scholarly journals Explicit Reduction Modulo p of Certain Two-Dimensional Crystalline Representations

2009 ◽  
Author(s):  
K. Buzzard ◽  
T. Gee
Keyword(s):  
Two Dimensional ◽  
Crystalline Representations ◽  
Reduction Modulo ◽  
Modulo P

scholarly journals An algorithm for computing the reduction of 2-dimensional crystalline representations of Gal(ℚ¯p/ℚp)

2018 ◽  
Vol 14 (07) ◽  
pp. 1857-1894 ◽  
Author(s):  
Sandra Rozensztajn
Keyword(s):  
Galois Representation ◽  
Langlands Correspondence ◽  
Dimension Formula ◽  
Crystalline Representations ◽  
Reduction Modulo

We describe an algorithm to compute the reduction modulo [Formula: see text] of a crystalline Galois representation of dimension [Formula: see text] of [Formula: see text] with distinct Hodge–Tate weights via the semi-simple modulo [Formula: see text] Langlands correspondence. We give some examples computed with an implementation of this algorithm in SAGE.

scholarly journals On the Supersingular Reduction of K3 Surfaces with Complex Multiplication

2018 ◽  
Vol 2020 (20) ◽  
pp. 7306-7346
Author(s):  
Kazuhiro Ito
Keyword(s):  
Comparison Theorem ◽  
Complex Multiplication ◽  
K3 Surfaces ◽  
Explicit Description ◽  
Brauer Group ◽  
K3 Surface ◽  
Good Reduction ◽  
Picard Number ◽  
Reduction Modulo ◽  
Modulo P

Abstract We study the good reduction modulo $p$ of $K3$ surfaces with complex multiplication. If a $K3$ surface with complex multiplication has good reduction, we calculate the Picard number and the height of the formal Brauer group of the reduction. Moreover, if the reduction is supersingular, we calculate its Artin invariant under some assumptions. Our results generalize some results of Shimada for $K3$ surfaces with Picard number $20$. Our methods rely on the main theorem of complex multiplication for $K3$ surfaces by Rizov, an explicit description of the Breuil–Kisin modules associated with Lubin–Tate characters due to Andreatta, Goren, Howard, and Madapusi Pera, and the integral comparison theorem recently established by Bhatt, Morrow, and Scholze.

scholarly journals Extensions of rank one (φ,Γ)-modules and crystalline representations

2010 ◽  
Vol 147 (2) ◽  
pp. 375-427 ◽  
Author(s):  
Seunghwan Chang ◽  
Fred Diamond
Keyword(s):  
Two Dimensional ◽  
Crystalline Representations ◽  
Graphic Representations ◽  
Rank One

AbstractLetKbe a finite unramified extension ofQp. We parametrize the (φ,Γ)-modules corresponding to reducible two-dimensional$\overline {\F }_p$-representations ofGKand characterize those which have reducible crystalline lifts with certain Hodge–Tate weights.

scholarly journals THE WEIGHT PART OF SERRE’S CONJECTURE FOR

2015 ◽  
Vol 3 ◽  
Author(s):  
TOBY GEE ◽  
TONG LIU ◽  
DAVID SAVITT
Keyword(s):  
Two Dimensional ◽  
Crystalline Representations ◽  
Serre's Conjecture ◽  
Weight Part ◽  
Local Methods ◽  
Totally Real ◽  
Totally Real Fields ◽  
Serre’S Conjecture

AbstractLet $p>2$ be prime. We use purely local methods to determine the possible reductions of certain two-dimensional crystalline representations, which we call pseudo-Barsotti–Tate representations, over arbitrary finite extensions of $\mathbb{Q}_{p}$. As a consequence, we establish (under the usual Taylor–Wiles hypothesis) the weight part of Serre’s conjecture for $\text{GL}(2)$ over arbitrary totally real fields.

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