Remark 3.2. Let us compare now Question 3.1, which will be denoted (pGPCw), with three classical conjectures which we recall here in an informal way (see [Reference AndréAnd04, VII] for more details). These three conjectures are the HC, the weak form of the Grothendieck period conjecture (GPCw) and the Fontaine–Messing p-adic variational HC (pHC).

1. (HC) A rational class in singular cohomology is algebraic if and only if it is in the right step of the de Rham filtration.

2. (GPCw) A rational class in de Rham cohomology is algebraic if and only if it is rational for singular cohomology.

3. (pHC) An algebraic class in crystalline cohomology lifts to characteristic zero if and only if it is in the right step of the de Rham filtration.

The above three conjectures together with Question 3.1 naturally fit into a table:

$$ \begin{align*} \begin{array}{c|c} (p{\textrm{GPC}w}) & ({\textrm{GPC}w})\\ \hline (p\textrm{HC}) & (\textrm{HC}) \end{array}. \end{align*} $$

The right part of the table is the characteristic zero part, the left part is the mixed characteristic one, the bottom part is the filtration part and the top part is the rationality part.

The conjecture (GPCw) is implied by the strong form of the Grothendieck period conjecture (concerning transcendence of periods); see [Reference AndréAnd04, Proposition 7.5.2.2]. (Notice that loc. cit. applies under our hypothesis; see §2(6).) Its p-adic counterpart is discussed in Section 8.

Remark 3.6. Question 3.4 already appeared in some form and is due to André [Reference AndréAnd95, Remark in §5]. As he observed later [Reference AndréAnd03, I.4.6.4 and Conjecture I.5.3.10], a ramification condition on p is necessary to avoid some counterexamples (we briefly outline his argument in Example 7.1). In particular, a similar condition is needed for Question 3.1 to be reasonable; we discuss it in Section 7. Let us just mention here that a variety over ${\overline {\mathbb {Q}}}$ verifies this condition for all but finitely many primes. The corrected version of Question 3.4 is Conjecture 8.6.

Remark 4.3. The functors $R_B, R_{\mathrm {dR}},R_{\mathrm {crys}}$ are monoidal, whereas $Z_0$ and $Z_p$ are only lax-monoidal because in general, for X and Y varieties, the product $X\times Y$ has more cycles than the product of cycles from X and Y. Moreover, the functors $R_B$ , $R_{\mathrm {dR}}$ and $R_{\mathrm {crys}}$ are exact and faithful, hence they define realization functors (or fiber functors) of their respective source categories turning them into Tannakian categories. Notice that $R_{\mathrm {dR}}$ and $R_B$ make the Tannakian category neutral whereas $R_{\mathrm {crys}}$ does not. On the other hand, $Z_0$ and $Z_p$ are not faithful as there are nonzero motives without cycles (e.g., $\mathfrak {h}^1(X)$ ).

Remark 4.5. As in the classical situation, we could define $\mathcal T_M$ , the torsor of $\mathrm {dR}$ - $\mathrm {crys}$ ¡¡periods¿¿ of M, as the functor of points associated to tensor isomorphisms:

$$\begin{align*}\mathcal T_M:={\mathrm{Isom}}^\otimes (R_{\mathrm{crys}}\circ {\mathrm{sp}} |_{\langle M\rangle}, R_{\mathrm{dR}}|_{\langle M\rangle}\otimes_{{\overline{\mathbb{Q}}}}{\overline{\mathbb{Q}}}_p).\end{align*}$$

Using Theorem 4.4(2), this variety has a (noncanonical) ${\overline {\mathbb {Q}}}$ -structure.

The Berthelot isomorphism (3.1) gives an element $c\in \mathcal T_M({\overline {\mathbb {Q}}}_p)$ . However, the position of the point $c\in \mathcal T_M({\overline {\mathbb {Q}}}_p)$ will depend on this noncanonical ${\overline {\mathbb {Q}}}$ -structure, and therefore, it will not have any special properties.

In order to have correct periods, one has to find a second canonical ${\overline {\mathbb {Q}}}$ -structure. We will see in the sequel that algebraic classes furnish it although several subtleties appear related to the fact that $Z_p$ is not, in general, a fiber functor (see Remark 4.3).

Remark 4.6. (Technical remark, not needed in the sequel.) There are traditionally three techniques to define Tannakian categories of motives which allow one to work with varieties for which $\sim _{\mathrm {hom}} = \sim _{\mathrm {num}}$ is not known. All of them are perfectly fine for the study of classical complex periods, but none of them is suitable to our context.

