroup of G such that $L=K^H$ , then $I_L = G/H$ as a G set.

This defines a semilinear action of G on $\mathrm{Ind}^K_LX$ , which we call $\varphi_G$ . This in turn gives us a variety $\mathrm{des}_{K/k}(\mathrm{Ind}^K_LX, \varphi_G)$ by Galois descent. By [Reference WeilWei12, Theorem 1.3.2], the variety $\mathrm{des}_{K/k}(\mathrm{Ind}^K_LX, \varphi_G)$ is isomorphic to $\mathrm{Res}_{L/k}(X)$ .

In order to show that maps into $K_0(\mathrm{Var}_k)$ are polynomial maps, it is helpful to note the following, which is probably well known.

By Lemma 4.7, it is enough to show that the assignment $\mathrm{Res}_{L/k}: [X] \mapsto [(\mathrm{Ind}^K_LX, \varphi_G)]$ satisfies the conditions to be a polynomial map whenever X is projective. That is, we need to show that for any projective varieties $Y_0, \ldots, Y_n$ , we have $\Delta_{Y_0} \Delta_{Y_1} \ldots \Delta_{Y_n} \mathrm{Res}_{L/k}([X]) = 0$ . Consider $\Delta_Y\mathrm{Res}_{L/k}: [X] \mapsto [(\mathrm{Ind}^K_L(X \amalg Y), \varphi_G)] - [(\mathrm{Ind}^K_LX, \varphi_G)]$ . Note that $\mathrm{Ind}^K_LX$ embeds as a closed subvariety into $\mathrm{Ind}^K_L(X \amalg Y)$ in a way that is compatible with the descent data on both varieties. Therefore, we can write $\Delta_Y \mathrm{Res}_{L/k}([X]) = [(X_Y, \varphi_G)]$ , where $(X_Y, \varphi_G)$ is the K variety with descent data given by the complement of $(\mathrm{Ind}^K_L(X), \varphi_G)$ in $(\mathrm{Ind}^K_L(X \amalg Y), \varphi_G)$ .

Therefore, $\Delta_Y \mathrm{Res}_{L/k}([X])$ is represented by the class of an actual K variety with descent data $(X_Y, \varphi_G)$ and so gives us a k variety $\mathrm{des}_{K/k}(X_Y, \varphi_G)$ . Iterating this process, if $Y_0, \ldots, Y_n$ are L varieties, we can write

\begin{align*}\Delta_{Y_0} \Delta_{Y_1} \ldots \Delta_{Y_n} \mathrm{Res}_{L/k} ([X]) = [(X_{Y_0, \ldots, Y_n}, \varphi_G)],\end{align*}

where $(X_{Y_0, \ldots, Y_n}, \varphi_G)$ is a K variety with descent data. We claim that $[(X_{Y_0, \ldots, Y_n}, \varphi_G)] = 0$ . Note that $[(X_{Y_0, \ldots, Y_n}, \varphi_G)]$ is identified with $[\mathrm{des}_{K/k}((X_{Y_0, \ldots, Y_n}), \varphi_G)]$ under the isomorphism $K_0(\mathrm{Var}_{K, \rm desc}) \cong K_0(\mathrm{Var}_k)$ , so it is enough to show that $[\mathrm{des}_{K/k}((X_{Y_0, \ldots, Y_n}), \varphi_G)] = 0$ . Since this is the class of a k variety, $\mathrm{des}_{K/k}((X_{Y_0, \ldots, Y_n}), \varphi_G)$ , this class vanishes if and only if the variety $\mathrm{des}_{K/k}((X_{Y_0, \ldots, Y_n}), \varphi_G)$ is the empty variety by Lemma 4.14.

We see that $\mathrm{des}_{K/k}((X_{Y_0, \ldots, Y_n}), \varphi_G) =\emptyset$ if and only if $\mathrm{des}_{K/k}((X_{Y_0, \ldots, Y_n}),\varphi_G)_K$ is trivial as a K variety. However, $[\mathrm{des}_{K/k}((X_{Y_0, \ldots, Y_n}), \varphi_G)_K] =[\Delta_{Y_0} \Delta_{Y_1} \ldots \Delta_{Y_n} \mathrm{Ind}^K_L(X)]$ , which vanishes since $\mathrm{Ind}^K_L$ is a polynomial map of degree $\leq n$ . In particular, $[\mathrm{des}_{K/k}((X_{Y_0, \ldots, Y_n}),\varphi_G)] = 0$ , and so $\Delta_{Y_0} \Delta_{Y_1} \ldots\Delta_{Y_n} \mathrm{Res}_{L/k} ([X])=0$ , as required.

Definition 4.16. Let q be a quadratic form over L, on a vector space, V. For $\iota \in I_L$ , define $q_{\iota}$ to be the quadratic form on $V \otimes_{L, \iota} K$ given by $q \otimes_{L, \iota} \langle 1 \rangle$ , where we embed L into K by $\iota$ . It is clear that $f_{\iota}: \widehat{\mathrm{W}}(L) \to \widehat{\mathrm{W}}(K)$ given by $[q] \mapsto [q_{\iota}]$ is a homomorphism. Define $\mathrm{Ind}^K_L(q) := \otimes_{\iota} q_\iota$ as a quadratic form on K. Note that $\mathrm{Ind}^K_L(q)$ has underlying vector space $\bigotimes_{\iota \in I_L} (V \otimes_{\iota} K)$ and that we may give this space a semilinear G action, where $g \in G$ acts by

\begin{align*}g(v \otimes \lambda)_{\iota} = (v \otimes g(\lambda))_{g \iota},\end{align*}

where $(v \otimes \lambda)_{\iota}$ means $v \otimes \lambda \in V \otimes_{\iota} K$ . We see that $\mathrm{Ind}^K_L(q)$ is a Galois-compatible quadratic form on $\bigotimes_{\iota \in I_L} (V \otimes_{\iota} K)$ with respect to this Galois action. Write $\mathrm{Ind}^K_L(q)$ for the Galois-compatible quadratic form on $\bigotimes_{\iota \in I_L} (V \otimes_{\iota} K)$ , and let $N_{L/k}(q)$ be the associated quadratic form over k that results from applying the equivalence of categories in Theorem 3.3 to $\mathrm{Ind}^K_L(q)$ .

By the above work, it is clear that $\mathrm{rank}(N_{L/k}(q)) = \mathrm{rank}(q)^{[L:k]}$ .

Corollary 4.21. Let $w \in \widehat{\mathrm{W}}(L)$ with $E_{\rm disc}(w) = (\alpha, \beta) \in E( (L^\times)/(L^\times)^2)$ . Then,

\begin{align*}E_{\rm disc}(N_{L/k}(w)) = (\alpha^{[L:k]}, N_{L/k}(\beta)).\end{align*}

Theorem 4.22. The following diagram is commutative.

