scholarly journals Higher-level canonical subgroups for p-divisible groups

2011 ◽  
Vol 11 (2) ◽  
pp. 363-419 ◽  
Author(s):  
Joseph Rabinoff
Keyword(s):  
Elliptic Curves ◽  
Geometric Structure ◽  
Valuation Ring ◽  
Mixed Characteristic ◽  
Tate Group ◽  
Divisible Groups

AbstractLet R be a complete rank-1 valuation ring of mixed characteristic (0, p), and let K be its field of fractions. A g-dimensional truncated Barsotti–Tate group G of level n over R is said to have a level-n canonical subgroup if there is a K-subgroup of G ⊗RK with geometric structure (Z/pnZ)g consisting of points ‘closest to zero’. We give a non-trivial condition on the Hasse invariant of G that guarantees the existence of the canonical subgroup, analogous to a result of Katz and Lubin for elliptic curves. The bound is independent of the height and dimension of G.

scholarly journals Arithmetic 𝒟-modules on the unit disk. With an appendix by Shigeki Matsuda

2011 ◽  
Vol 148 (1) ◽  
pp. 227-268 ◽  
Author(s):  
Richard Crew
Keyword(s):  
Unit Disk ◽  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Mixed Characteristic ◽  
Complete Discrete Valuation Ring ◽  
Canonical Extensions

AbstractLet 𝒱 be a complete discrete valuation ring of mixed characteristic. We classify arithmetic 𝒟-modules on Spf(𝒱[[t]]) up to certain kind of ‘analytic isomorphism’. This result is used to construct canonical extensions (in the sense of Katz and Gabber) for objects of this category.

scholarly journals The smooth locus in infinite-level Rapoport–Zink spaces

2020 ◽  
Vol 156 (9) ◽  
pp. 1846-1872
Author(s):  
Alexander B. Ivanov ◽  
Jared Weinstein
Keyword(s):  
Elliptic Curves ◽  
Open Subset ◽  
Formal Group ◽  
Dense Open Subset ◽  
Additional Structure ◽  
Connected Component ◽  
Deformation Spaces ◽  
Modular Curve ◽  
Divisible Groups ◽  
Smooth Locus

Rapoport–Zink spaces are deformation spaces for $p$-divisible groups with additional structure. At infinite level, they become preperfectoid spaces. Let ${{\mathscr M}}_{\infty }$ be an infinite-level Rapoport–Zink space of EL type, and let ${{\mathscr M}}_{\infty }^{\circ }$ be one connected component of its geometric fiber. We show that ${{\mathscr M}}_{\infty }^{\circ }$ contains a dense open subset which is cohomologically smooth in the sense of Scholze. This is the locus of $p$-divisible groups which do not have any extra endomorphisms. As a corollary, we find that the cohomologically smooth locus in the infinite-level modular curve $X(p^{\infty })^{\circ }$ is exactly the locus of elliptic curves $E$ with supersingular reduction, such that the formal group of $E$ has no extra endomorphisms.

scholarly journals Stabilité de l’holonomie sur les variétés quasi-projectives

2011 ◽  
Vol 147 (6) ◽  
pp. 1772-1792 ◽  
Author(s):  
Daniel Caro
Keyword(s):  
Valuation Ring ◽  
Residue Field ◽  
Discrete Valuation Ring ◽  
Mixed Characteristic ◽  
Complete Discrete Valuation Ring ◽  
The Stability

AbstractLet 𝒱 be a mixed characteristic complete discrete valuation ring with perfect residue field k. We solve Berthelot’s conjectures on the stability of the holonomicity over smooth projective formal 𝒱-schemes. Then we build a category of F-complexes of arithmetic 𝒟-modules over quasi-projective k-varieties with bounded and holonomic cohomology. We obtain its stability under Grothendieck’s six operations.

scholarly journals -neighborhoods and comparison theorems

2015 ◽  
Vol 151 (10) ◽  
pp. 1945-1964 ◽  
Author(s):  
Piotr Achinger
Keyword(s):  
Valuation Ring ◽  
Open Neighborhood ◽  
Residue Field ◽  
Discrete Valuation Ring ◽  
Universal Cover ◽  
Comparison Theorems ◽  
Main Ingredient ◽  
Generic Fiber ◽  
Mixed Characteristic ◽  
Original Approach

