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On the behaviour of root numbers in families of elliptic curves
, 2004
"... Let E be a oneparameter family of elliptic curves over Q. We prove that the average root number is zero for a large class of families of elliptic curves of fairly general type. Furthermore, we show that any family E with at least one point of multiplicative reduction over Q(T) has average root numb ..."
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Let E be a oneparameter family of elliptic curves over Q. We prove that the average root number is zero for a large class of families of elliptic curves of fairly general type. Furthermore, we show that any family E with at least one point of multiplicative reduction over Q(T) has average root number 0, provided that two classical arithmetical conjectures hold for two polynomials constructed explicitly in terms of E. The behaviour of the root number in any family E without multiplicative reduction over Q(T) is shown to be rather regular and nonrandom; we give expressions for the average root number in this case.
Root numbers and the parity problem
, 2003
"... Let E be a oneparameter family of elliptic curves over a number field K. It is natural to expect the average root number of the curves in the family to be zero. All known counterexamples to this folk conjecture occur for families obeying a certain degeneracy condition. We prove that the average roo ..."
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Cited by 6 (3 self)
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Let E be a oneparameter family of elliptic curves over a number field K. It is natural to expect the average root number of the curves in the family to be zero. All known counterexamples to this folk conjecture occur for families obeying a certain degeneracy condition. We prove that the average root number is zero for a large class of families of elliptic curves of fairly general type. Furthermore, we show that any nondegenerate family E has average root number 0, provided that two classical arithmetical conjectures hold for two homogeneous polynomials with integral coefficients constructed explicitly in terms of E. The first such conjecture – commonly associated with Chowla – asserts the equidistribution of the parity of the number of primes dividing the integers represented by a polynomial. More precisely: given a homogeneous polynomial f ∈ Z[x, y], it is believed that µ(f(x, y)) averages to zero. This conjecture can be said to represent the parity problem in its pure form, while covering the same notional ground as the BunyakovskySchinzel and HardyLittlewood conjectures taken together. For deg f = 1 and deg f = 2, Chowla’s conjecture is essentially equivalent to the prime number theorem. For deg f> 2, the conjecture has been unproven up to now; the traditional approaches by means of analysis and sieve theory fail. We prove the conjecture for deg f = 3. There remains to state the second arithmetical conjecture referred to previously. It is believed that any nonconstant homogeneous polynomial f ∈ Z[x, y] yields to a squarefree sieve. We sharpen the existing bounds on the known cases by a sieve refinement and a new approach combining height functions, sphere packings and sieve methods. iii ��������������������� � ��������� � ����� � �����
ON THE SQUAREFREE SIEVE
, 2004
"... A squarefree sieve is a result that gives an upper bound for how often a squarefree polynomial may adopt values that are not squarefree. More generally, we may wish to control the behavior of a function depending on the largest square factor of P(x1,..., xn), where P is a squarefree polynomial. ..."
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A squarefree sieve is a result that gives an upper bound for how often a squarefree polynomial may adopt values that are not squarefree. More generally, we may wish to control the behavior of a function depending on the largest square factor of P(x1,..., xn), where P is a squarefree polynomial.
Powerfree values, large deviations, and integer points on irrational curves
 JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUX
, 2006
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ON WEYL’S INEQUALITY, HUA’S LEMMA, AND EXPONENTIAL SUMS OVER BINARY FORMS
 VOL. 100, NO. 3 DUKE MATHEMATICAL JOURNAL
, 1999
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The density of rational lines on cubic hypersurfaces
 Trans. Amer. Math. Soc
"... Abstract. We provide a lower bound for the density of rational lines on the hypersurface dened by an additive cubic equation in at least 57 variables. In the process, we obtain a result on the paucity of nontrivial solutions to an associated system of Diophantine equations. 1. ..."
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Abstract. We provide a lower bound for the density of rational lines on the hypersurface dened by an additive cubic equation in at least 57 variables. In the process, we obtain a result on the paucity of nontrivial solutions to an associated system of Diophantine equations. 1.
Number fields with large class groups
"... After a review of the quadratic case, a general problem about the existence of number fields of a fixed degree with extremely large class numbers is formulated. This problem is solved for abelian cubic fields. Then some conditional results proven elsewhere are discussed about totally real number fie ..."