In particular, if $X/L$ is a variety such that $\mathrm{Res}_{L/k}(X)$ exists, then

\begin{align*}\chi^{\rm mot}_k(\mathrm{Res}_{L/k}(X))= N_{L/k}(\chi^{\rm mot}_L(X)).\end{align*}

Proof. The composition of two multiplicative polynomial functions is again a multiplicative polynomial function. Therefore, if for X a smooth projective variety over L, we have

\begin{align*}\chi^{\rm mot}(\mathrm{Res}_{L/k}(X)) = N_{L/k}(\chi^{\rm mot}(X)),\end{align*}

then $\chi^{\rm mot} \circ \mathrm{Res}_{L/k}$ and $N_{L/k} \circ \chi^{\rm mot}$ define the same polynomial function from $K_0(\mathrm{Var}_L)$ to $\widehat{\mathrm{W}}(k)$ by Lemma 4.7. Since $\mathrm{Res}_{L/k}(X) = \mathrm{des}_{K/k}( (\mathrm{Ind}^K_L(X), \varphi_G))$ , and $N_{L/k}(\chi^{\rm mot}_L(X))$ is the image of $\mathrm{Ind}^K_L( \chi^{\rm mot}_L(X)^{\rm desc})$ under the isomorphism $\widehat{\mathrm{W}}(k) \to \widehat{\mathrm{W}}(K)^{\rm desc}$ , the theorem follows if we can show that the following diagram is commutative.

Therefore, we need to show that

\begin{align*}\chi^{\rm mot, desc}( \mathrm{Ind}^K_LX) = \prod_{\iota} g_{\iota}(\chi^{\rm mot}(X)) \in \widehat{\mathrm{W}}(k),\end{align*}

which is clear by naturality of the motivic Euler characteristic, the semilinear actions on both sides, and the fact that Galois descent is compatible with the motivic Euler characteristic as in Corollary 3.7.

5. Fubini theorems

Historically, since the motivic Euler characteristic is a motivic measure, we might think of it as a volume. Indeed, the motivic Gauss–Bonnet formula of [Reference Levine and RaksitLR20] says that we may obtain the motivic Euler characteristic as the integral of an Euler class across our space. Let X,Y be varieties over k, and suppose that we have a morphism $f: X \to Y$ . For the purposes of this paper, a Fubini theorem is a theorem that allows us to compute properties of $\chi^{\rm mot}(X)$ from the motivic Euler characteristic $\chi^{\rm mot}(Y)$ as well as the motivic Euler characteristics of the fibres $\chi^{\rm mot}(X_y)$ . For example, suppose that our base field is $\mathbb{C}$ , so that $\chi^{\rm mot}$ is the compactly supported Euler characteristic applied to the complex points. By the standard Fubini theorem, $\chi_c(X_y(\mathbb{C}))=0$ for all y implies that $\chi_c(X(\mathbb{C}))=0$ . This section is concerned with proving certain Fubini theorems for the motivic Euler characteristic modulo J. We first look at the situation where $f: X \to Y$ satisfies that $\chi^{\rm mot}(X_y) = 0$ for every closed point y of Y. The principle that ‘if an invariant is zero on the fibres, then it is zero on the total space’ will be referred to as a naïve Fubini theorem. It does not tend to hold in our context, and we obtain a Fubini theorem for the $\ell$ -adic determinant of cohomology in Corollary 5.19, which does not satisfy this naïve property. However, in Corollary 5.27, we show that we may bound the ramification of the character that we obtain by looking at the $\ell$ -adic determinant of cohomology of the total space, which allows us to restrict the possibilities of this character to a finite set. We then turn to looking at a different Fubini theorem, one for maps whose fibres are algebraic groups, where we can use the structure of algebraic groups in order to prove a naïve Fubini theorem for torsors in Corollary 5.31.

5.1 The Fubini theorem over real closed fields

The goal of this subsection is to prove the first part of our Fubini theorem.

Theorem 5.1. Suppose that X and Y are varieties over k, and let $f: X \to Y$ be a morphism such that for every y a closed point of Y, we have that $\chi^{\rm mot}(X_y)$ is torsion. Then $\chi^{\rm mot}(X)$ is torsion.

We call this the real closed Fubini theorem as, by Pfister’s local–global principle, we have that $\chi^{\rm mot}(X)$ is torsion if and only if $\mathrm{rank}(\chi^{\rm mot}(X))=0$ and $\mathrm{sign}(\chi^{\rm mot}(X))=0$ . This is therefore is a true naïve Fubini theorem. This holds if we can show the corresponding statement for the rank and sign of motivic Euler characteristics. We first prove this statement for real closed fields and their algebraic closures. Until specified otherwise, let k be either a real closed field or the algebraic closure of a real closed field.

Definition 5.2. Let $\mathcal{SAS}_k$ be the category of semi-algebraic sets over k. Define $K_0(\mathcal{SAS}_k)$ to be the ring generated by isomorphism classes of semi-algebraic sets over k subject to the relations:

1. (1) $[\emptyset]=0$ ;

2. (2) for U a closed semi-algebraic subset of X, we have $[X] = [X \setminus U] + [U]$ ; and

3. (3) $[X][Y] = [X \times Y]$ .

There are canonical homomorphisms $K_0(\mathrm{Var}_k) \rightarrow K_0(\mathcal{SAS}_k)$ , given by sending a variety X to X(k) or to $X(\overline{k})$ .

Let X be a semi-algebraic set. Write $H^i_c(X;k)$ to mean the $i{\text{th}}$ real closed singular cohomology with compact support, and coefficients in k. Note that the semi-algebraic simplicial cohomology is isomorphic to the sheaf cohomology with coefficients in the constant sheaf $\underline{k}$ . This is proven in the non-compactly supported case in [Reference Knebusch and DelfsKD82, Theorem 3.3]. This isomorphism extends to the compactly supported case, using that the compactly supported cohomology is given by

\begin{align*}H^*_c(X; k) = \varinjlim H^*(X, X\setminus V; k),\end{align*}

where V runs over all of the compact subsets of X. Since we have canonical isomorphisms

\begin{align*}H_{\rm sheaf}^*(X,X\setminus V; \underline{k}) \cong H_{\rm sing}^*(X, X\setminus V; k),\end{align*}

this isomorphism carries over to the limit.

Definition 5.3. Let X be a semi-algebraic set over k. Define the real closed Euler characteristic $\chi^{\rm rc}(X) := \sum_{i=0}^{\infty} (-1)^i \mathrm{dim}_k( H^i_c(X;k)) \in \mathbb{Z}$ .

Lemma 5.4. The real closed Euler characteristic $\chi^{\rm rc}$ , as defined above, furnishes a ring homomorphism $\chi^{\rm rc}: K_0(\mathcal{SAS}_k) \rightarrow \mathbb{Z}$ .

