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11

The anticipated analogue is as follows: There is a map $f: \mathcal X \to \prod_{v \in S} \mathcal X_v,$ where $\mathcal X$ is the stack of $p$-adic representations of $G_{E,S}$ into ${}^LG$ (the $L$-group over $E$ of some group $G$ that is part of a Shimura datum, with reflex field $E$) unramified outside $S$, and each $\mathcal X_v$ is the corresponding ...

4

By Proposition 5.1(i) in Mazur and Roberts, Local Euler characteristics, Invent. Math. 9 (1970), 201-234, $G$ fits in an exact sequence $0 \to G \to A \to B \to 0$, where $A$ and $B$ are smooth commutative affine group schemes over $R$ of the same relative dimension, say $n$. Then the cotangent space $e^* \Omega_{G/R}$ is isomorphic to the cokernel of the ...

13

Plenty! $R$ is generated, as a ring, by $F$. So its structure as a ring is going to be $\mathbb Q(\alpha)/f(\alpha)$, where $f$ is the minimal polynomial of $F$. Because you are using homological equivalence, $f$ is just the least common multiple of the minimal polynomials of the action of $F$ on the various cohomology groups of $X$. By Deligne's RH, these ...

5

Yes, the $\ell$-adic weight filtration is compatible with the weight filtration in mixed Hodge theory under the comparison isomorphism. These facts go back to Deligne, and are described in his announcement Poids dans la cohomologie des variétiés algébriques ICM 1974. Finding a detailed proof is bit harder though... Added remarks You can take a look at Huber'...

3

How did we end up with the such complicated constructions of $B$? To add to Laurent's answer remark that "these rings did not, however, come out of nowhere", I believe that in the early 80s, Fontaine noticed that two seemingly very different constructions naturally lead to very similar (classes of) rings: The Honda systems he had introduced in ...

6

These rings of periods were constructed by Fontaine in the 70's and 80's, based on earlier work of Tate (on $p$-divisible groups). The constructions are indeed quite complicated, and it's all the more remarkable that they cut out the right categories. These rings did not, however, come out of nowhere. For instance, $B_{cris}$ is $H^0_{cris}(O_{\overline{Q}_p}... 11 First question (do non-strictly-arithmetic subgroups exist?): Any "strictly arithmetic" subgroup in your sense will, in particular, be a congruence subgroup, i.e. the intersection of$G(\mathbb{Q})$with an open compact subgroup in$G(\mathbb{A}_f)$. Since non-congruence subgroups exist in$SL_2 / \mathbb{Q}$, and in lots of other groups too, these ... 4 Let$R$be$\mathbb Z_p$(or any other dvr). Let$S$be obtained by gluing two copies of$\mathbb P^1_{\mathbb Z_p}$away from the$0$-point in the special fiber, i.e. away from the vanishing locus of the ideal$(p,x)$in local coordinates away from$\infty$. Let$X$be obtained by gluing two copies of$\mathbb P^1_{\mathbb Z_p}$away from the$0$-section, i.... 4 I will focus attention on smooth projective varieties$X$over$k$with$\mathrm{Pic}(X_{\bar{k}})$a free finitely generated abelian group, as they illustrate all the essential behaviour relevant to your question. Here$\mathrm{Pic}^0(X_{\bar{k}})$is trivial so it is certainly not true in general that the map$H^1(k,\mathrm{Pic}^0(X_{\bar{k}})) \rightarrow ...

4

Yes. There exists a "nice" cohomology theory for $p$-adic varieties, taking values in vector spaces over $\mathbb Q$, defined by fixing an embedding $\mathbb Q_p \to \mathbb C$, base-changing along this embedding, and taking singular cohomology. Every morphism of varieties over $\mathbb Q_p$ induces a map on cohomology groups in this theory (by the ...

7

It's conjectured -- see e.g. this question -- that the Frobenius is always semisimple, so its minimal polynomial is the radical of its characteristic polynomial (the product of its distinct linear factors, each with multiplicity 1). So $m_{F_i}$ should be independent of $\ell$. This also shows that $m_{F_i}$ is different from $P_i$ iff $P_i$ has a root of ...

4

I am just posting my comment as an answer, mostly to correct the mistake identified by @LaurentMoret-Bailly. The property of being universally closed depends only on the underlying reduced scheme. Thus, there are counterexamples coming from a nilradical that is quasi-coherent, yet not coherent. For one example, let $S$ be $\text{Spec}(R)$ and let $X$ be ...

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