The first is the construction of motives avoiding algebraic cycles due to Nori. The problem here, among other things, is that we do not know yet how to express algebraic classes in Nori’s category of motives. This presents an impediment to finding a ${\overline {\mathbb {Q}}}$ -structure to crystalline cohomology.

The second is André’s category of motivated cycles. Here, the field of coefficients of the category, which is conjecturally $\mathbb {Q}$ , may a priori be too large (for example, it might contain transcendental numbers).

Finally, one could work with numerical motives and use André–Kahn’s section [Reference AndréAnd04, Section 9]). In this case, it is not clear if one can define the specialization map $ {\mathrm {sp}}\colon {{\mathcal M ot}({\overline {\mathbb {Q}}})^{\textrm {gr}}} \rightarrow {\mathcal M ot}({\overline {\mathbb {F}}}_p)$ . Also, the analogous of Question 3.1 would not be reasonable. For example, let X be a hypothetical variety which verifies $\sim _{\mathrm {hom}} = \sim _{\mathrm {num}}$ in positive characteristic but not in characteristic zero. We have the following diagram

where $s_0$ and $s_p$ are the André–Kahn sections. The inclusion

$$\begin{align*}s_0(Z_0^{\mathrm{num}}(X_0)) \subset s_p(Z_p^{\mathrm{num}}(X_p))\cap H_{\mathrm{dR}}(X/{\overline{\mathbb{Q}}})\end{align*}$$

is strict as the latter contains at least $Z_0^{\hom }(X_0)$ .

Remark 5.2. The elements of $\mathcal {B}$ are linearly independent in $R_{\mathrm {crys}}( {\mathrm {sp}}(M))$ because homological and numerical equivalence coincide on M by hypothesis (see Section 4). In general, they do not span the whole space, this is why the matrix is only rectangular (contrary to the classical case of complex periods where it is a square matrix).

The matrix ${\mathcal M at}_{\mathcal {B},\mathcal {B}'}(M)$ depends on the chosen bases $\mathcal {B}$ and $\mathcal {B}'$ , but its ${\overline {\mathbb {Q}}}$ -span inside ${\overline {\mathbb {Q}}}_p$ does not.

Remark 5.4. All coordinates of the matrix ${\mathcal M at}_{\mathcal {B},\mathcal {B}'}(M)$ are examples of such $[M,v,\xi ]$ . Conversely, as soon as v and $\xi $ are nonzero, one can find bases $\mathcal {B}$ and $\mathcal {B}' $ such that $[M,v,\xi ] $ appears as an entry in ${\mathcal M at}_{\mathcal {B},\mathcal {B}'}(M)$ .

Remark 5.6. (p-adic periods contain coefficients of Frobenius) Some authors usually call p-adic periods the coefficients of the Frobenius matrix [Reference FurushoFur07, Reference BrownBro17] acting on the de Rham cohomology. Let us explain that André’s p-adic periods include these ones.

Consider a variety X over $\mathbb {Q}$ (or a number field) with good reduction $X_p$ at p. Through Berthelot’s comparison isomorphism (1.1), one can consider the Frobenius of $X_p$ acting on $\mathrm H^i_{\mathrm {dR}}(X/\mathbb {Q})\otimes \mathbb {Q}_p$ and take its matrix with respect to a basis coming from $\mathrm H^i_{\mathrm {dR}}(X/\mathbb {Q})$ for various $i\ge 1$ . The coefficients of this matrix, or the $\mathbb {Q}$ -algebra they generate, are called the p-adic periods of X in [Reference FurushoFur07, Reference BrownBro17]. Let us call them Frobenius periods.

In order to see them as special cases of André’s p-adic periods, consider for various $i\ge 1$ the internal End motive $M:={\mathcal {E}}\mathrm {nd}(\mathfrak {h}^i(X),\mathfrak {h}^i(X))$ . Then the Frobenius of $X_p$ gives a natural element $\gamma _{\operatorname {Frob}}$ in $Z_p(M)$ . Notice that the realization $R_{\mathrm {dR}}(M)$ coincides with $ {\mathrm {End}}(\mathrm H^i_{\mathrm {dR}}(X/\mathbb {Q}))$ and the coefficients of $\gamma _{\operatorname {Frob}}$ in a basis of $R_{\mathrm {dR}}(M)$ are precisely the Frobenius periods that we introduced above.

In general, as we point out in Remark 5.17, the motive $M_p$ could have many more cycles than the Frobenius endomorphism and all of them give p-adic periods as we defined them.