Proof. We only need to show that the three relations defining $K_0(\mathcal{SAS}_k)$ are satisfied. The relation that $\chi^{\rm rc}([\emptyset])=0$ is trivial. By [Reference Edmundo and PrelliEP15, Theorem 4.1], we see that the sheaf cohomology with compact support and coefficients in $\underline{R}$ satisfies the exactness and excision axioms. Therefore, for U an open subset of X, we have a long exact sequence

\begin{align*}\ldots \rightarrow H^n_c(U;k) \rightarrow H^n_c(X;k) \rightarrow H^n_c(X \setminus U; k) \rightarrow H^{n+1}_c(U;k) \rightarrow \ldots.\end{align*}

This immediately tells us that $\chi^{\rm rc}(X) = \chi^{\rm rc}(U) +\chi^{\rm rc}(X \setminus U)$ . Multiplicativity of $\chi^{\rm rc}$ follows from the Künneth formula [Reference Edmundo and PrelliEP20, Theorem 3.40].

Theorem 5.5. Let $f: W \to Z$ be a morphism of semi-algebraic sets over k. Suppose that for each $p \in Z$ , the real closed Euler characteristic of the fibre $\chi^{\rm rc}(W_p)=q$ . Then $\chi^{\rm rc}(W) = q \chi^{\rm rc}(Z)$ .

Proof. By [Reference Bochnak, Coste and RoyBCR98, Lemma 9.3.2], there exists a stratification of Z into constructible subsets $Z_1, \ldots, Z_n$ , such that for each i, we have an isomorphism $f^{-1}(Z_i) \cong F_i \times W_i$ for some $F_i$ fitting into the commutative diagram

where the morphism $F_i \times Z_i \rightarrow Z_i$ is the canonical projection. This means that for $p \in Z_i$ , there is an isomorphism $W_p \cong F_i$ , so $\chi^{\rm rc}(F_i) = \chi^{\rm rc}(W_p)=q$ . In $K_0(\mathcal{SAS}_k)$ we have $[W] = \sum_{i=1}^n [F_i][Z_i]$ , so applying $\chi^{\rm rc}$ and noting that $\chi^{\rm rc}(F_i)=q$ , we obtain that $\chi^{\rm rc}(W)=q\chi^{\rm rc}(Z)$ .

Lemma 5.6. For X a k variety, we have an equality in $\widehat{\mathrm{W}}(k)$ :

\begin{align*}\chi^{\rm mot}(X) = \tfrac12 (\chi^{\rm rc}(X(\overline{k}))+\chi^{\rm rc}(X(k)) ) \langle 1 \rangle + \tfrac12 ( \chi^{\rm rc}(X(\overline{k}))-\chi^{\rm rc}(X(k))) \langle -1 \rangle.\end{align*}

Proof. By Theorem 2.14, we have that $\mathrm{rank}(\chi^{\rm mot}(X)) = e(X) = \chi^{\rm rc}(X(\overline{k}))$ . Moreover, we have that $\mathrm{sign}(\chi^{\rm mot}(X)) = \epsilon(X) = \chi^{\rm rc}(X(k))$ by Theorem 2.20. The result follows since if q is any element of $\widehat{\mathrm{W}}(k)$ , we have that

\begin{align*} q = \tfrac12 (\mathrm{rank}(q) +\mathrm{sign}(q))\langle 1\rangle + \tfrac12 (\mathrm{rank}(q) -\mathrm{sign}(q))\langle-1\rangle.\end{align*}

Theorem 5.7. Let X and Y be k varieties, and let $f: X \rightarrow Y$ be a morphism. For p a closed point of Y let $X_p$ be the fibre of f over p. Then, if $\chi^{\rm mot}_k(X_p)$ is trivial for all p, then so is $\chi^{\rm mot}(X)$ .

Proof. By Lemma 5.6,

\begin{align*}\chi^{\rm mot}(X) = \tfrac12 (\chi^{\rm rc}(X(\overline{k}))+\chi^{\rm rc}(X(k)) ) \langle 1 \rangle + \tfrac12 (\chi^{\rm rc}(X(\overline{k}))-\chi^{\rm rc}(X(k)) ) \langle -1 \rangle,\end{align*}

so it is enough to prove the result for $\chi^{\rm rc}(X(\overline{k}))$ and $\chi^{\rm rc}(X)$ . First, suppose that k is a real closed field. If p has residue field k, then $\chi^{\rm rc}(X_p(k)) = \mathrm{sign}(\chi^{\rm mot}(X_p))=0$ by assumption. If p has residue field $\overline{k}$ , then $X_p(k)=\emptyset$ , so $\chi^{\rm rc}(X_p(k))=0$ , and therefore $\chi^{\rm rc}(X_p(k))=0$ for all p. Theorem 5.5 gives that $\chi^{\rm rc}(X(k))=0$ . Similarly, $\chi^{\rm rc}(X_p(\overline{k}))=\mathrm{rank}(\chi^{\rm mot}(X_p))$ , so Theorem 5.5 gives $\chi^{\rm rc}(X(\overline{k}))=0$ , as required.

If k is the algebraic closure of a real closed field, then $\chi^{\rm rc}(X(k))=\chi^{\rm mot}(X)$ , so we again apply Theorem 5.5 to X(k) and obtain $\chi^{\rm mot}(X)=0$ .

We wish to extend the above result to our real closed Fubini theorem for any field of characteristic 0. We first need the following lemma.

Lemma 5.8. Let n be a non-negative integer. Then $\mathrm{Spr}(\mathbb{Q}(t_1, \ldots, t_n)) \neq \emptyset$ .

Proof. We prove this by induction, with the case $n=0$ being trivial. Suppose that it is true for $n' \lt n$ , let $K = \mathbb{Q}(t_1,\ldots, t_{n-1})$ , and let $\prec \in \mathrm{Spr}(K)$ . As in [Reference Bochnak, Coste and RoyBCR98, Example 1.1.2], we may define an order $\lt $ on $K(t_n) =\mathbb{Q}(t_1, \ldots, t_n)$ as follows. Let $p(t_n) \in K[t_n]$ be non-zero. Then $p(t_n) = a_k t_n^k + \ldots + a_{m} t_n^m$ for some non-negative integers $k\leq m$ with $a_k \neq 0$ . Then say $0 \lt p(t_n)$ if $0 \prec a_k$ . For a general element $[{p(t_n)}]/[{q(t_n)}]$ of $K(t_n)$ , we say that $0 \lt [{p(t_n)}]/[{q(t_n)}]$ if and only if $0 \lt p(t_n)q(t_n)$ . As in [Reference Bochnak, Coste and RoyBCR98, Example 1.1.2], this defines a total order on $K(t_n)$ extending $\prec$ , as required.

Now suppose that k is an arbitrary field of characteristic 0.

