Effective convergence bounds for Frobenius structures on connections

$q$-deformation with ($φ, Γ$) structure of the de Rham cohomology of the Legendre family of elliptic curves

In the late '60s, B. Dwork studied a Frobenius structure compatible with the classical hypergeometric differential equation with parameters $\left(\frac{1}{2},\frac{1}{2} ; 1 \right)$ by analyzing behavior of solutions of the differential equation under Frobenius transformation. Recently, P. Scholze conjectured the existence of $q$-de Rham cohomology groups for any $\mathbb{Z}$-scheme. In this paper, we give a Frobenius structure compatible with the $q$-hypergeometric differential equation with parameters $(q^{\frac12},q^{\frac12};q)$ by showing a $q$-analogue of some results of Dwork. This co…

Sur les π-exponentielles I : Vecteurs de Witt annulés par Frobenius et Algorithme de (leur rayon de convergence)

(Quoted from the article) Our object is the theory of "π-exponentials" Pulita developed in his thesis [...] We start with an abstract algebra statement about the structure of the kernel of iterations of the Frobenius endomorphism on the ring of Witt vectors with coordinates in the ring of integers of an ultrametric extension of $\mathbf{Q}_p$. Provided sufficiently (ramified) roots of unity are available, it is, unexpectedly simply, a principal ideal with respect to an explicit generator essentially given by Pulita's π-exponential. This result is a consequence and a reformulation of core facts…

Explicit Frobenius lifts on elliptic curves

In this article we give explicit formulae for a lift of the relative Frobenius morphism between elliptic curves and show how one can compute this lift in the case of ordinary reduction in odd characteristic. Our theory can also be used in the case of supersingular reduction. By means of the explicit formulae that describe a Frobenius lift, we are able to generalize Mestre's 2-adic arithmetic geometric mean (AGM) sequence of elliptic curves to odd characteristic, and prove its convergence. As an application, we give an efficient point counting algorithm for ordinary elliptic curves which is bas…

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。

特征 $p$ 概形的棱镜晶体

Prismatic Crystals for schemes in characteristic $p$

构造棱镜包络层，证明棱镜晶体与可积拓扑拟幂零 $p$-联络等价，并识别棱镜上同调为 de Rham 复形；通过提升的 Ogus-Vologodsky 函子给出局部等价，进而得到 l.c.i. 概形上棱镜晶体与晶体晶体的整体等价。

Introduction to p-adic q-difference equations (weak Frobenius structure and transfer theorems)

Inspired by the theory of p-adic differential equations, this paper introduces an analogous theory for q-difference equations over a local field, when |q|=1. We define some basic concepts, for instance the generic radius of convergence, introduce technical tools, such as a twisted Taylor formula for analytic functions, and prove some fundamental statements, such as an effective bound theorem, the existence of a weak Frobenius structure and a transfer theorem in regular singular disks.

Convergence polygons for connections on nonarchimedean curves

This is a survey article on ordinary differential equations over nonarchimedean fields based on the author's lecture at the 2015 Simons Symposium on nonarchimedean and tropical geometry. Topics include: the convergence polygon associated to a differential equation (or a connection on a curve); links to the formal classification of differential equations (Turrittin-Levelt); index formulas for de Rham cohomology of connections; ramification of finite morphisms; relations with the Oort lifting problem on automorphisms of curves. The appendices include some new technical results and an extensive t…

Densities of eigenspaces of Frobenius and distributions of R-modules

For any polynomial $p\left(x\right)$ over $\mathbb{F}_{l}$ we determine the asymptotic density of hyperelliptic curves over $\mathbb{F}_{q}$ of genus $g$ for which $p\left(x\right)$ divides the characteristic polynomial of Frobenius acting on the $l$-torsion of the Jacobian, and give an explicit formula for this density. We prove this result as a consequence of more general density theorems for quotients of Tate modules of such curves, viewed as modules over the Frobenius. The proof involves the study of measures on $R$-modules over arbitrary rings $R$ which are finite $\mathbb{Z}_{l}$-algebra…

Radius of convergence of p-adic connections and the Berkovich ramification locus

We apply the theory of the radius of convergence of a p-adic connection to the special case of the direct image of the constant connection via a finite morphism of compact p-adic curves, smooth in the sense of rigid geometry. In the case of an etale covering of curves with good reduction, we get a lower bound for that radius and obtain a new geometric proof of a variant of the p-adic Rolle theorem of Robert and Berkovich. We take this opportunity to clarify the relation between our notion of radius of convergencand the more intrinsic one used by Kedlaya

有理数域上椭圆曲线具有固定弗罗贝尼乌斯域的无穷多素数

Infinitely many primes with a fixed Frobenius field for an elliptic curve over $\mathbb{Q}$

该研究针对有理数域上特定非复乘椭圆曲线与虚二次域，证明存在无穷多素数使得椭圆曲线在该素数处的弗罗贝尼乌斯域等于给定虚二次域，建立了相关计数函数的下界与上界，方法涉及复乘理论等。