scholarly journals SQUARE-FREE DISCRIMINANTS OF FROBENIUS RINGS

2010 ◽  
Vol 06 (06) ◽  
pp. 1391-1412 ◽  
Author(s):  
CHANTAL DAVID ◽  
JORGE JIMÉNEZ URROZ
Keyword(s):  
A Priori ◽  
Congruence Class ◽  
Complex Multiplication ◽  
Trivial Case ◽  
Frobenius Rings ◽  
Reduction Modulo ◽  
Modulo P ◽  
Imaginary Field ◽  
Ring Of Endomorphisms ◽  
Precise Asymptotic

Let E be an elliptic curve over ℚ. We know that the ring of endomorphisms of its reduction modulo an ordinary prime p is an order of the quadratic imaginary field generated by the Frobenius element πp. However, except in the trivial case of complex multiplication, very little is known about the fields that appear as algebras of endomorphisms when p varies. In this paper, we study the endomorphism ring by looking at the arithmetic of [Formula: see text], the discriminant of the characteristic polynomial of πp. In particular, we give a precise asymptotic for the function counting the number of primes p up to x such that [Formula: see text] is square-free and in certain congruence class fixed a priori, when averaging over elliptic curves defined over the rationals. We discuss the relation of this result with the Lang–Trotter conjecture, and some other questions on the curve modulo p.

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 POWER RESIDUES OF FOURIER COEFFICIENTS OF ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION

2009 ◽  
Vol 05 (01) ◽  
pp. 109-124
Author(s):  
TOM WESTON ◽  
ELENA ZAUROVA
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Fourier Coefficient ◽  
Fourier Coefficients ◽  
Complex Multiplication ◽  
Roots Of Unity ◽  
Modulo P

Fix m greater than one and let E be an elliptic curve over Q with complex multiplication. We formulate conjectures on the density of primes p (congruent to one modulo m) for which the pth Fourier coefficient of E is an mth power modulo p; often these densities differ from the naive expectation of 1/m. We also prove our conjectures for m dividing the number of roots of unity lying in the CM field of E; the most involved case is m = 4 and complex multiplication by Q(i).

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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