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Let K / F be a finite Galois extension of number fields and let σ be an absolutely irreducible, self-dual, complex valued representation of Gal ( K / F ) . Let p be an odd prime and consider two elliptic curves E 1 , E 2 defined over Q with good, ordinary reduction at primes above p and equivalent mod-p Galois representations. In this article, we study the variation of the parity of the multiplicities of σ in the representation space associated to the p ∞ -Selmer groups of E 1 and E 2 over K. We also compare the root numbers for the twists of E 1 and E 2 over F by σ and show that the p-parity conjecture holds for the twist of E 1 / F by σ if and only if it holds for the twist of E 2 / F by σ. We also express Mazur-Rubin-Nekovař's arithmetic local constants in terms of certain local Iwasawa invariants.

Let φ be a Drinfeld A-module of arbitrary rank and arbitrary characteristic over a finitely generated field K, and set GK=Gal(Ksep/K). Let E=EndK(φ). We show that for almost all primes p of A the image of the group ring A[GK] in EndA(Tp(φ)) is the commutant of E. In the special case E=A it follows that the representation of GK on the p-torsion points φ[p](Ksep) of φ is absolutely irreducible for almost all p.

Let p be a prime number, let F ¯ p be the algebraic closure of F p = Z / p Z , let C be an absolutely irreducible curve in A r ( F ¯ p ) and h = ( h 1 , … , h s ) a rational map defined on the curve C . We investigate the distribution in the s-dimensional unit cube ( R / Z ) s of the images through h of the F p -points of C , after a suitable embedding.

Resume Dans cet article, nous definissons et etudions la notion de corps P-reduisant qui generalise celle de corps pythagoricien. Nous montrons que si P designe un polynome absolument irreductible de Q (T1, …, Tn)[X], alors aucune extension finie et stricte de la cloture P-reduisante dans d'un corps K hilbertien de caracteristique 0 n'est P-reduisante. Dans une deuxieme partie, nous regardons un cas particulier de corps P-reduisant: les corps ultra-n-fermatiens. Nous montrons que si p est un nombre premier impair et si K est un corps de caracteristique nulle contenant les racines p2-iemes de l'unite, alors le groupe de Galois, Gal(Ku−fermp/K), (ou Ku−fermp designe la cloture ultra-p-fermatienne de K) est un pro-p-groupe sans torsion. Copyright 2001 Academic Press. This paper is devoted to the notion of P-reducing field, which generalizes the notion of a Pytagorean field. We show that if P is an absolutely irreducible polynomial of Q (T1, …, Tn)[X], then there is no proper finite P-reducing extension of the P-reducing closure of an Hilbertian field of characteristic 0. In the second part, we study a particular case of P-reducing fields: ultra-n-Fermatian fields. We show that if p is an odd prime number and K is a field of characteristic 0 containing all the p2th roots of unity, then the Galois groups, Gal(Ku−fermp/K) (where Ku−fermp is the ultra-p-Fermation closure of K) is a torsion-free pro-p-group.

Let f ( x , y ) be a polynomial defined over Z in two variables of total degree d ⩾2, and let Vp ={( x , y )∈ C p : f ( x , y )≡0 (mod p )} for each prime p , where C p ={( x , y )∈ Z 2 : 0⩽ x p and 0⩽ y p }. In this paper, we show that if f ( x , y ) is absolutely irreducible modulo p for all sufficiently large p , then we have the following distribution formula for the zeros of f ( x , y ) modulo p ,[formula]where δ is any given real number with 0⩽ δ ⩽1, and[formula]Here ‖ θ ‖ denotes the nearest distance from θ to integer, and β is a real number with 0⩽ β ⩽ 1 2 .

We characterize the polynomials P(X, Y) that are irreducible over a number field K and such that for some t ∈ K, the specialized polynomials P(tm, Y) are reducible in K[Y] for infinitely many integers m. As a consequence, we show for example that if P is absolutely irreducible and if t is neither a strict power in K nor of the form −4w4 or −w2 with w ∈ K, then P(tm, Y) is irreducible in K[Y] for infinitely many integers m (cf. Corollary 1.8).

A smooth, projective, absolutely irreducible curve of genus 19 over F 2 admitting an infinite S -class field tower is presented. Here S is a set of four F 2 -rational points on the curve. This is shown to imply that A (2) = limsup # X ( F 2 )/ g(X) ≥ 4/(19 − 1) ≈ 0.222. Here the limit is taken over curves X over F 2 of genus g ( X ) → ∞.

A method for obtaining very precise results along the lines of the Hilbert Irreducibility Theorem is described and then applied to a special case. In addition, the relationship of the irreducibility theorem to other tools of diophantine analysis is investigated. In particular, we give a proof of the irreducibility theorem that uses only Noether's lemma and the fact that an absolutely irreducible curve has a rational point over a finite field of large order.

We consider absolutely irreducible polynomials f ∈ Z [ x , y ] with deg x f = m , deg y f = n , and height H . We show that for any prime p with p >[ m ( n +1) n 2 +( m +1)( n −1) m 2 ] mn +( n −1)/2 · H 2 mn + n −1 the reduction f mod p is also absolutely irreducible. Furthermore if the Bouniakowsky conjecture is true we show that there are infinitely many absolutely irreducible polynomials f ∈ Z [ x , y ] which are reducible mod p where p is a prime with p ⩾ H 2 m .

Ostrowski established in 1919 that an absolutely irreducible integral polynomial remains absolutely irreducible modulo all sufficiently large prime numbers. We obtain a new lower bound for the size of such primes in terms of the number of integral points in the Newton polytope of the polynomial, significantly improving previous estimates for sparse polynomials.

Given an absolutely irreducible horizontal hypersurface Z in a projective space over the ring of integers R of a number field, we give an explicit bound for the product of the norms of the prime ideals of R over which the fibre of Z becomes reducible. This bound is given as a function of a projective height of Z and is obtained using arithmetic intersection theory, in particular, an arithmetic Bezout theorem.

Let f ( X 1 ,…, X n ) be an absolutely irreducible polynomial with coefficients in a finite field. Elementary methods are used to derive an explicit lower bound for the number of zeros of f .

It is shown that an absolutely irreducible homogeneous cubic polynomial f ∈ Z [ x 0 , x 1 , x 2 ] is also absolutely irreducible mod p if p >1248 H 6 where H is the height of f . Modulo a general number theoretic conjecture an example shows that the result is best possible.