Remark 5.7. Contrary to the classical situation of complex periods, it is not true that the p-adic periods of M contain the p-adic periods of every object $N\in \langle M\rangle $ .

For singular and de Rham cohomology, all classes on N are polynomials on classes on M due to the Künneth formula. A slightly different way of saying this is that taking singular cohomology is a fiber functor. This is false for algebraic classes and intimately related to Remark 4.3. Similarly, one can see this as the failure of algebraic classes to provide a fiber functor.

For instance, it could happen that M has no algebraic cycles whereas the motive $M\otimes M^\vee \in \langle M\rangle $ always has at least one algebraic class, namely the identity of M. But even if a motive has algebraic classes, it is not true in general that they generate all the algebraic classes of objects in $\langle M\rangle $ (see Example 6.19).

Remark 5.9. Clearly, the algebra $\mathcal P_p(M)$ is finitely generated by construction. It is shown in Remark 6.9 that the bigger algebra $\mathcal P_p(\langle M\rangle )$ is also finitely generated. It follows that the ${\overline {\mathbb {Q}}}$ -subalgebra of ${\overline {\mathbb {Q}}}_p$ generated by all $\mathcal P_p(\langle M\rangle )$ , with M varying in ${\mathcal M ot}({\overline {\mathbb {Q}}})$ , is countable. Hence, not all p-adic numbers are periods.

Remark 5.11. In the classical case of complex periods $\mathcal P_{\mathbb {C}}(M)$ , one has the inequality $ {\mathrm {trdeg}}_{\overline {\mathbb {Q}}}(\mathcal P_{\mathbb {C}}(M))\le \dim G_{\mathrm {dR}}(M) $ . (This inequality is expected to be an equality by the Grothendieck period conjecture.) In Theorem 5.10, the upper bound is instead $\dim G_{\mathrm {dR}}(M) - \dim G_{\mathrm {crys}}(M)$ . One may interpret this difference as related to the fact that the p-adic period matrix is only rectangular (Remark 5.2).

Remark 5.12. As in the classical case, the dimensions of $G_{\mathrm {dR}}(M)$ and $ G_{\mathrm {crys}}(M)$ are sometimes computable because they are (expected to be) related to Tannakian groups of purely cohomological nature.

Indeed, if M comes from a motive with coefficients in $\mathbb {Q}$ , then the HC predicts the equality $\dim G_{\mathrm {dR}}(M) = \dim {\mathrm {MT}}(M) $ , where $ {\mathrm {MT}}(M)$ is the Mumford–TateFootnote ² group of M. Similarly, the Tate conjecture predicts the equality $ \dim G_{\mathrm {crys}}(M)= \dim (\varphi \vert R_{\mathrm {crys}}(M))$ , where $(\varphi \vert R_{\mathrm {crys}}(M))\le \operatorname {GL}(R_{\mathrm {crys}}(M)$ is the Zariski closure of the subgroup generated by the Frobenius morphism $\varphi $ acting on $R_{\mathrm {crys}}(M))$ , for a suitable model of M defined over a finite field. (Note that the groups $ {\mathrm {MT}}(M) $ and $(\varphi \vert R_{\mathrm {crys}}(M))$ are often computable.)

These conjectures are widely open but known in some already interesting cases, for example, products of elliptic curves (see [Reference ImaiImai76] over $\mathbb {C}$ and [Reference SpießSpi99] over finite fields).

Remark 5.17. (Frobenius matrix vs p-adic periods) We showed in Remark 5.6 that André’s p-adic periods contain the coefficients of Frobenius. Let us further argue that this inclusion should be strict.

Consider $M = {\mathcal {E}}\mathrm {nd}(\mathfrak {h}^1(E))$ , where E is a non-CM elliptic curve with supersingular reduction $E_p$ as in the above example. Recall that a cycle $\gamma \in Z_p(M_p)$ is the same thing as an endomorphism f of $E_p$ and its p-adic periods are precisely the coefficients of f with respect to a basis of $\mathrm H^1_{\mathrm {dR}}(E/{\overline {\mathbb {Q}}})$ . Arguing as in Example 5.13, we have that the transcendence degree of its periods is at most $2$ . In particular, the coefficients of the Frobenius endomorphism have transcendence degree at most 2. On the other hand, Conjecture 8.1 predicts that the inequality $ {\mathrm {trdeg}}(\mathcal P_p(M))\le 3$ is actually an equality. In particular, there should be more p-adic periods than just the coefficients of the Frobenius matrix.