Lemma 5.9. Let $f: X \to Y$ be a morphism such that for every y a closed point of Y, we have $\mathrm{rank}(\chi^{\rm mot}_k(X_y))=0$ . Then $\mathrm{rank}(\chi^{\rm mot}(X))=0$ .

Proof. This lemma follows by reducing to the case of Theorem 5.7, and so we need to reduce to the case where k is the algebraic closure of a real closed field. We first reduce to the case where our base field is the algebraic closure of a finitely generated extension of $\mathbb{Q}$ .

Since X,Y are varieties over k, we may assume that X,Y are defined over a field F that is finitely generated over $\mathbb{Q}$ . Let $X_0$ and $Y_0$ be varieties over F such that $X = X_0 \times_F \mathrm{Spec}(k)$ and $Y=Y_0 \times_F \mathrm{Spec}(k)$ . Pick an algebraic closure $\overline{F}$ of F, and for every finite extension $F'/F$ , pick compatible embeddings $F' \hookrightarrow \overline{F}$ . Let $\overline{X} := X_0 \times_F \mathrm{Spec}(\overline{F})$ , and similarly for $\overline{Y}$ . Then, by Lemma 4.2, we see that $\mathrm{rank}(\chi^{\rm mot}_k(X)) = \mathrm{rank}(\chi^{\rm mot}_F(X_0)) = \mathrm{rank}(\chi^{\rm mot}_{\overline{F}}(\overline{X}))$ . We claim that for every $\overline{y}$ a closed point of $\overline{Y}$ , we have $\mathrm{rank}( \chi^{\rm mot}_{\overline{F}}(\overline{X}_{\overline{y}}))=0$ . Let $y \in Y$ be the closed point of Y lying under $\overline{y}$ under the morphism $\overline{Y} \to Y$ . Let $F_y$ denote the residue field at y, and let $n := [F_y:F]$ . Note that $\overline{X}_{\overline{y}}$ is defined over $F_y$ , and $\overline{X}_{\overline{y}} = X_y \times_{\mathrm{Spec}(F_y)} \mathrm{Spec}(\overline{F})$ . By Lemma 4.2, we see that $\mathrm{rank}(\chi^{\rm mot}_{\overline{F}}(\overline{X}_{\overline{y}})) = \mathrm{rank}( \chi^{\rm mot}_{F_y}(X_y))$ . Lemma 4.5 then gives $\chi^{\rm mot}_F(X_y) = \mathrm{Tr}_{F_y/F}(\chi^{\rm mot}_{F_y}(X_y))$ , so taking ranks, we see that $0 = n_y \cdot \mathrm{rank}(\chi^{\rm mot}_{F_y}(X_y))$ . Therefore, $\mathrm{rank}( \chi^{\rm mot}_{F_y}(X_y))=0$ , as required.

Therefore, it suffices to prove the statement for $\overline{X} \to \overline{Y}$ , where $\overline{X}$ and $\overline{Y}$ are varieties over $\overline{F}$ such that the fibres of the morphism have trivial motivic Euler characteristic. Let n denote the transcendence degree of $F/\mathbb{Q}$ . By Noether’s normalisation lemma (see, for example, [Reference LangLan02, Chapter 8, Theorem 2.1]), the field F is an finite extension of $\mathbb{Q}(t_1, \ldots, t_n)$ . By Lemma 5.8, we have $\mathrm{Spr}(\mathbb{Q}(t_1, \ldots, t_n)) \neq \emptyset$ . Pick $\lt \in \mathrm{Spr}(\mathbb{Q}(t_1, \ldots, t_n))$ , and let $F^{\rm real}$ be the real closure of $\mathbb{Q}(t_1, \ldots, t_n)$ with respect to this ordering. Then $\overline{F}$ is the algebraic closure of $F^{\rm real}$ , so the extension $\overline{F}/F^{\rm real}$ is a degree-2 field extension. We may therefore think of $\overline{X}, \overline{Y}$ and the fibres $\overline{X}_y$ as varieties over $F^{\rm real}$ . The result follows by Theorem 5.7.

Corollary 5.10. Let k now be any characteristic 0 field. Let X and Y be varieties over k, and let $f: X \rightarrow Y$ be a morphism. Let p be a closed point of Y, and let $X_p$ be the fibre of f over p. If $\chi^{\rm mot}(X_p)$ is torsion in $\widehat{\mathrm{W}}(k)$ for all p, then $\chi^{\rm mot}(X)$ is torsion.

Proof. By Pfister’s local–global principle, we see that $\chi^{\rm mot}(X_p)$ is torsion if and only if we have $\mathrm{rank}(\chi^{\rm mot}(X_p))=0$ and $\chi^{\rm mot}(X_{k_\lt , p})=0$ for all $\lt \in\mathrm{Spr}(k)$ , and similarly for $\chi^{\rm mot}(X)$ . If $\mathrm{Spr}(k) = \emptyset$ , then the only conditions are on the rank, so the result follows by Lemma 5.9. Suppose then that $\mathrm{Spr}(k) \neq \emptyset$ . Let $\lt \in \mathrm{Spr}(k)$ . Since $\chi^{\rm mot}$ commutes with base change, we have $\chi^{\rm mot}(X_{k_\lt ,y})=0$ for all closed points y of Y. Theorem 5.7 gives $\chi^{\rm mot}(X_{k_\lt })=0$ , as required.

5.2 The determinant of cohomology

Now suppose that k is an arbitrary field of characteristic 0. In this subsection, we show a Fubini theorem for the $\ell$ -adic determinant of cohomology that is not a naïve Fubini theorem. Let U be a smooth scheme over k and $\mathcal{F}$ be a smooth $\ell$ -adic sheaf for $\ell\neq\mathrm{char}(k)$ . Consider the one-dimensional $\ell$ -adic representation of $\mathop{\mathrm{Gal}}\nolimits_k$ :

\begin{align*}\mathop{\mathrm{det}}\nolimits R\Gamma_{c}(U_{\overline{k}}, \mathcal{F})=\bigotimes_{i}\mathop{\mathrm{det}}\nolimits H_{c}^{i}(U_{\overline{k}}, \mathcal{F})^{\otimes(-1)^i}.\end{align*}

In order to prove our Fubini theorem, we make use of the following theorem, which is the main theorem of [Reference SaitoSai94].

Theorem 5.11. Assume that there is a smooth projective variety $X/k$ that contains U such that $D= X \setminus U$ is a divisor with simple normal crossings. Moreover, assume that the ramification of $\mathcal{F}$ on D is tame. Finally, assume that $\mathcal{F}$ is defined on a model of U that is defined over a ring of finite type over $\mathbb Z$ . Then we have

\begin{align*}\mathop{\mathrm{det}}\nolimits R\Gamma_{c}(U_{\overline{k}},\mathcal{F})\otimes\mathop{\mathrm{det}}\nolimits R\Gamma_{c}(U_{\overline{k}}, \mathbb{Q}_{\ell})^{\otimes-\mathrm{rank}\mathcal F}=c_{X,U/k}^{*}(\mathop{\mathrm{det}}\nolimits \mathcal{F})\otimes J_{D,\mathcal{F}}^{\otimes(-1)}\end{align*}

as one-dimensional $\ell$ -adic representations of $\mathrm{Gal}_k$ .

