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There is an excellent book by Neal Koblitz "p-adic numbers, p-adic analysis and zeta-functions" were the Dwork's proof is stated in a very detailed way, including all preliminaries from p-adic analysi …

Let's assume that $X$ admits a $K$-point $x$ and use the corresponding geometric point as the base point. The existence of a rational point is in fact necessary for a positive answer, as explained by …

Not quite an answer, but a heuristics from the point of view of Weil philosophy about why the rationality mod $p$ is much easier. Weil reduced the rationality of zeta-function to the existence of a g …

There are counterexamples (at least for some $p$) even if we assume that $X$ lifts all the way to a (non-algebraizable) formal scheme over $\mathbb{Z}_p$. See e.g. Theorem 4.1 in https://arxiv.org/pdf …

$\newcommand{\cA}{\mathcal{A}}\newcommand{\cB}{\mathcal{B}}\newcommand{\bZ}{\mathbb{Z}}$No, that does not imply that $\cA$ splits over $R$. In fact, if $\cA_1=\cA\times_R R/p$ is isogenous to a produc …

Let's first figure out why the definition given in Berthelot-Ogus coincides with the one from the Stacks project. Unraveling the definition 5.8.3 we see that for a sheaf $G$ on $(Y/W)_{cris}$ the in …

$\newcommand{\Spec}{\mathrm{Spec}}\newcommand{\oL}{\overline{L}}\newcommand{\bG}{\mathbb{G}}\newcommand{\bZ}{\mathbb{Z}}\newcommand{\cL}{\mathcal{L}}$Not in general. The sequence of $p$-divisible grou …

Let $X$ be a curve of genus $g\geq 2$ over a number field $K$. If $\mathrm{rk} \,\mathrm{Jac}\, X$ is less than $g$ there is a $p$-adic method of bounding $\# X(K)$ due to Chabauty and Coleman (see ht …

1)Yes, such decomposition follows from the fact that Frobenius on the de Rham-Witt differential forms acts in a way that slopes on $H^i(X, W\Omega^j)[1/p]$ are in the interval $[j,j+1)$. This forces t …