Search Results

There is a piece of data missing from the question. As explained in this question, in order to get the embedding of Shimura data, you have to choose an element $\gamma \in B^\times$ such that $\gamma^ …

Here's a class of examples which isn't in your list as far as I know. Take an even unimodular lattice, e.g. the $E_8$ lattice. The corresponding orthogonal group is a reductive group over $\mathbf{Q}$ …

Your belief is correct. A $\mathbb{Z}$-point has to reduce to an $\mathbb{F}_p$-point for all $p$, which kills examples with gcd > 1. If you want to make this precise, try writing down an explicit d …

You seem to be asking what the reason is for many papers on $p$-adic Selmer groups to assume throughout that $p > 2$: is because the case $p = 2$ is less interesting, or because it is more difficult? …

You seem to be starting with the answer and trying to deduce the question. By Langlands philosophy every motive should correspond to an automorphic form. [...] What makes $M^1_n$ automorphic? I thin …

This has nothing to do with Witt vectors, smoothness over $\mathbf{Z}_p$ etc: formation of the $\mathfrak{g}_n$ commutes with base-extension to $\mathbf{Q}_p$ or even $\overline{\mathbf{Q}}_p$, so we …

Let $S$ be some base scheme, $H$ a finite flat group scheme over $S$, and $\alpha: \mu_p \to H$ a homomorphism of group schemes ($p$ a prime). Is the kernel of $\alpha$ necessarily flat over $S$? (I …

No, this does not work; we need analytic, not just meromorphic, continuation. If meromorphic continuation were enough, then we would know modularity of elliptic curves in a great deal more generality …

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 …

Let $X$ be a scheme which is smooth and quasi-projective over $\operatorname{Spec} \mathbf{Z}[1/N]$, and let $\ell$ be a prime dividing $N$ (hence invertible on $X$). Then then there is a regulator ma …

Let $X$ be a smooth projective algebraic variety over a field $k$, of dimension $n$, and let $Z$ be a smooth closed subvariety of dimension $m$, with $i: Z \hookrightarrow X$ the inclusion map. For a …

Let $X$ be a smooth algebraic variety (over some field of char 0), $Z$ a smooth closed subvariety of codimension 1, $i : Z \hookrightarrow X$ the inclusion, and $j : U \hookrightarrow X$ the complemen …

(Synthesis of answers from comments, posted as community-wiki answer for convenience.) If $k = \mathbb{C}$ then algebraic de Rham cohomology, defined a la Hartshorne using the completion of $X$ alon …

For simplicity, let's take $K = \mathbf{Q}_p$. One of the few things about $\mathbf{B}_{\mathrm{cris}}$ that one can straightforwardly prove directly from its definition is that it contains $\widehat …

For the first question: it can happen that $\pm \Gamma$ is congruence but $\Gamma$ is not; there is a beautiful paper on this phenomenon, with lots of examples, by Kiming, Schütt and Verril here. Fo …