scholarly journals On the locus of 2-dimensional crystalline representations with a given reduction modulo p

2020 ◽  
Vol 14 (3) ◽  
pp. 643-700
Author(s):  
Sandra Rozensztajn
Keyword(s):  
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.

On the Local Structure of Mahler Systems

2020 ◽  
Author(s):  
Julien Roques
Keyword(s):  
Differential Equations ◽  
Local Structure ◽  
Alternative Proof ◽  
Reduction Modulo ◽  
Modulo P ◽  
Almost All ◽  
The Given ◽  
Algebraic Solutions

Abstract This paper is a 1st step in the direction of a better understanding of the structure of the so-called Mahler systems: we classify these systems over the field $\mathscr{H}$ of Hahn series over $\overline{{\mathbb{Q}}}$ and with value group ${\mathbb{Q}}$. As an application of (a variant of) our main result, we give an alternative proof of the following fact: if, for almost all primes $p$, the reduction modulo $p$ of a given Mahler equation with coefficients in ${\mathbb{Q}}(z)$ has a full set of algebraic solutions over $\mathbb{F}_{p}(z)$, then the given equation has a full set of solutions in $\overline{{\mathbb{Q}}}(z)$ (this is analogous to Grothendieck’s conjecture for differential equations).

scholarly journals Reduction modulo p of cyclic group actions of order pr

1999 ◽  
Vol 141 (1) ◽  
pp. 37-58
Author(s):  
Keiko Kinugawa ◽  
Masayoshi Miyanishi
Keyword(s):  
Cyclic Group ◽  
Group Actions ◽  
Reduction Modulo ◽  
Modulo P
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