arXiv subjects

Bryden Cais

Publications and source records attributed to Bryden Cais.

At least 19 recordsLinked to original sources

arXiv2026

Minimal $a$-numbers of Artin--Schreier covers of ordinary curves

Let $k$ be a perfect field of characteristic $p>0$, and let $d$ be a positive integer not divisible by $p$. We define a non-empty Zariski open subset $U$ of the space of polynomials of degree $d$, and for $f(x)\in U(k)$, we compute the $a$-number of the curve defined by $y^p-y=f(x)$. This $a$-number realizes a lower bound given by Booher and Cais, so the latter is tight. Our result also implies that the bound of Booher and Cais for minimal $a$-numbers of Artin-Schreier covers of ordinary curves is tight.

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arXiv2025

Class field theory for function fields and finite abelian torsors

Let $U$ be a smooth and connected curve over an algebraically closed field of positive characteristic, with smooth compactification $X$. We generalize classical Geometric Class Field theory to provide a classification of fppf $G$-torsors over $U$ in terms of isogenies of generalized Jacobians, for any finite abelian group scheme $G$. We then apply this classification to give a novel description of the abelianized Nori fundamental group scheme of $U$ in terms of the Serre--Oort fundamental groups of generalized Jacobians of $X$; when $U=X$ is projective, we recover a well known description of the abelianized fundamental group scheme of $X$ as the projective limit of all torsion subgroup schemes of its Jacobian.

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arXiv2024

Iwasawa theory of Frobenius-torsion class group schemes

We establish a new Iwasawa theory for the kernel of Frobenius on Jacobians of curves in geometric $\mathbf{Z}_p$-towers over the projective line in characteristic $p$, thereby proving several of the main conjectures of [arXiv:2107.12555].

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arXiv2023

$p$-torsion for unramified Artin--Schreier covers of curves

Let $Y\to X$ be an unramified Galois cover of curves over a perfect field $k$ of characteristic $p>0$ with $\mathrm{Gal}(Y/X)\cong\mathbb{Z}/p\mathbb{Z}$, and let $J_X$ and $J_Y$ be the Jacobians of $X$ and $Y$ respectively. We consider the $p$-torsion subgroup schemes $J_X[p]$ and $J_Y[p]$, analyze the Galois-module structure of $J_Y[p]$, and find restrictions this structure imposes on $J_Y[p]$ (for example, as manifested in its Ekedahl--Oort type) taking $J_X[p]$ as given.

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arXiv2022

Iwasawa Theory for $p$-torsion Class Group Schemes in Characteristic $p$

We investigate a novel geometric Iwasawa theory for $\mathbf{Z}_p$-extensions of function fields over a perfect field $k$ of characteristic $p>0$ by replacing the usual study of $p$-torsion in class groups with the study of $p$-torsion class group schemes. That is, if $\cdots \to X_2 \to X_1 \to X_0$ is the tower of curves over $k$ associated to a $\mathbf{Z}_p$-extension of function fields totally ramified over a finite non-empty set of places, we investigate the growth of the $p$-torsion group scheme in the Jacobian of $X_n$ as $n\rightarrow \infty$. By Dieudonné theory, this amounts to studying the first de Rham cohomology groups of $X_n$ equipped with natural actions of Frobenius and of the Cartier operator $V$. We formulate and test a number of conjectures which predict striking regularity in the $k[V]$-module structure of the space $M_n:=H^0(X_n, Ω^1_{X_n/k})$ of global regular differential forms as $n\rightarrow \infty.$ For example, for each tower in a basic class of $\mathbf{Z}_p$-towers we conjecture that the dimension of the kernel of $V^r$ on $M_n$ is given by $a_r p^{2n} + λ_r n + c_r(n)$ for all $n$ sufficiently large, where $a_r, λ_r$ are rational constants and $c_r : \mathbf{Z}/m_r \mathbf{Z} \to \mathbf{Q}$ is a periodic function, depending on $r$ and the tower. To provide evidence for these conjectures, we collect extensive experimental data based on new and more efficient algorithms for working with differentials on $\mathbf{Z}_p$-towers of curves, and we prove our conjectures in the case $p=2$ and $r=1$.

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arXiv2022

Iwasawa Dieudonné theory of function fields

Let $k$ be a perfect field of characteristic $p$ and $Γ$ an infinite, first countable pro-$p$ group. We study the behavior of the $p$-primary part of the "motivic class group", i.e. the full $p$-divisible group of the Jacobian, in any $Γ$-tower of function fields over $k$ that is unramified outside a finite (possibly empty) set of places $Σ$, and totally ramified at every place of $Σ$. When $Σ=\emptyset$ and $Γ$ is a torsion free $p$-adic Lie group, we obtain asymptotic formulae which show that the $p$-torsion class group schemes grow in a remarkably regular manner. In the ramified setting $Σ\neq\emptyset$, we obtain a similar asymptotic formula for the $p$-torsion in "physical class groups", i.e. the $k$-rational points of the Jacobian, which generalizes the work of Mazur and Wiles, who studied the case $Γ=\mathbf{Z}_p$.