See also Remark 9.8 for a detailed comparison between Frobenius coefficients and André’s p-adic periods.

Remark 6.2.

1. (1) In the above definition, we did not use any ${\overline {\mathbb {Q}}}$ -structure of crystalline cohomology, for example the one coming from Theorem 4.4(2), hence this object is canonically associated to M.

2. (2) The spaces $\mathcal H_M$ and $\mathcal H^{\prime }_M$ are endowed with a natural action of $G_{\mathrm {dR}}(M)$ .

Remark 6.6. In order to prove representability of $\mathcal H_M$ , we needed to factorize $R_{\mathrm {crys}}$ through ${\overline {\mathbb {Q}}}$ (as in Theorem 4.4(2)). It would be nice to bypass it and prove the representability directly.

Remark 6.9. (1) The map $\mathrm {eval}_B$ is the p-adic analog of the integration

$$\begin{align*}\int\colon \mathcal P^{\mathrm{mot}}_{\mathbb{C}}(M)\to \mathbb{C}\end{align*}$$

from the classical complex case.

(2) By its very construction, the image of $\mathrm {eval}_B$ is the algebra $\mathcal P_p(\langle M\rangle )$ from Definition 5.8 which shows that $\mathcal P_p(\langle M\rangle )$ is finitely generated.

Remark 6.12. Consider the map of algebras

$$\begin{align*}\mathrm{eval}_{B}\colon \mathcal{P}^{\mathrm{mot}}_p(M)\to {\overline{\mathbb{Q}}}_p\end{align*}$$

given by evaluating functions at the point $\mathfrak {c}_{\mathrm {Bert}}\in \mathcal H_M({\overline {\mathbb {Q}}}_p)$ . Then one has

$$\begin{align*}\mathrm{eval}_{B} ([N,v,\xi]^{\mathrm{mot}}) = [N,v,\xi]\end{align*}$$

by its very construction; see Definition 5.3. In particular, we have

$$\begin{align*}\mathrm{eval}_{B} (\mathcal{P}^{\mathrm{mot}}_p(N)) = \mathcal P_p(N);\end{align*}$$

see Definition 5.5.

Notice also that the algebra $\mathcal {P}^{\mathrm {mot}}_p(N)$ is finitely generated. Indeed it is generated by the functions $[N,v,\xi ]^{\mathrm {mot}}$ , with v and $\xi $ varying, respectively, in two fixed basis of the vector spaces $ Z_p( {\mathrm {sp}}(N))$ and $ R_{\mathrm {dR}}(N)^{\vee }$ .

Remark 6.14. The function $[N,v,\xi ]^{\mathrm {mot}}$ introduced in Definition 6.11 makes sense both in $\mathcal O(\mathcal H_N) $ and in $\mathcal O(\mathcal H_M) $ . By construction, the natural map $ \mathcal O(\mathcal H_N) \hookrightarrow \mathcal O(\mathcal H_M) $ sends $[N,v,\xi ]^{\mathrm {mot}}\in \mathcal O(\mathcal H_N)$ to $[N,v,\xi ]^{\mathrm {mot}}\in \mathcal O(\mathcal H_M)$ . For this reason, we did not put any symbol M in the definition of $[N,v,\xi ]^{\mathrm {mot}}\in \mathcal O(\mathcal H_M)$ . Notice also that the injectivity of the natural map $ \mathcal O(\mathcal H_N) \hookrightarrow \mathcal O(\mathcal H_M) $ implies that the relations between the motivic p-adic periods do not vary if we enlarge the category.

Remark 6.18. We briefly interrupt the series of examples to highlight a particular point about matrix coefficients which might clarify the difference between the algebras $\mathcal {P}^{\mathrm {mot}}_p(M)$ and $\mathcal {P}^{\mathrm {mot}}_p(\langle M\rangle )$ .

Let G be a reductive group over k, an algebraically closed field of characteristic zero. For any representation V of G, we have the matrix coefficients map

$$ \begin{align*} m_V\colon &V\boxtimes V^\vee \to \mathcal O(G)\\ & v\boxtimes \xi\mapsto (g\mapsto \xi(gv)). \end{align*} $$

The Peter–Weyl theorem gives an isomorphism of $G\times G$ -modules

(6.4) $$ \begin{align} \bigoplus_{V\text{ irr}} V\boxtimes V^\vee \simeq \mathcal O(G), \end{align} $$

where V runs over all irreducible representations of G. Moreover, for any faithful irreducible representation V, the summand $V\boxtimes V^\vee $ generates $\mathcal O(G)$ as an algebra (see [Reference WaterhouseWat12, §3.5 Theorem]).