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After a review of the quadratic case, a general problem about the existence of number fields of a fixed degree with extremely large class numbers is formulated. This problem is solved for abelian cubic fields. Then some conditional results proven elsewhere are discussed about totally real number fields of a fixed degree, each of whose normal closure has the symmetric group as Galois group. 1 Introduction. It was Littlewood who first addressed the question of how large the class number h of an imaginary quadratic field Q ( √ d) can be as a function of d  as d → − ∞ through fundamental discriminants. In 1927 [14] he showed, assuming the generalized Riemann hypothesis (GRH), that for all fundamental d < 0
Sieve Methods
"... called undeniable signature schemes require prime numbers of the form 2p 1 such that p is also prime. Sieve methods can yield valuable clues about these distributions and hence allow us to bound the running times of these algorithms. In this treatise we survey the major sieve methods and their i ..."
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called undeniable signature schemes require prime numbers of the form 2p 1 such that p is also prime. Sieve methods can yield valuable clues about these distributions and hence allow us to bound the running times of these algorithms. In this treatise we survey the major sieve methods and their important applications in number theory. We apply sieves to study the distribution of squarefree numbers, smooth numbers, and prime numbers. The first chapter is a discussion of the basic sieve formulation of Legendre. We show that the distribution of squarefree numbers can be deduced using a squarefree sieve 1 . We give an account of improvements in the error term of this distribution, using known results regarding the Riemann Zeta function. The second chapter deals with Brun's Combinatorial sieve as presented in the modern language of [HR74]. We apply the general sieve to give a simpler
Journal de Théorie des Nombres
"... Solving conics over function fields par Mark van HOEIJ et John CREMONA Résumé. Soit F un corps de caractéristique différente de 2, et K = F (t). Nous donnons un algorithme simple pour trouver, étant donné a, b, c ∈ K ∗ , une solution nontriviale dans K (si elle existe) à l’équation aX 2 + bY 2 + cZ ..."
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Solving conics over function fields par Mark van HOEIJ et John CREMONA Résumé. Soit F un corps de caractéristique différente de 2, et K = F (t). Nous donnons un algorithme simple pour trouver, étant donné a, b, c ∈ K ∗ , une solution nontriviale dans K (si elle existe) à l’équation aX 2 + bY 2 + cZ 2 = 0. Dans certains cas, l’algorithme a besoin d’une solution d’une equation similaire à coefficients dans F; nous obtenons alors un algorithme recursif pour résoudre les coniques diagonales sur Q(t1,..., tn) (en utilisant les algorithmes existants pour telles équations sur Q) et sur Fq(t1,..., tn). Abstract. Let F be a field whose characteristic is not 2 and K = F (t). We give a simple algorithm to find, given a, b, c ∈ K ∗ , a nontrivial solution in K (if it exists) to the equation aX 2 + bY 2 + cZ 2 = 0. The algorithm requires, in certain cases, the solution of a similar equation with coefficients in F; hence we obtain a recursive algorithm for solving diagonal conics over Q(t1,..., tn) (using existing algorithms for such equations over Q) and over Fq(t1,..., tn). 1.
Powerfree values of polynomials and integer points on irrational curves
 JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUX
, 2004
"... Let f ∈ Z[x] be a polynomial of degree r ≥ 3 without roots of multiplicity r or (r − 1). Assume that f(x) ̸ ≡ 0 mod p r−1 has a solution in (Z/p r−1) ∗ for every p. Erdős conjectured that f(p) is then free of (r − 1)th powers for infinitely many primes p. This is proved here for every f a root of ..."
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Let f ∈ Z[x] be a polynomial of degree r ≥ 3 without roots of multiplicity r or (r − 1). Assume that f(x) ̸ ≡ 0 mod p r−1 has a solution in (Z/p r−1) ∗ for every p. Erdős conjectured that f(p) is then free of (r − 1)th powers for infinitely many primes p. This is proved here for every f a root of which generates its splitting field, and for some other f as well. The proof takes advantage of a certain kind of repulsion among integer points in curves of positive genus. Probabilistic arguments also come into play.