An analogue of the field-of-norms functor and the Grothendieck Conjecture

Packets in Grothendieck's Section Conjecture

Using the identification of sections of the Galois group of the ground field into the arithmetic fundamental group with neutral fiber functors of the category of finite connections, we define the "packets" in Grothendieck's section conjecture and show their properties predicted by him.

The fundamental groupoid scheme and applications

We define a linear structure on Grothendieck's arithmetic fundamental group $π_1(X, x)$ of a scheme $X$ defined over a field $k$ of characteristic 0. It allows us to link the existence of sections of the Galois group ${\rm Gal}(\bar k/k)$ to $π_1(X, x)$ with the existence of a neutral fiber functor on the category which linearizes it. When applied to Grothendieck section conjecture, it allows us to find a $k$-structure on the universal covering, and $k$-rational pro-points at finite and infinite distance which lift given $k$-rational points. Changes as compared to the first version: (some) typ…

On the section conjecture of Grothendieck

For a given arithmetic scheme, in this paper we will introduce and discuss the monodromy action on a universal cover of the étale fundamental group and the monodromy action on an \emph{sp}-completion constructed by the graph functor, respectively; then by these results we will give a proof of the section conjecture of Grothendieck for arithmetic schemes.

有限域上曲线的 Grothendieck 猜想的数域类比

A number field analogue of the Grothendieck conjecture for curves over finite fields

本文以最大分圆扩张类比常数域扩张，建立有限域上双曲曲线 Grothendieck 猜想的数域版本，并在无分歧素数情形证实相关猜想。

Arithmetic theory of q-difference equations. The q-analogue of Grothendieck-Katz's conjecture on p-curvatures

Grothendieck's conjecture on p-curvatures predicts that an arithmetic differential equation has a full set of algebraic solutions if and only if its reduction in positive characteristic has a full set of rational solutions for almost all finite places. It is equivalent to Katz's conjectural description of the generic Galois group. In this paper we prove an analogous statement for arithmetic q-difference equation.

Modified proof of a local analogue of the Grothendieck conjecture

A local analogue of the Grothendieck Conjecture is an equivalence of the category of complete discrete valuation fields $K$ with finite residue fields of characteristic $p\ne 0$ and the category of absolute Galois groups of fields $K$ together with their ramification filtrations. The case of characteristic 0 fields $K$ was considered by Mochizuki several years ago. Then the author proved it by different method if $p>2$ (but $\operatorname{char}K=0$ or $p$). This paper represents a modified approach: it covers the case $p=2$, contains considerable technical simplifications and replaces the Galo…