Now, let $H\le G$ be a closed subgroup. The map (6.4) restricts to an isomorphism of G-modules

$$\begin{align*}\bigoplus_{V\text{ irr}} V^H\boxtimes V^\vee \simeq \mathcal O(G/H).\end{align*}$$

However, now a summand $V^H\boxtimes V^\vee $ might not generate the algebra $\mathcal O(G/H)$ . An extreme example is for irreducible representations V of G such that $V^H=\{0\}$ . This is the case for $\mathbb {G}_m\le \operatorname {SL}(2)$ and any even dimensional irreducible representation. A less extreme example is mentioned below for $\mathbb {G}_m\le \operatorname {PGL}(2)$ and $ {\mathrm {Sym}}^4k^2$ . We do not know for which irreducible representations V the matrix coefficients of $V^H\boxtimes V^\vee $ generate the algebra $\mathcal O(G/H)$ .

In the motivic setting, the algebra $\mathcal {P}^{\mathrm {mot}}_p(\langle M\rangle )$ is of the form $\mathcal O(G/H)$ whereas $\mathcal {P}^{\mathrm {mot}}_p(M)$ corresponds to the subalgebra generated by $V^H\boxtimes V^\vee $ , where $V=R_{\mathrm {dR}}(M)$ viewed as a $G=G_{\mathrm {dR}}(M)$ -module (see Definition 6.11).

This phenomenon does not happen in the classical case of complex periods because $\mathcal P_{\mathbb {C}}^{\mathrm {mot}}(\langle M\rangle )$ is of the form $\mathcal O(G)$ .

Remark 7.2. (Relation to gamma functions) In the classical complex setting, periods of abelian CM motives are related to products of special values of the gamma function (the Gross–Deligne conjecture [Reference GrossG78, p.209]). There are explicit relations between such special values [Reference DeligneDel79, Appendix] which are conjectured to be the only ones. Such a prediction is coherent with the Grothendieck period conjecture.

In the p-adic context, the situation is more subtle as Example 7.1 shows. There seems to be a discrepancy between the ramified and the unramified case, namely in the ramified situation the algebricity of the p-adic periods is not fully explained by motives. The reason seems to boil down to the fact that the p-adic periods have different expressions in terms of special values of the p-adic gamma function in the two cases (see Example 5.13 for the two expressions).

Remark 7.3. (André’s Betti lattices and ramification.) In [Reference AndréAnd95] (and later [Reference AndréAnd03, I.5.3]), André investigated the possibility of constructing a second ${\overline {\mathbb {Q}}}$ -structure to crystalline cohomology – especially for the $H_{\mathrm {crys}}^1(A)$ , with A an abelian variety – to be compared to the ${\overline {\mathbb {Q}}}$ -structure given by the de Rham cohomology $H^1_{\mathrm {dR}}(\tilde {A})$ of a given lifting $\tilde {A}$ . Such a canonical structure should be called ‘Betti’, and denoted $H^1_B$ . It should come with two properties: First, the endomorphisms of A should act on $H^1_B$ and second any algebraic classes on any power of A should be rational with respect to this Betti structure. He proposed a construction in two cases:

1. (1) If A is ordinary [Reference AndréAnd95, §5]. In this case, one can take the de Rham cohomology of the Serre–Tate lifting. (This is interesting if one started with a non-CM lift $\tilde {A}$ – in particular not the Serre–Tate lifting itself.)

2. (2) If A is supersingular. In this case he shows that there exists a unique Betti structure $H^1_B(A)$ with the property that the endomorphisms of A preserve it and that $\wedge ^2H^1_B(A)(1)\subset H_{\mathrm {crys}}^2(A)(1)$ is the space of algebraic classes [Reference AndréAnd03, Proposition I.5.3.3].

The proposed ${\overline {\mathbb {Q}}}$ -structure $H^1_B$ is related to the one we are studying in an indirect way; see Example 5.14 for the supersingular reduction case and Example 5.16 for the ordinary reduction case.

In [Reference AndréAnd95, Remark in §5], André speculates (at least in the case of elliptic curves) that the two ${\overline {\mathbb {Q}}}$ -structures of $H_{\mathrm {crys}}^1(A)$ , Betti and de Rham, should not coincide. In loc. cit., he pointed out that if A is supersingular and $\tilde {A}$ is CM, then these two ${\overline {\mathbb {Q}}}$ -structures are related through values of the p-adic gamma function (by [Reference OgusOgu90], based on computations of Coleman [Reference ColemanCol90]).