The first term on the right-hand side is the pullback of the determinant character $\mathop{\mathrm{det}}\nolimits \mathcal{F}$ of $\pi^{\acute{e}t}_{1}(U)^{\rm ab,tame}$ to $\mathrm{Gal}_k$ by the pairing with the relative canonical class $c_{X,U/k}\in CH_{0}(X, D)$ . In this paper, we only need that $c^*_{X,U/k}$ respects tensor products, and we will only ever consider this in the case when $\mathrm{det}(\mathcal{F})$ is trivial, so this term will always vanish. We refer to the bottom of [Reference SaitoSai94, p. 414] for the precise construction of $c_{X,U/k}^*$ . We will, however, require the definition of $J_{D,\mathcal{F}}$ , as set out in the paragraph before [Reference SaitoSai94, Theorem 1, p. 424], as well as the definition on [Reference SaitoSai94, p. 416], and also [Reference SaitoSai94, Proposition 2, p. 417]. We will recall these below for the convenience of the reader.

Definition 5.12. Let A denote a regular noetherian ring, let $S=\mathrm{Spec}(A)$ , and suppose that S is connected with function field k. Following [Reference SaitoSai94], define a Jacobi datum on S to be a triple $(T, \chi, n)$ such that:

1. (1) $T=(T_{i})_{i\in I}$ is a finite family of finite étale covers $T_i \to S$ ;

2. (2) $\chi=(\chi_{i})_{i\in I}$ is a family of characters $\chi_{i}:\mu_{d_i}(T_i)\to \overline{\mathbb{Q}_{l}}^\times$ of the group of $d_i$ th roots of unity for some integer $d_i$ invertible in A such that $\mu_{d_i}\simeq\mathbb Z/d_i\mathbb{Z}$ on $T_i$ ; and

3. (3) $n=(n_{i})_{i\in I}$ is a family of integers satisfying the norm condition below.

Write $T_i= \coprod_{j \in J_i} T_{ij}$ , where each $T_{ij}$ is connected. Let the norm $N_{T_{ij}/S}(\chi_i)$ denote the product of the conjugates, and let the norm $N_{T_i/S}$ be given by

\begin{align*}N_{T_i/S}(\chi_i):\mu_{d_i}(T_i)&\to \overline{\mathbb{Q}_{\ell}}^\times\\ \zeta: &\mapsto \prod_{j} \chi_i(N_{T_{ij}/S}(\zeta)).\end{align*}

By construction, each $N_{T_i/S}(\chi_i)$ factors through some $\mu_{d_i'}(T_i)$ such that $d_i' | d_i$ and $\mu_{d_i'} = \mathbb{Z}/d_i'\mathbb{Z}$ on S. Let d be a common multiple of all of the $d'_i$ , and regard $N_{T_i/S}(\chi_i)$ as a character on $\mu_d$ via the composition

\begin{align*}\mu_d &\to \mu_{d'_i} , \\ \zeta &\mapsto \zeta^{\frac{d}{d'_i}}.\end{align*}

The norm condition for our Jacobi datum is that we require that $\prod_{i\in I}N_{T_i/S}(\chi_i)^{n_i}=1$ , where this product is considered as a product of characters of $\mu_d$ on S.

If A’ is another ring equipped with a morphism $f:\mathrm{Spec}(A') \to \mathrm{Spec}(A)$ , we may pull back the Jacobi datum $(T,\chi,n)$ on $\mathrm{Spec}(A)$ to $\mathrm{Spec}(A')$ to obtain $f^*(T, \chi, n) = (f^*T, f^*\chi, n)$ , where $f^*(T) = (f^*T_i)$ is given by the pullback of the finite étale covers $T_i \to \mathrm{Spec}(A)$ along the morphism f, and $f^*\chi_i$ is the composition $\chi_i: \mu_{d_i}(f^*T_i) \to \mu_{d_i}(T_i) \to \mathbb{Q}_{\ell}^\times$ . We say that the Jacobi datum $(T',\chi',n')$ on A’ is defined on A if there exists a Jacobi datum $(T,\chi,n)$ on A such that we may obtain $(T',\chi',n')$ by pulling back the Jacobi datum $(T,\chi,n)$ along the morphism $\mathrm{Spec}(A') \to \mathrm{Spec}(A)$ . If s is a closed point of A with residue field $\kappa(s)$ , we will say that the reduction of $(T,\chi,n)$ at s is the Jacobi datum on $\kappa(s)$ given by pulling back $(T,\chi,n)$ along the closed immersion corresponding to s.

Definition 5.13. Let $\mathbb{F}$ be a finite field of order q. Let $\chi$ be a multiplicative character, i.e. a group homomorphism $\chi: \mathbb{F}^\times \to \overline{\mathbb{Q}_{l}}^\times$ . Let $\psi$ be an additive character, i.e. a group homomorphism $\psi: \mathbb{F} \to \overline{\mathbb{Q}_{\ell}}^\times$ . Define the Gauss sum

\begin{align*}G_{\mathbb{F}}(\chi, \psi) := -\sum_{a \in \mathbb{F}^\times} \chi^{-1}(a)\psi(a).\end{align*}

Definition 5.14. Let $\mathbb{F}$ be a finite field of order q. Then, if we have a Jacobi datum $(T, \chi, n)$ on $\mathrm{Spec}(\mathbb{F})$ , we see that each $T_i = \coprod_{j \in \overline{I}_i} \mathrm{Spec}(\mathbb{F}_{ij})$ for some $\mathbb{F}_{ij}/\mathbb{F}$ a finite extension and $\overline{I}_i$ a finite index set. Define the Jacobi sum $j_{\chi}=j_{T,\chi,n}$ for a Jacobi datum $(T, \chi, n)$ on $\mathrm{Spec}(\mathbb{F})$ by

\begin{align*}j_{\chi}:= \prod_{i\in I} \prod_{j \in \overline{I}_i}(G_{\mathbb{F}_{ij}}(\overline{\chi}_{ij}, \psi_0\circ\mathrm{Tr}_{\mathbb{F}_{ij}/\mathbb{F}}))^{n_i}.\end{align*}

Here, $\psi_0$ is a non-trivial additive character of k, and if $\mathbb{F}_{ij}$ is of order $q_{ij}$ then $\overline{\chi}_{ij}$ is a multiplicative character of $\mathbb{F}_i$ defined by

\begin{align*}\overline{\chi}_{i}(a):=\chi_{i}(a^{(q_{ij}-1)/d_i}).\end{align*}

As mentioned on [Reference SaitoSai94, p. 417], we can consider characters on $\mathbb{F}^\times$ as characters of $\mu_{q-1}(\mathbb{F})$ . Moreover, as characters of $\mu_{q-1}(\mathbb{F})$ , we have $\prod_{j \in \overline{I}_i} \overline{\chi}_i |_{\mathbb{F}^\times} = N_{T_i/\mathrm{Spec}(S)}(\chi_i)$ . As mentioned on [Reference SaitoSai94, p. 417], [Reference DeligneDel77, Proposition 4.15] tells us that the product $j_{\chi}$ is independent of our choice of $\psi_{0}$ , since $\prod_{i\in I}N_{\mathbb{F}_i/\mathbb{F}}(\chi_{i})^{n_i}=1$ .

