Analytic Geometry and Hodge-Frobenius Structure Continued

Frobenius distribution for pairs of elliptic curves and exceptional isogenies

Let E and E' be two elliptic curves over a number field. We prove that the reductions of E and E' at a finite place p are geometrically isogenous for infinitely many p, and draw consequences for the existence of supersingular primes. This result is an analogue for distributions of Frobenius traces of known results on the density of Noether-Lefschetz loci in Hodge theory. The proof relies on dynamical properties of the Hecke correspondences on the modular curve.

Frobenius modules over multivariate Robba rings

We describe a class of multivariate series rings generalizing the usual Robba ring over a p-adic field, and give a basic development of phi-modules over such rings. This makes it possible to give a unified survey of a number of recent developments in p-adic Hodge theory.

Frobenius splitting and ordinarity

We examine the relationship between the notion of Frobenius splitting and ordinarity for varieties. We show the following: a) The de Rham-Witt cohomology groups $H^i(X, W({\mathcal O}_X))$ of a smooth projective Frobenius split variety are finitely generated over $W(k)$. b) we provide counterexamples to a question of V. B. Mehta that Frobenius split varieties are ordinary or even Hodge-Witt. c) a Kummer $K3$ surface associated to an Abelian surface is $F$-split (ordinary) if and only if the associated Abelian surface is $F$-split (ordinary). d) for a $K3$-surface defined over a number field, t…

From Hodge Index Theorem to the number of points of curves over finite fields

We push further the classical proof of Weil upper bound for the number of rational points of an absolutely irreducible smooth projective curve $X$ over a finite field in term of euclidean relationships between the Neron Severi classes in $X\times X$ of the graphs of iterations of the Frobenius morphism. This allows us to recover Ihara's bound, which can be seen as a {\em second order} Weil upper bound, to establish a new {\em third order} Weil upper bound, and using {\tt magma} to produce numerical tables for {\em higher order} Weil upper bounds. We also give some interpretation for the defect…

The Geometry of Hida Families and Λ-adic Hodge Theory

We construct Λ-adic de Rham and crystalline analogues of Hida's ordinary Λ-adic etale cohomology, and by exploiting the geometry of integral models of modular curves over the cyclotomic extension of \Q_p, we prove appropriate finiteness and control theorems in each case. We then employ integral p-adic Hodge theory to prove Λ-adic comparison isomorphisms between our cohomologies and Hida's etale cohomology. As applications of our work, we provide a "cohomological" construction of the family of (ϕ,Γ)-modules attached to Hida's ordinary Λ-adic etale cohomology by Dee, and we give a new and purely…

Frobenius 在上同调上作用的对称性与奇偶性

Symmetry and parity in Frobenius action on cohomology

本文证明 Frobenius 在晶体上同调上的 Newton 多边形具有对称性，并据此推出真光滑簇奇数次 Betti 数为偶数，回答 Serre 的问题，同时推广至有限域任意簇。

Appendix to "Congruences for rational points on varieties over finite fields" by N. Fakhruddin and C. S. Rajan

We give a positive answer to the questions 4.4 on Hodge type and 4.2 on divisibility of the eigenvalues of Frobenius of the article by N. Fakhruddin and C. S. Rajan (NT/0402230).

Weights in arithmetic geometry

The concept of weights on the cohomology of algebraic varieties was initiated by fundamental ideas and work of A. Grothendieck and P. Deligne. It is deeply connected with the concept of motives and appeared first on the singular cohomology as the weights of (possibly mixed) Hodge structures and on the etale cohomology as the weights of eigenvalues of Frobenius. But weights also appear on algebraic fundamental groups and in p-adic Hodge theory, where they become only visible after applying the comparison functors of Fontaine. After rehearsing various versions of weights, we explain some more re…

棱柱晶体与特征p下的光滑方案中的Frobenius提升模p²

Prismatic crystals for smooth schemes in characteristic $p$ with Frobenius lifting mod $p^2$

本文研究了特征p下光滑方案的棱柱晶体与p-连接之间的等价性，并扩展了Ogus的工作，同时讨论了Hodge-Tate丛的平凡化问题。

CM-lifts for Isogeny Classes of Shimura F-crystals over Finite Fields

We extend to large contexts pertaining to Shimura varieties of Hodge type a result of Zink on the existence of lifts to characteristic 0 of suitable representatives of certain isogeny classes of abelian varieties endowed with Frobenius and other endomorphisms over $\dbF_{p^q}$, whose $p$-divisible groups in mixed characteristic $(0,p)$ are with complex multiplication.

On logarithmic nonabelian Hodge theory of higher level in characteristic p

Given a natural number $m$ and a log smooth integral morphism $X\to S$ of fine log schemes of characteristic $p>0$ with a lifting of its Frobenius pull-back $X'\to S$ modulo $p^{2}$, we use indexed algebras ${\cal A}_{X}^{gp}$, ${\cal B}_{X/S}^{(m+1)}$ of Lorenzon-Montagnon and the sheaf ${\cal D}_{X/S}^{(m)}$ of log differential operators of level $m$ of Berthelot-Montagnon to construct an equivalence between the category of certain indexed ${\cal A}_{X}^{gp}$-modules with ${\cal D}_{X/S}^{(m)}$-action and the category of certain indexed ${\cal B}_{X/S}^{(m+1)}$-modules with Higgs field. Our …

Hodge filtration和晶格表示的GL_n

Hodge filtration and crystalline representations of $\mathrm{GL}_n$

本文研究了GL_n的晶格表示与Hodge滤波器之间的关系，证明了局部解析表示中存在显式的有限长度子表示，扩展了之前的结果。

A generalization of Colmez-Greenberg-Stevens formula

In this paper we study the derivatives of Frobenius and the derivatives of Hodge-Tate weights for families of Galois representations with triangulations. We give a generalization of the Fontaine-Mazur L-invariant and use it to build a formula which is a generalization of the Colmez-Greenberg-Stevens formula.

de Rham 复形的不可分解性与 Sen 算子的非半单性

Non-decomposability of the de Rham complex and non-semisimplicity of the Sen operator

本文研究了特征 $p$ 下 de Rham 复形的分解障碍，证明了存在 Hodge-to-de Rham 谱序列不退化的可提升光滑射影簇，并基于交换代数与 Steenrod 运算揭示了 Sen 算子的非半单性及 $E_{\infty}$-代数形式性条件。

关于继续分数展开的维度谱结构

On the structure of the dimension spectrum for continued fraction expansions

本文研究了继续分数展开系数受限于无限子集时的维度谱结构，证明了某些集合的维度谱完整，并探讨了不同生成方式下的维度谱特性。

Mochizuki环结构的anabelian变形、perfectoid几何与形式群的代数化

Algebraization of Mochizuki's anabelian variation of ring structures, perfectoid geometry and formal groups

本文证明存在配备乘法幺半群作用的泛一维形式群规则，并揭示其在p-adic域、perfectoid环及Mochizuki anabelian思想中的应用，为anabelian几何与perfectoid几何现象提供统一途径。

关于 $\operatorname{Spec}\mathbf{Z}$ 的绝对几何

On the Absolute Geometry of $\operatorname{Spec}\mathbf{Z}$

通过拉回算术站点的F1-结构层构造绝对F1-算术曲线，统一p进Hodge理论、复解析几何与adelic scaling站点中的基本结构，并给出Scholze启发式原理的普适几何实现。