A technical ingredient in Faltings’ original approach to$p$-adic comparison theorems involves the construction of$K({\it\pi},1)$-neighborhoods for a smooth scheme$X$over a mixed characteristic discrete valuation ring with a perfect residue field: every point$x\in X$has an open neighborhood$U$whose generic fiber is a$K({\it\pi},1)$scheme (a notion analogous to having a contractible universal cover). We show how to extend this result to the logarithmically smooth case, which might help to simplify some proofs in$p$-adic Hodge theory. The main ingredient of the proof is a variant of a trick of Nagata used in his proof of the Noether normalization lemma.

scholarly journals Elliptic curves with large Shafarevich–Tate group

2013 ◽  
Vol 133 (2) ◽  
pp. 369-374 ◽  
Author(s):  
Iftikhar A. Burhanuddin ◽  
Ming-Deh A. Huang
Keyword(s):  
Elliptic Curves ◽  
Tate Group

scholarly journals Semistable models of elliptic curves over residue characteristic 2

2020 ◽  
pp. 1-9
Author(s):  
Jeffrey Yelton
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Mixed Characteristic ◽  
Semistable Model ◽  
Characteristic 2 ◽  
Residue Characteristic ◽  
Legendre Form

Abstract Given an elliptic curve E in Legendre form $y^2 = x(x - 1)(x - \lambda )$ over the fraction field of a Henselian ring R of mixed characteristic $(0, 2)$ , we present an algorithm for determining a semistable model of E over R that depends only on the valuation of $\lambda $ . We provide several examples along with an easy corollary concerning $2$ -torsion.

Berkeley Lectures on p-adic Geometry

2020 ◽  
Author(s):  
Peter Scholze ◽  
Jared Weinstein
Keyword(s):  
Moduli Spaces ◽  
Reductive Group ◽  
Arithmetic Geometry ◽  
Mixed Characteristic ◽  
Algebraic Spaces ◽  
New Ideas ◽  
The Subject ◽  
Perfectoid Spaces ◽  
Divisible Groups ◽  
The University

This book presents an important breakthrough in arithmetic geometry. In 2014, this book's author delivered a series of lectures at the University of California, Berkeley, on new ideas in the theory of p-adic geometry. Building on his discovery of perfectoid spaces, the author introduced the concept of “diamonds,” which are to perfectoid spaces what algebraic spaces are to schemes. The introduction of diamonds, along with the development of a mixed-characteristic shtuka, set the stage for a critical advance in the discipline. This book shows that the moduli space of mixed-characteristic shtukas is a diamond, raising the possibility of using the cohomology of such spaces to attack the Langlands conjectures for a reductive group over a p-adic field. The book follows the informal style of the original Berkeley lectures, with one chapter per lecture. It explores p-adic and perfectoid spaces before laying out the newer theory of shtukas and their moduli spaces. Points of contact with other threads of the subject, including p-divisible groups, p-adic Hodge theory, and Rapoport-Zink spaces, are thoroughly explained.

scholarly journals Good reduction of K3 surfaces

2017 ◽  
Vol 154 (1) ◽  
pp. 1-35 ◽  
Author(s):  
Christian Liedtke ◽  
Yuya Matsumoto
Keyword(s):  
Cohomology Group ◽  
Valuation Ring ◽  
Residue Field ◽  
Discrete Valuation Ring ◽  
Study Group ◽  
K3 Surface ◽  
Good Reduction ◽  
Mixed Characteristic ◽  
Galois Action ◽  
Stable Reduction

Let $K$ be the field of fractions of a local Henselian discrete valuation ring ${\mathcal{O}}_{K}$ of characteristic zero with perfect residue field $k$. Assuming potential semi-stable reduction, we show that an unramified Galois action on the second $\ell$-adic cohomology group of a K3 surface over $K$ implies that the surface has good reduction after a finite and unramified extension. We give examples where this unramified extension is really needed. Moreover, we give applications to good reduction after tame extensions and Kuga–Satake Abelian varieties. On our way, we settle existence and termination of certain flops in mixed characteristic, and study group actions and their quotients on models of varieties.

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