On the other hand, as he afterwards noticed, those special values of the gamma function might turn out to be algebraic by the Gross–Koblitz formula [Reference AndréAnd03, I.4.6.4]. He conjectured that this phenomenon should be pathological and formulated the following conjecture [Reference AndréAnd03, Conjecture I.5.3.10] (the section of the conjecture is written for elliptic curves ‘for simplicity’, as the author says in loc. cit.). If A is supersingular and $\tilde {A}$ is not CM (or if the prime number p does not ramify in the CM field), then the two ${\overline {\mathbb {Q}}}$ -structures of $H_{\mathrm {crys}}^1(A)$ , Betti and de Rham, should not coincide.

Remark 7.5. (Not needed in the sequel.) The standard conjectures predict that if the motive N is a CM motive with the above definition, then $ {\mathrm {End}}(N)$ has a positive involution, hence it should indeed be a CM number field.

Remark 7.7. In the Examples 5.14 and 5.16, the prime p is not ramified in $\langle M\rangle $ . The reason is that the Tannakian groups are $\operatorname {PGL}(2)$ and $\operatorname {GL}(2)$ , and so the only CM objects in $\langle M\rangle $ are the Tate twists (see, for example, the proof of the next proposition). For Tate twists, the ramification condition is empty as .

Remark 8.2. We do not know if the ramification condition is optimal, but it makes sense to impose it in order to avoid the situation from Example 7.1.

Remark 8.8. Conjecture 8.6 with Proposition 8.7 corrects Question 3.1. Notice that when we discussed Question 3.1 the ramification condition was not yet introduced. The relation of this conjecture with other classical conjectures is sketched in Remark 3.2.

Remark 8.10. As for classical periods, there is no converse of the above proposition. Namely, in the abstract Tannakian setting of the proof, (2) does not imply (1) in general.

For an example, consider the trivial subgroup $H= 1 < \operatorname {GL}(2)(k)$ . Notice that $\mathcal H = G$ . Consider $a,b\in K$ algebraically independent over k, and consider the matrix

$$\begin{align*}\alpha:=\begin{pmatrix} a & 1\\ 1 & b \end{pmatrix}\in G(K). \end{align*}$$

We claim that for all representations $V\in \operatorname {Rep}_k(G)$ we have

$$\begin{align*}\alpha(V)\cap V=\{0\} \text{ inside } V\otimes_k K.\end{align*}$$

Notice first that it is enough to check it on irreducible representations. For $\operatorname {GL}(2)$ , they are all of the form $\det ^s\otimes {\mathrm {Sym}}^r k^2$ . A nonenlightening computation shows that for such representations the claim is true. Thus, (2) holds in this example.

However, $\alpha $ is not generic in G as its Zariski closure has dimension two, hence (1) does not hold.

Remark 8.11. Proposition 8.9 has an analog for classical periods; see, for example, [Reference AndréAnd04, Proposition 7.5.2.2]. (Notice that loc. cit. applies under our hypothesis; see Section 2(6).) The standard way of showing it is to consider the de Rham–Betti realization, which lands in the category of triples $(V,W,c)$ , where V and W are ${\overline {\mathbb {Q}}}$ -vector spaces and c is a comparison isomorphism $V\otimes \mathbb {C} \simeq W\otimes \mathbb {C}.$ A key point is that this category is Tannakian.

In the p-adic setting, one would like to consider the category of triples $(V,W,i)$ , where V and W are ${\overline {\mathbb {Q}}}$ -vector spaces and i is an injection $V\otimes {\overline {\mathbb {Q}}}_p \hookrightarrow W\otimes {\overline {\mathbb {Q}}}_p.$ However, this new category fails to be Tannakian (it is not even abelian because cokernels do not exist), hence the classical approach does not apply to Proposition 8.9.

Remark 9.1. There exists a triangulated tensor subcategory $\mathrm {DM}^{B}(\mathcal O_{\mathfrak {p}})$ of $\mathrm {DM}(\mathcal O_{\mathfrak {p}})$ containing the motives of smooth proper schemes and for which the Berthelot comparison theorem (3.1) holds. Indeed by [Reference VezzaniVez22, Remark 4.5] there is a natural monoidal transformation (denoted by $\alpha $ in loc. cit.) between the crystalline and de Rham realization functors. In particular, the subcategory of $\mathrm {DM}(\mathcal O_{\mathfrak {p}})$ on which the transformation is invertible is indeed a tensor triangulated subcategory. The fact that it contains motives of smooth proper schemes is again in [Reference VezzaniVez22, Remark 4.5].