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arXiv2020

a-Numbers of Curves in Artin-Schreier Covers

Let $π: Y \to X$ be a branched $\mathbf{Z}/p \mathbf{Z}$-cover of smooth, projective, geometrically connected curves over a perfect field of characteristic $p>0$. We investigate the relationship between the $a$-numbers of $Y$ and $X$ and the ramification of the map $π$. This is analogous to the relationship between the genus (respectively $p$-rank) of $Y$ and $X$ given the Riemann-Hurwitz (respectively Deuring--Shafarevich) formula. Except in special situations, the $a$-number of $Y$ is not determined by the $a$-number of $X$ and the ramification of the cover, so we instead give bounds on the $a$-number of $Y$. We provide examples showing our bounds are sharp. The bounds come from a detailed analysis of the kernel of the Cartier operator.

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arXiv2019

Breuil-Kisin Modules via crystalline cohomology

For a perfect field $k$ of characteristic $p>0$ and a smooth and proper formal scheme $\mathscr{X}$ over the ring of integers of a finite and totally ramified extension $K$ of $W(k)[1/p]$, we propose a cohomological construction of the Breuil-Kisin modules attached to the $p$-adic étale cohomology $H^i_{\mathrm{ét}}(\mathscr{X}_{\overline{K}},\mathbf{Z}_p)$. We then prove that our proposal works when $p>2$, $i < p-1$, and the crystalline cohomology of the special fiber of $\mathscr{X}$ is torsion-free in degrees $i$ and $i+1$.

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arXiv2016

The Geometry of Hida Families I: $Λ$-adic de Rham cohomology

We construct the $Λ$-adic de Rham analogue of Hida's ordinary $Λ$-adic étale cohomology and of Ohta's $Λ$-adic Hodge cohomology, and by exploiting the geometry of integral models of modular curves over the cyclotomic extension of $\mathbf{Q}_p$, we give a purely geometric proof of the expected finiteness, control, and $Λ$-adic duality theorems. Following Ohta, we then prove that our $Λ$-adic module of differentials is canonically isomorphic to the space of ordinary $Λ$-adic cuspforms. In the sequel to this paper, we construct the crystalline counterpart to Hida's ordinary $Λ$-adic étale cohomology, and employ integral $p$-adic Hodge theory to prove $Λ$-adic comparison isomorphisms between all of these cohomologies. As applications of our work in this paper and the sequel, we will be able to provide a "cohomological" construction of the family of $(φ,Γ)$-modules attached to Hida's ordinary $Λ$-adic étale cohomology by the work of Dee, as well as a new and purely geometric proof of Hida's finitenes and control theorems. We are also able to prove refinements of theorems of Mazur-Wiles and of Ohta.

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arXiv2016

Dieudonne crystals and Wach modules for p-divisible fgroups

Let $k$ be a perfect field of characteristic $p>2$ and $K$ an extension of $F=\mathrm{Frac} W(k)$ contained in some $F(μ_{p^r})$. Using crystalline Dieudonné theory, we provide a classification of $p$-divisible groups over $\mathscr{O}_K$ in terms of finite height $(φ,Γ)$-modules over $\mathfrak{S}:=W(k)[[u]]$. Although such a classification is a consequence of (a special case of) the theory of Kisin--Ren, our construction gives an independent proof and allows us to recover the Dieudonné crystal of a $p$-divisible group from the Wach module associated to its Tate module by Berger--Breuil or by Kisin--Ren.

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arXiv2015

On F-crystalline representations

We extend the theory of Kisin modules and crystalline representations to allow more general coefficient fields and lifts of Frobenius. In particular, for a finite and totally ramified extension $F/\mathbb Q_p$, and an arbitrary finite extension $K/F$, we construct a general class of infinite and totally wildly ramified extensions $K_\infty/K$ so that the functor $V\mapsto V|_{G_{K_\infty}}$ is fully-faithfull on the category of $F$-crystalline representations $V$. We also establish a new classification of $F$-Barsotti-Tate groups via Kisin modules of height 1 which allows more general lifts of Frobenius.

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arXiv2014

The Geometry of Hida Families II: $Λ$-adic $(φ,Γ)$-modules and $Λ$-adic Hodge Theory

We construct the $Λ$-adic crystalline and Dieudonné analogues of Hida's ordinary $Λ$-adic étale cohomology, and employ integral $p$-adic Hodge theory to prove $Λ$-adic comparison isomorphisms between these cohomologies and the $Λ$-adic de Rham cohomology studied in the prequel to this paper as well as Hida's $Λ$-adic étale 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 étale cohomology by the work of Dee, and we give a new and purely geometric proof of Hida's finitenes and control theorems. We also prove suitable $Λ$-adic duality theorems for each of the cohomologies we construct.

