On the reduction modulo p of an absolutely irreducible polynomial f (x, y)

1997 ◽  
Vol 68 (2) ◽  
pp. 129-138 ◽  
Author(s):  
Umberto Zannier
Keyword(s):  
Irreducible Polynomial ◽  
Reduction Modulo ◽  
Modulo P ◽  
Absolutely Irreducible Polynomial

scholarly journals CONCENTRATION OF POINTS ON MODULAR QUADRATIC FORMS

2011 ◽  
Vol 07 (07) ◽  
pp. 1835-1839 ◽  
Author(s):  
ANA ZUMALACÁRREGUI
Keyword(s):  
Quadratic Form ◽  
Upper Bound ◽  
Quadratic Forms ◽  
Irreducible Polynomial ◽  
Number Of Solutions ◽  
Modulo P ◽  
Absolutely Irreducible Polynomial ◽  
Mod P

Let Q(x, y) be a quadratic form with discriminant D ≠ 0. We obtain non-trivial upper bound estimates for the number of solutions of the congruence Q(x, y) ≡ λ ( mod p), where p is a prime and x, y lie in certain intervals of length M, under the assumption that Q(x, y) - λ is an absolutely irreducible polynomial modulo p. In particular, we prove that the number of solutions to this congruence is Mo(1) when M ≪ p1/4. These estimates generalize a previous result by Cilleruelo and Garaev on the particular congruence xy ≡ λ( mod p).

scholarly journals Integer points on algebraic curves with exceptional units

1997 ◽  
Vol 63 (2) ◽  
pp. 145-164 ◽  
Author(s):  
Dimitrios Poulakis
Keyword(s):  
Elliptic Equations ◽  
Algebraic Curves ◽  
Irreducible Polynomial ◽  
Algebraic Number Field ◽  
Genus Zero ◽  
Integer Points ◽  
Regular Functions ◽  
Affine Curves ◽  
Absolutely Irreducible Polynomial ◽  
Integral Solutions

AbstractLet F(X, Y) be an absolutely irreducible polynomial with coefficients in an algebraic number field K. Denote by C the algebraic curve defined by the equation F(X, Y) = 0 and by K[C] the ring of regular functions on Cover K. Assume that there is a unit ϕ in K[C] − K such that 1 − ϕ is also a unit. Then we establish an explicit upper bound for the size of integral solutions of the equation F(X, Y) = 0, defined over K. Using this result we establish improved explicit upper bounds on the size of integral solutions to the equations defining non-singular affine curves of genus zero, with at least three points at ‘infinity’, the elliptic equations and a class of equations containing the Thue curves.

scholarly journals Constructing irreducible polynomials with prescribed level curves over finite fields

2001 ◽  
Vol 27 (4) ◽  
pp. 197-200
Author(s):  
Mihai Caragiu
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Irreducible Polynomial ◽  
Irreducible Polynomials ◽  
Level Curves ◽  
Curves Over Finite Fields ◽  
Irreducibility Criterion ◽  
Absolutely Irreducible Polynomial

We use Eisenstein's irreducibility criterion to prove that there exists an absolutely irreducible polynomialP(X,Y)∈GF(q)[X,Y]with coefficients in the finite fieldGF(q)withqelements, with prescribed level curvesXc:={(x,y)∈GF(q)2|P(x,y)=c}.

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.

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