Remark 9.3. An example of $\mathcal {A}$ as in Notation 9.2 is the category of mixed Tate motives [Reference LevineLev93]. It is conjectured that the whole $\mathrm {DM}$ should be the derived category of an abelian category and that the realization functors should be exact [Reference AndréAnd04, 21.1]. It is also conjectured that such an abelian category should be semisimple over finite fields.

Remark 9.5. The definitions and the proofs in Theorem 9.4 are identical to the pure case. Let us just mention that in the proof of Theorem 6.4 we used that $G_{\mathrm {crys}}(M) $ is reductive. Notice that it is again the case here (even though $G_{\mathrm {dR}}(M) $ will not be reductive in general) because the category of mixed motives over ${\overline {\mathbb {F}}}_p$ is semisimple (under our assumptions from Notation 9.2).

Remark 9.7. (Pure versus mixed periods.) In the conjectural abelian category of mixed motives, any motive is expected to have a weight filtration which should split over finite fields. In particular, there should be plenty of nontrivial extensions

(with N being a motive of negative weight) that split when reduced modulo p. This leads to algebraic classes modulo p that are not liftable, just as the example above, hence they provide a source of nontrivial p-adic periods. In the same spirit as the Kummer example, they are related to the coefficients of the matrix of the absolute Frobenius acting on crystalline cohomology.

Let us give another concrete example along these lines. Let E be an elliptic curve over $\mathbb {Q}$ , O be its origin and P be a rational nontorsion point. The localization triangle in the triangulated category of motives gives a nontrivial extension that should correspond in the abelian category of motives to a short exact sequence

which we dualize

to have the suited form as above (with $N=H^1( E )^{\vee }$ and $M= H^1(E,\{O,P\})^{\vee }).$

The Frobenius matrix is of the shape

$$\begin{align*}\varphi(H^1(E,\{O,P\})^{\vee})=\begin{pmatrix} a & b & c \\ d & e & f \\ 0 & 0 & 1 \end{pmatrix}, \end{align*}$$

and the p-adic period matrix is a vector of the shape

$$\begin{align*}\mathcal{P}_p(H^1(E,\{O,P\})^{\vee})=\begin{pmatrix} \alpha \\ \beta \\ 1 \end{pmatrix}.\end{align*}$$

The relation between the two is given by

$$\begin{align*}\begin{pmatrix} a & b \\ d & e \end{pmatrix} \cdot \begin{pmatrix} \alpha \\ \beta \end{pmatrix} = \begin{pmatrix} c \\ f \end{pmatrix}. \end{align*}$$

The general picture would be the following. Let M be a motive, and, for all integers n, let $M_n$ be its associated n-th graded piece for the weight filtration. As all extensions split modulo p, one has

$$\begin{align*}{\mathrm{sp}}(M)=\bigoplus_n {\mathrm{sp}}(M_n).\end{align*}$$

Moreover, for weight reasons, one has $Z_p( {\mathrm {sp}}(M))= Z_p( {\mathrm {sp}}(M_0))$ .

Fix a basis for each $R_{\mathrm {dR}}(M_n)$ and a splitting of the weight filtration of $R_{\mathrm {dR}}(M)$ , and consider the induced basis of $R_{\mathrm {dR}}(M)$ . For a given cycle $\gamma \in Z_p( {\mathrm {sp}}(M))$ , its coordinates with respect to this basis have the form

$$\begin{align*}\mathcal{P}_p(\gamma)= \begin{pmatrix} \vdots \\ v_n \\ \vdots \end{pmatrix},\end{align*}$$

where $v_n $ is a vector living in weight n.

Notice that the coordinates of $v_0$ are the pure p-adic periods of $\gamma $ seen as an element in $Z_p( {\mathrm {sp}}(M_0))$ and the total vector $ \mathcal {P}_p(\gamma )$ is the only Frobenius invariant vector whose zeroth coordinate is precisely $v_0$ .

In particular, mixed p-adic periods are generated by pure p-adic periods and the coefficients of the matrix of the Frobenius. (Recall also that there are more pure p-adic periods than just the coefficients of the Frobenius matrix; see Remark 5.17 and Example 5.16.)

Remark 9.8. (Frobenius matrix vs p-adic periods.) As the above remarks and examples explained, the coefficients of the Frobenius matrix are intimately related to André’s p-adic periods. Some authors, especially in the context of mixed Tate motives, usually call p-adic periods those coefficients [Reference FurushoFur07, Reference BrownBro17]. Let us explain why André’s p-adic periods are actually better p-adic analogues of classical complex periods.