Definition 5.15. As in [Reference SaitoSai94, Proposition 2], we can associate an $\ell$ -adic representation $J_{(T,\chi,n)}$ of $\mathop{\mathrm{Gal}}\nolimits_k$ to each Jacobi datum $(T, \chi, n)$ on a field k as follows. A Jacobi datum on k is defined on a regular ring A of finite type over $\mathbb Z$ , since each of the $T_i$ are varieties over k. The representation $J_{(T,\chi,n)}$ is the pullback of the representation of $\pi_{1}^{\acute{e}t}(\mathrm{Spec}(A))^{ab}$ characterised by the following condition: for each closed point s of $\mathrm{Spec}(A)$ , the action of the geometric Frobenius $\mathrm{Frob}_s$ at s is given by the multiplication by the Jacobi sum $j_{\chi}(s)$ defined by the reduction of the Jacobi datum $(T, \chi, n)$ at s. By [Reference SaitoSai94, Proposition 2], this character both exists and is unique. Say that $J_{(T,\chi,n)}$ is the associated $\ell$ -adic character of the Jacobi datum $(T, \chi, n)$ .

Definition 5.16 (Saito [Reference SaitoSai94]). Let U and $\mathcal{F}$ be as in Theorem 5.11. We can define a Jacobi datum on k associated to the ramification of $\mathcal{F}$ along D. Let $D_i$ be the irreducible components of D with constant field $k_i$ . Let $\rho$ be the $\ell$ -adic representation of $\pi_1 (U,\overline{x})^{\rm tame}$ corresponding to $\mathcal{F}$ . The kernel $\pi^{\acute{e}t}_1(U,\overline{x})^{\rm tame}\to\pi_{1}^{\acute{e}t}(X,\overline{x})$ is the normal subgroup topologically generated by the local monodromy groups $\widehat{\mathbb Z}'(1)_{D_i}$ along the $D_{i}$ where $\widehat{\mathbb Z}'(1)=\varprojlim\mu_d$ with d invertible in k. Let $\rho_{i}$ be the restriction of $\rho$ to $\widehat{\mathbb Z}'(1)_{D_{1}}\simeq\widehat{\mathbb Z}'(1)_{k_i}$ .

Since we have a model of finite type over $\mathbb Z$ , Grothendieck’s monodromy theorem [Reference DeligneDel72, Thàreme de monodromie 1.2] tells us that the $\rho_{i}$ are quasi-unipotent. Therefore, the action of $\widehat{\mathbb Z}'(1)$ on the semi-simplification $\rho_{i}^{\rm ss}$ factors through a finite quotient. This means that we can decompose

\begin{align*}\rho_{i}^{\rm ss}\simeq\bigoplus_{j\in I_{i}}\mathrm{Tr}_{k_{ij}/k_{i}}(\chi_{ij}).\end{align*}

Here, $k_{ij}$ is the finite extension of $k_{i}$ obtained by adjoining the $d_{ij}$ th roots of unity, $\chi_{ij}$ is a character of $\mu_{d_{ij}}(k_{ij})$ of order $d_{ij}$ , and we define $\mathrm{Tr}_{k_{ij}/k_i}$ to be the $[k_{ij}:k_i]$ dimensional $\widehat{\mathbb Z}'(1)$ representation

\begin{align*}\mathrm{Tr}_{k_{ij}/k}(\chi_{ij}) := \bigoplus_{\sigma \in \mathop{\mathrm{Gal}}\nolimits(k_{ij}/k_i)} \chi_{ij} \circ \sigma.\end{align*}

For $i\in I$ , let $D_{i}^{*}=D_{i}- \bigcup_{J\neq i}D_{j}$ and let $c_{i}$ be the compactly supported Euler characteristic of $(D_{i}^{*}\times_{k_i} \mathrm{Spec}(\overline{k_i}))$ . Define our Jacobi datum $(T, \chi, n)$ by $\overline{I}=\coprod_{i}I_{i},$ $T=(k_{ij}),$ $\chi=(\chi_{ij})$ and $n=(n_{ij})$ , with $n_{ij}=c_{i}$ for $i\in I$ and $j\in I_{i}$ . Define the term $J_{D,\mathcal{F}}$ to be the $\ell$ -adic representation $J_{(T, \chi, n)}$ .

Definition 5.17. Let $f: W \to Y$ be a projective morphism of positive relative dimension. Assume that there is a smooth and projective variety Z over k containing Y such that the complement $D=Z-Y$ is a divisor with simple normal crossings, that $R^if_*\mathbb{Q}_{\ell}$ is smooth over Y, and that the ramification along D is tame for all i. We define

\begin{align*}\prod_{i=0}^{2(\mathop{\mathrm{dim}}\nolimits(W)-\mathop{\mathrm{dim}}\nolimits(Y))}J_{D,R^if_*\mathbb{Q}_{\ell}}^{\otimes(-1)^{i+1}}\end{align*}

to be the Saito factor of f at D.

Theorem 5.18. Let $f: W \to Y$ be a projective morphism of positive relative dimension. Assume that there is a projective and smooth variety Z over k containing Y such that the complement $D=Z-Y$ is a divisor with simple normal crossings, that $R^if_*\mathbb{Q}_{\ell}$ is smooth, and that the ramification is tame for all i. Moreover, assume that $\sum_i (-1)^i \mathrm{rank}(R^if_*\mathbb{Q}_{\ell})=0$ , $\bigotimes_i \mathop{\mathrm{det}}\nolimits R^if_*\mathbb{Q}_{\ell}^{(-1)^i}$ is trivial, and f is defined on a model over a ring of finite type over $\mathbb{Z}$ . Then,

\begin{align*}\mathrm{det}_{\ell}(W)=\prod_{i=0}^{2(\mathop{\mathrm{dim}}\nolimits(W)-\mathop{\mathrm{dim}}\nolimits(Y))}J_{D,R^if_*\mathbb{Q}_{\ell}}^{\otimes(-1)^{i+1}}.\end{align*}

