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Manin proves Mordel's conjecture for function fields in characteristic zero.his proof has a gap but Coleman fill this gap and restate Manin proof in a more modern language.both of them work over chara …

assume that $Spf\,A\to Spf\,\mathbb{Z}_p[[t_1,...,t_n]]$ is a closed immersion of flat integral formal schemes over $\mathbb{Z_p}$. I see Kisin several time use that if $Spf\,A(O_K)\subset SPf\,\mathb …

I want to read Manin proof of Mordell Conjecture over function fields.I understand most of the article but I have problems with "kernel theorem"and it's proof: consider $A$ is an abelian variety over …

There is a functor from the category of higher displays over $k$ of type $\mu$ to the category of isocrystals over $k$ where $k$ is an algebraically closed field where you forget about the filtration …

Chevalley’s affineness criterion says that if $f: X\to Y$ is surjective and finite, $Y$ is Noetherian and $X$ is affine then $Y$ is also affine. The usual proof uses Serre's criterion and Noetherian i …

I'm reading Laurie's note about Fargues-Fontaine Curve and I think he uses a different definition of $B_{\mathrm{cris}}$. Usually when $R$ is a perfect ring of characteristic $p$, $A_{\mathrm{cris}}(R …

I know that by using Hodge decomposition and the fact that Schubert cells are Hodge cycles you can compute the Hodge numbers of Grassmanian but is there a more elementary way to compute sheaf cohomolo …

Deligne's idea was that Shimura varieties should be understood as moduli space of motives(with extra structures). lot's of Shimura varieties of abelian type can be understood as moduli space of abelia …

for your first question, the idea is basically what MikhailBorovoi said but of course you have to be careful to get a model outside $S$: add all the prime divisors of dominators of the equations defin …

We can associate two $\mathbb Q_p$ vector spaces to a $p$-divisible group, and I'm a little confused about the relation between these two groups. First of all, I think part of my problem is that when …

I think if you can read french the best source is Deligne book but there are several interesting articles by katz you can read for example:On the differentiation of de rham cohomology classes with res …

I'm trying to read Scholze's article "Etale cohomology of diamonds" (arXiv link) and both in this article and in Berkeley notes, the diamonds are defined as sheaves on the category of characteristic $ …

I'm reading Manin's article on formal groups and I have a problem with Lemma 1.1. Consider $k$ a prefect ring of characteristic $p$ and $(A,m,k)$ a noetherian complete local ring of the same characte …

In the lecture note of Bhatt from Arizona winter school 2017, there is an exercise which claims if X is a proper smooth scheme defined over $\mathbb{Z}[1/N]$ and if there is a polynomial $P$ such that …

Assume that $X=Spf R$ is p-adic formal scheme over $O_{C_p}$ with generic fiber $X_{\eta}$. I want to know why the nearby cycles map $Ru^\star \mathbb{Z/p}$ is equal to $R\Gamma_{et}(spec R[1/p],\math …