First, Example 9.6 shows that the p-adic numbers appearing as André’s p-adic periods have a more striking analogy with the corresponding complex ones than the coefficients of the Frobenius have.

Second, recall that for motives other than mixed Tate motives there are more André p-adic periods than just the coefficients of the Frobenius matrix (Remark 5.17).

Third, and more important, the motivic interpretation of André’s p-adic periods seems to behave as in the complex case, contrary to the coefficients of the Frobenius. Indeed, we expect that the evaluation map $\mathrm {eval}_{B}\colon \mathcal O(\mathcal H_M)\to {\overline {\mathbb {Q}}}_p$ is injective; see Proposition 8.4. This is the analogous of the fact that the classical evaluation map for complex periods $\mathcal O( {\mathrm {Isom}}(R_{\mathrm {dR}}, R_B)) \longrightarrow \mathbb {C}$ is expected to be injective. On the other hand, the evaluation map $ \mathcal O ( {\mathrm {Aut}} (R_{\mathrm {dR}})) \longrightarrow \mathbb {Q}_p$ considered by [Reference FurushoFur07, Reference BrownBro17] is well known to be noninjective at least because the characteristic polynomial of the Frobenius has integral coefficients, and this gives relations between the entries of the Frobenius that cannot be seen in $ {\mathrm {Aut}} (R_{\mathrm {dR}})$ . For example, if one takes a non-CM elliptic curve, the group $ {\mathrm {Aut}} (R_{\mathrm {dR}})$ has dimension $4$ whereas the four coefficients of the Frobenius are related at least by trace and determinant; see Example 5.14. In our setting, we do not know if $\mathrm {eval}_{B}\colon \mathcal O(\mathcal H_M)\to {\overline {\mathbb {Q}}}_p$ is injective, but we do know at least that the relations induced by the characteristic polynomial have motivic origins as shown in the following proposition.

Remark 9.10. (Continuing Remark 9.8.) In the same vein as the above proposition and remark, notice that our constructions give a much better bound to the transcendence degree of the coefficients of the Frobenius matrix than the classical ones. If M is a motive and $d_M$ is the transcendence degree of the algebra generated by those coefficients, one gets $d_M \leq \dim G_{\mathrm {dR}}(M)$ from the constructions in [Reference FurushoFur07, Reference BrownBro17] whereas one gets

(9.1) $$ \begin{align} d_M \leq \dim G_{\mathrm{dR}}(M\otimes M^\vee) & - \dim G_{\mathrm{crys}}(M\otimes M^\vee) \le \nonumber\\ & \leq \dim G_{\mathrm{dR}}(M) - \dim G_{\mathrm{crys}}(M)\le \dim G_{\mathrm{dR}}(M) \end{align} $$

from Theorem 5.10. All the above inequalities are, in general, strict. The first one is an equality if and only if the p-adic periods of the graded pieces of the weight filtration of M are trivial (see Remark 9.7). The last one is an equality when $\dim G_{\mathrm {crys}}(M)=0$ , namely if and only if M becomes, modulo p, a sum of copies of the unit motive. The second inequality is more subtle: The motive $M\otimes M^\vee $ corresponds to a representation of $G_{\mathrm {dR}}(M)$ which might not be faithful. If its kernel is K, then we have

$$ \begin{align*} G_{\mathrm{dR}}(M\otimes M^\vee) &= G_{\mathrm{dR}}(M)/K \quad \text{ and} \\ G_{\mathrm{crys}}(M\otimes M^\vee) &= G_{\mathrm{crys}}(M)/(K\cap G_{\mathrm{crys}}(M)). \end{align*} $$

Hence, we have equality if and only if the kernel K is included in $G_{\mathrm {crys}}(M)$ .

Remark 10.4. One main difference between these questions and their complex or p-adic counterparts is the presence of the choice of $\sigma $ . In particular, the left-hand side in the inclusion in Question 10.2 and in the inequality in Question 10.3 depend a priori on $\sigma $ whereas the right-hand sides do not. As we will see below, beside some exceptional cases, the left-hand sides will actually depend on $\sigma $ which will force the above questions to have negative answers in some cases.

Remark 10.6. Given a simple CM abelian variety over a number field, the set of prime numbers p such that its reduction modulo p is simple has density one [Reference Murty and PatankarMP08, Theorem 3.1].