Proof. By the Leray spectral sequence,

\begin{align*}\mathrm{det}_{\ell}(W)=\!\!\!\!\!\!\!\!\!\!\!\!\!\!\prod_{i=0}^{2(\mathop{\mathrm{dim}}\nolimits(W)-\mathop{\mathrm{dim}}\nolimits(Y))}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\mathop{\mathrm{det}}\nolimits R\Gamma_c(Y_{\overline{k}},R^if_*\mathbb{Q}_{\ell})^{(-1)^i}.\end{align*}

Since f is defined on a model over a ring of finite type over $\mathbb Z$ , the same is true for the sheaves $R^if_*\mathbb{Q}_{\ell}$ , so we apply Saito’s theorem (Theorem 5.11) to obtain

\begin{align*}\prod_i (\mathop{\mathrm{det}}\nolimits R\Gamma_c(Y_{\overline{k}},R^if_*\mathbb{Q}_{\ell})^{(-1)^i} ) &\otimes\mathrm{det}_{\ell}(Y)^{\otimes-\sum_i(-1)^i\mathrm{rank}R^if_*\mathbb{Q}_{\ell}}=\\&\quad c_{Y,D/k}^{*}\Big(\bigotimes_i(\mathop{\mathrm{det}}\nolimits R^if_*\mathbb{Q}_{\ell})^{(-1)^i}\Big)\otimes\prod_iJ_{D,R^if_*\mathbb{Q}_{\ell}}^{\otimes(-1)^{i+1}},\end{align*}

using the bilinearity of the pairing $CH_{0}(X, D)\times\pi^{\acute{e}t}_{1}(U)^{\rm ab,tame}\to\mathrm{Gal}_k$ . By our assumptions on the rank and determinants of $R^if_*\mathbb{Q}_{\ell}$ , we see that

\begin{align*}\mathrm{det}_{\ell}(Y)^{\otimes-\sum_i(-1)^i\mathrm{rank}R^if_*\mathbb{Q}_{\ell}} &= 1 , \\ c_{Y,D/k}^{*}\Big(\bigotimes_i(\mathop{\mathrm{det}}\nolimits R^if_*\mathbb{Q}_{\ell})^{(-1)^i}\Big)&=1,\end{align*}

where 1 denotes the trivial representation of $\mathop{\mathrm{Gal}}\nolimits_k$ . Since the sheaves $R^if_*\mathbb{Q}_\ell$ vanish whenever $i \gt 2(\mathrm{dim}(W)-\mathrm{dim}(Y))$ , this gives the result.

Corollary 5.19. Let $f: W \to Y$ be a projective morphism of positive relative dimension g. Assume that there is a projective and smooth variety Z over k containing Y such that the complement $D=Z-Y$ is a divisor with simple normal crossings, that $R^if_*\mathbb{Q}_{\ell}$ is smooth, and that the ramification along D is tame for all i. Moreover, assume that for every closed point of Y the fibre, $W_y$ has $\chi^{\rm mot}(W_y) \equiv 0 \pmod{J}$ . Then the motivic Euler characteristic modulo J satisfies $\chi^{\rm mot}(W) \equiv \kappa_W \pmod{J}$ , where $\kappa_W$ is an element of $\widehat{\mathrm{W}}(k)/J$ such that $\mathrm{rank}(\kappa_W)=0$ , $\mathrm{sign}(\kappa_W)=0$ , and $\mathrm{disc}(\kappa_W)=\mathrm{det}_{\ell}(W) = \prod_{i=0}^{2g}J_{D,R^if_*\mathbb{Q}_{\ell}}^{\otimes(-1)^{i+1}}$ .

Proof. The fact that $\mathrm{rank}(\kappa_{\mathcal{C}})$ and $\mathrm{sign}(\kappa_{\mathcal{C}})$ both vanish is due to our real closed Fubini theorem, Corollary 5.10. Consider the condition $\chi^{\rm mot}(W_y) \equiv 0 \pmod{J}$ for all y. Taking ranks tells us that $\mathrm{rank}(\chi^{\rm mot}(W_y))=0$ for all y, and applying proper base change tells us that $\sum_i (-1)^i \mathrm{rank}(R^if_*\mathbb{Q}_{\ell})=0$ . Similarly, we see that $\mathrm{det}_{\ell}(W_y)=0$ for all y, which is equivalent to saying that $\bigotimes_i \mathop{\mathrm{det}}\nolimits (R^if_*\mathbb{Q}_{\ell})^{(-1)^i}$ is trivial as well. The above theorem gives us the result.

Remark 5.20. This Fubini theorem reduces our computation of $\chi^{\rm mot}\pmod{J}$ to Saito factors associated to the boundary of our morphism, thought of as characters in $\mathop{\mathrm{Hom}}\nolimits(\mathop{\mathrm{Gal}}\nolimits_k, \overline{\mathbb{Q}}_{\ell}^\times)$ . Since $\mathrm{disc}(\kappa_W)=\mathrm{det}_{\ell}(W) = \prod_{i=0}^{2g}J_{D,R^if_*\mathbb{Q}_{\ell}}^{\otimes(-1)^{i+1}}$ , and $\mathrm{disc}(\kappa_W)$ is a character of order 2 by construction, the Saito factor $\prod_{i=0}^{2g}J_{D,R^if_*\mathbb{Q}_{\ell}}^{\otimes(-1)^{i+1}}$ will also be a character of order 2. For the rest of this section, we show that these characters of $\mathop{\mathrm{Gal}}\nolimits_k$ naturally factor through the absolute Galois group of a number field, and we can control the ramification of these characters.

Definition 5.21. For k a field, let $k^{\rm const}$ denote the algebraic closure of its prime subfield in k. In particular, if k is a field that is finitely generated over $\mathbb{Q}$ , the field $k^{\rm const}$ is a number field.

Lemma 5.22. Let $(T,\chi,n)$ be a Jacobi datum on k, a field of characteristic 0. Let d denote the lowest common multiple of the $d_j$ , and let $A := \mathrm{Spec}(\mathcal{O}_{k^{\rm const}}[\frac1d])$ . Then there exists a Jacobi datum $(T',\chi',n')$ on $\mathrm{Spec}(A)$ such that the character $J_{(T,\chi,n)}$ is given by the composition

\begin{align*}\mathop{\mathrm{Gal}}\nolimits_k \to \pi_1^{\acute{e}t}(\mathrm{Spec}(A)) \xrightarrow{J_{(T',\chi',n')}} \overline{\mathbb{Q}_{\ell}}^\times.\end{align*}

Proof. This follows instantly by the second half of the corollary on [Reference SaitoSai94, p. 418].

Corollary 5.23. The character $J_{(T,\chi,n)}$ factors as a composition $\mathop{\mathrm{Gal}}\nolimits_k \twoheadrightarrow \mathop{\mathrm{Gal}}\nolimits_{k^{\rm const}} \to \overline{\mathbb{Q}_{\ell}}^\times$ .

