A Dictionary between Fontaine-Theory and its Analogue in Equal Characteristic

Notes on an analogue of the Fontaine-Mazur conjecture

We estimate the proportion of function fields satisfying certain conditions which imply a function-field analogue of the Fontaine-Mazur conjecture. As a byproduct, we compute the fraction of abelian varieties (or even Jacobians) over a finite field which have a rational point of order l.

Simple connectivity of Fargues--Fontaine curves

We show that the Fargues--Fontaine curve associated to an algebraically closed field of characteristic p is geometrically simply connected; that is, its base extension from Q_p to any complete algebraically closed overfield admits no nontrivial connected finite etale covering. We then deduce from this an analogue for perfectoid spaces (and some related objects) of Drinfeld's lemma on the fundamental group of a product of schemes in characteristic p.

丰塔内晶体函子的积分类似物

An integral analogue of Fontaine's crystalline functor

研究在完美域\(k\)上维特向量\(W\)上的光滑形式概型\(\mathfrak{X}\)，构造函子\(\mathbb{D}_{\mathrm{crys}}\)，证明其在特定条件下的性质及诱导范畴等价关系，还用它加强\(p -\)可除群的棱柱形迪厄多内理论研究。

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

The paper contains a construction of an analogue of the Fontaine-Wintenberger field-of-norms functor for higher dimensional local fields. This construction is done completely in terms of the ramification theory of such fields. It is applied to deduce the mixed characteristic case of a local analogue of the Grothendieck Conjecture for these fields from its characteristic p case, which was proved earlier by the author.

单变量$p$进函数与${\rm GL}_2(\mathbf{Q}_p)$的表示

Fonctions d'une variable $p$-adique et représentations de ${\rm GL}_2(\mathbf{Q}_p)$

本研究拓展了Fontaine环与$p$进泛函分析的对应关系，给出了${\rm GL}_2(\mathbf{Q}_p)$主系列表示的$p$进局部Langlands对应的精细化结果。

棱镜上同调与de Rham-Witt形式

Prismatic cohomology and de Rham-Witt forms

本文为任意棱镜构造Fontaine映射的类似物，在完美情形下定义de Rham-Witt形式到棱镜上同调的典范映射并证明其为同构，进而给出$p$-完备多项式代数棱镜上同调的显式描述。