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arXiv2014

A characterization of strictly APF extensions

Let K denote a finite extension of Qp. We give necessary and sufficient conditions for an infinite totally wildly ramified extension L/K to be strictly APF in the sense of Fontaine-Wintenberger. Our conditions are phrased in terms of the existence of a certain tower of intermediate subfields. These conditions are well-suited to producing examples of strictly APF extensions, and in particular, our main theorem proves that the phi-iterate extensions previously considered by the first two authors are strictly APF.

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arXiv2013

Canonical Cohen rings for norm fields

Fix $K/\mathbf{Q}_p$ a finite extension and let $L/K$ be an infinite, strictly APF extension in the sense of Fontaine--Wintenberger. Let $X_K(L)$ denote its associated norm field. The goal of this paper is to associate to $L/K$, in a canonical and functorial way, a $p$-adically complete subring $\mathbf{A}_{L/K}^+ \subset \widetilde{\mathbf{A}}^+$ whose reduction modulo~$p$ is contained in the valuation ring of $X_K(L)$. When the extension $L/K$ is of a special form, which we call a $φ$-iterate extension, we prove that $X_K(L)$ is (at worst) a finite purely inseparable extension of the fraction field of $\mathbf{A}_{L/K}^+/(p)$. The class of $φ$-iterate extensions includes all Lubin--Tate extensions, as well as many other extensions such as the non-Galois ``Kummer" extension occurring in work of Faltings, Breuil, and Kisin. In particular, our work provides a canonical and functorial construction of every characteristic zero lift of the norm fields that have thus far played a foundational role in (integral) $p$-adic Hodge theory, as well as many other cases which have yet to be studied.

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arXiv2013

On the U_p-operator in characteristic p

For a perfect field κof characteristic p>0, a positive ingeger N not divisible by p, and an arbitrary subgroup Γof GL_2(Z/NZ), we prove (with mild additional hypotheses when p\le 3) that the U-operator on the space M_k(Γ/κ) of (Katz) modular forms for Γover κinduces a surjection U:M_{k}(Γ/κ)\rightarrow M_{k'}(Γ/κ) for all k\ge p+2, where k'=(k-k_0)/p + k_0 with 2\le k_0\le p+1 the unique integer congruent to k modulo p. When κ=F_p, p\ge 5, N\neq 2,3, and Γis the subgroup of upper-triangular or upper-triangular unipotent matrices, this recovers a recent result of Dewar.

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arXiv2012

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 geometric proof of Hida's finitenes and control theorems. We are also able to prove refinements of theorems of Mazur-Wiles and of Ohta; in particular, we prove that there is a canonical isomorphism between the module of ordinary Λ-adic cuspforms and the part of the crystalline cohomology of the Igusa tower on which Frobenius acts invertibly.

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arXiv2012

Random Dieudonne modules, random p-divisible groups, and random curves over finite fields

We describe a probability distribution on isomorphism classes of principally quasi-polarized p-divisible groups over a finite field k of characteristic p which can reasonably be thought of as "uniform distribution," and we compute the distribution of various statistics (p-corank, a-number, etc.) of p-divisible groups drawn from this distribution. It is then natural to ask to what extent the p-divisible groups attached to a randomly chosen hyperelliptic curve (resp. curve, resp. abelian variety) over k are uniformly distributed in this sense. For instance, one can ask whether the proportion of genus-g curves over F_p whose Jacobian is ordinary approaches the limit that such a heuristic would predict. This heuristic is analogous to conjectures of Cohen-Lenstra type for fields k of characteristic other than p, in which case the random p-divisible group is defined by a random matrix recording the action of Frobenius. Extensive numerical investigation reveals some cases of agreement with the heuristic and some interesting discrepancies. For example, plane curves over F_3 appear substantially less likely to be ordinary than hyperelliptic curves over F_3.

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arXiv2009

Canonical extensions of Néron models of Jacobians

Let A be the Néron model of an abelian variety A_K over the fraction field K of a discrete valuation ring R. Due to work of Mazur-Messing, there is a functorial way to prolong the universal extension of A_K by a vector group to a smooth and separated group scheme over R, called the canonical extension of A. In this paper, we study the canonical extension when A_K=J_K is the Jacobian of a smooth proper and geometrically connected curve X_K over K. Assuming that X_K admits a proper flat regular model X over R that has generically smooth closed fiber, our main result identifies the identity component of the canonical extension with a certain functor Pic^{\natural,0}_{X/R} classifying line bundles on X that have partial degree zero on all components of geometric fibers and are equipped with a regular connection. This result is a natural extension of a theorem of Raynaud, which identifies the identity component of the Néron model J of J_K with the functor Pic^0_{X/R}. As an application of our result, we prove a comparison isomorphism between two canonical integral structures on the de Rham cohomology of X_K.

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