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Funnily enough I was at a talk about this only last week. One doesn't get independence in the global function field case because there's an obstruction coming from the determinant; but there is a stat …

If $\rho$ is a representation with this property, then by taking $a$ to run through a basis of $\mathbf{Z}_p^n$ and intersecting the corresponding $H$'s, there must be a finite-index subgroup of $G$ w …

For Q1: the problem is that Chow motives are a little too restrictive, because they are all (conjecturally at least) semi-simple, whereas non-semisimple objects are hugely important for arithmetic. E. …

One major application of research in integral $p$-adic Hodge theory is in proving modularity results, e.g. for elliptic curves. Here one wants to understand liftings of global mod p Galois representat …

The converse is false. See the lecture notes by Chandan Singh Dalawat at http://arxiv.org/abs/math/0605326, which give some examples of varieties over finite extensions of $\mathbb{Q}_p$ whose $\ell$- …

Yes, this can happen. Here is a counterexample (which is probably not the simplest possible, but it's the one that first came to mind). There are not very many 2-dimensional representations of the G …

We have $(V \otimes B_e)^{G_K} = D_{\mathrm{cris}}(V)^{\varphi = 1}$. So any representation which is crystalline, but such that $\mathbf{D}_{\mathrm{cris}}(V)$ has zero $\varphi$-invariants, is an exa …

Since you mention the term "de Rham" in your question, you are clearly aware of the existence of p-adic Hodge theory; so I am surprised that you do not realise that this theory allows you to write dow …

Here are two arguments for why such a representation $\rho$ cannot exist. Automorphic argument: Fontaine and Mazur have conjectured that any irreducible $n$-dimensional geometric representation $\rh …

It's important to be clear that this map on $H^1_{\mathrm{dR}}$ overlies a highly non-trivial map on the base-ring $R$. You can imagine a case where $R$ is something like $\mathbf{Z}_p\langle X \rangl …

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 fac …

The question of order of vanishing is quite elementary: the L-function of a Hecke character (i.e. 1-dimensional Artin representation) over any number field has an Euler product, which is convergent an …

Everything you could wish for is true :-). Passing to the inverse limit in Milne's theorem 21.1 one gets an isomorphism $$ H^i_{et}(X, \mathbf{Z}_\ell) = H^i_{sing}(X(\mathbf{C}), \mathbf{Z}) \otimes …

This is essentially the same as your other recent CM-theory question, in a mild disguise; for both questions the point is that $\psi(I)$ is a generator of $I$. This follows easily from the fact that $ …

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 $G(\mathbb{Q})$ w …