Proof. First, note that the character $J_{(T,\chi,n)}$ factors through the map $\mathop{\mathrm{Gal}}\nolimits_k \to \pi_1^{\acute{e}t}(\mathrm{Spec}(A))$ , which in turn factors as $\mathop{\mathrm{Gal}}\nolimits_k \to \mathop{\mathrm{Gal}}\nolimits_{k^{\rm const}} \to \pi_1^{\acute{e}t}(\mathrm{Spec}(A))$ . The map $\mathop{\mathrm{Gal}}\nolimits_k \to \mathop{\mathrm{Gal}}\nolimits_{k^{\rm const}}$ is surjective since $k^{\rm const}$ is algebraically closed in k by construction.

Definition 5.24. Since $\mathop{\mathrm{Gal}}\nolimits_k \to \mathop{\mathrm{Gal}}\nolimits_{k^{\rm const}}$ is surjective, the character $\mathop{\mathrm{Gal}}\nolimits_{k^{\rm const}} \to \overline{\mathbb{Q}_{\ell}}^\times$ appearing in the above corollary is unique. Call this character $J_{(T,\chi,n)}^{\rm const}$ .

Corollary 5.25. Let p be a prime number coprime to $d_j$ for every j, and let $\mathfrak{p}$ be a prime of $k^{\rm const}$ with residue characteristic p. Then the character $J_{(T,\chi,n)}^{\rm const}$ is unramified at $\mathfrak{p}$ .

Proof. By construction, $J_{(T,\chi,n)}^{\rm const}$ factors as $\mathop{\mathrm{Gal}}\nolimits_{k^{\rm const}} \to \pi_1^{\acute{e}t}(\mathrm{Spec}(A)) \to \overline{\mathbb{Q}_{\ell}}^\times$ . For $\mathfrak{p}$ as above, we see that $\mathfrak{p} \in \mathrm{Spec}(A)$ . Therefore, the inertia group of $\mathfrak{p}$ is in the kernel of the morphism $\mathop{\mathrm{Gal}}\nolimits_{k^{\rm const}} \to \pi_1^{\acute{e}t}(\mathrm{Spec}(A))$ . In particular, it is in the kernel of $J_{(T,\chi,n)}^{\rm const}$ , as required.

Corollary 5.26. Suppose that we are now in the situation of Theorem 5.11, and let $J_{D,\mathcal{F}}$ be the $\ell$ -adic character appearing in the theorem. Suppose that p is a prime number satisfying

\begin{align*}p \gt \mathrm{rank}(\mathcal{F}) \cdot [k^{\rm const}:\mathbb{Q}] + 1,\end{align*}

and let $\mathfrak{p}$ be a prime of $k^{\rm const}$ with residue characteristic p. Then $J_{D,\mathcal{F}}^{\rm const}$ is unramified at $\mathfrak{p}$ .

Proof. By the above corollary, we only need to show that p does not divide any $d_j$ . The $d_j$ appear through the decomposition of the semi-simplification of the monodromy representation associated to $\mathcal{F}$ ,

\begin{align*}\rho^{\rm ss} = \bigoplus_j \mathrm{Tr}_{k_j/k}(\chi_j),\end{align*}

where $\chi_j$ is a representation on $\mu_{d_j}(k_j)$ and, moreover, $\mathrm{dim}(\rho^{\rm ss}) = \mathrm{rank}(\mathcal{F})$ . Note that

\begin{align*}\mathrm{dim}\mathrm{Tr}_{k_j/k}(\chi_j) = [k_j:k] \geq [k(\zeta_{d_j}):k] = [k^{\rm const}(\zeta_{d_j}):k^{\rm const}],\end{align*}

where $\zeta_{d_j}$ denotes a primitive $d_j{\text{th}}$ root of unity. We also see that

\begin{align*} [k^{\rm const}(\zeta_p):\mathbb{Q}] &= [k^{\rm const}(\zeta_p):k^{\rm const}] \cdot [k^{\rm const}:\mathbb{Q}],\\ [k^{\rm const}(\zeta_p):\mathbb{Q}] &\geq [\mathbb{Q}(\zeta_p):\mathbb{Q}] = p-1. \end{align*}

Putting this together gives that $[k^{\rm const}(\zeta_p):k^{\rm const}] \geq \frac{p-1}{[ k^{\rm const}:\mathbb{Q}]}$ , and by the assumption on p, this gives $[k^{\rm const}(\zeta_p):k^{\rm const}]\gt \mathrm{rank}(\mathcal{F})$ . Suppose that p divides $d_j$ for some $d_j$ , seeking a contradiction. Then,

\begin{align*}\mathrm{dim}(\mathrm{Tr}_{k_j/k}(\chi_j)) \geq [k^{\rm const}(\zeta_{d_j}):k^{\rm const}] \geq [k^{\rm const}(\zeta_p):k^{\rm const}] \gt \mathrm{rank}(\mathcal{F}) = \mathrm{dim}(\rho^{\rm ss}),\end{align*}

which is a contradiction, since $\mathrm{Tr}_{k_j/k}(\chi_j)$ is a factor in the direct-sum decomposition of $\rho^{\rm ss}$ .

Corollary 5.27. Suppose that we are in the situation of Corollary 5.19. Then the character $\mathrm{disc}(\kappa_W)$ factors as

\begin{align*}\mathop{\mathrm{Gal}}\nolimits_k \twoheadrightarrow \mathop{\mathrm{Gal}}\nolimits_{k^{\rm const}} \xrightarrow{\kappa_W^{\rm const}} \{\pm 1\} \subseteq \mathbb{Q}_{\ell}^\times,\end{align*}

and the character $\mathrm{disc}(\kappa_W^{\rm const})$ is unramified at all primes of $k^{\rm const}$ whose residue characteristic is larger than $\mathrm{rank}(R^if_*\mathbb{Q}_{\ell}) \cdot [k^{\rm const}:\mathbb{Q}] + 1$ for all i.

Proof. We see that $\mathrm{disc}(\kappa_W^{\rm const})$ is given by $\prod_i J_{D,R^if_*\mathbb{Q}_{\ell}}^{(-1)^i}$ . Each $J_{D,R^if_*,\mathbb{Q}_{\ell}}$ is unramified at primes whose residue characteristic is larger than $\mathrm{rank}(R^if_*\mathbb{Q}_{\ell}) \cdot [k^{\rm const}:\mathbb{Q}] + 1$ , so if a prime has residue characteristic larger than $\mathrm{rank}(R^if_*\mathbb{Q}_{\ell}) \cdot [k^{\rm const}:\mathbb{Q}] + 1$ for all i, it will be unramified for all of the characters $J_{D,R^if_*\mathbb{Q}_{\ell}}$ , and so will be unramified for $\mathrm{disc}(\kappa_W)$ .