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Elliptic curves with a given number of points over finite fields
"... Abstract. Given an elliptic curve E and a positive integer N, we consider the problem of counting the number of primes p for which the reduction of E modulo p possesses exactly N points over Fp. On average (over a family of elliptic curves), we show bounds that are significantly better than what is ..."
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Abstract. Given an elliptic curve E and a positive integer N, we consider the problem of counting the number of primes p for which the reduction of E modulo p possesses exactly N points over Fp. On average (over a family of elliptic curves), we show bounds that are significantly better than what is trivially obtained by the Hasse bound. Under some additional hypotheses, including a conjecture concerning the short interval distribution of primes in arithmetic progressions, we obtain an asymptotic formula for the average. 1.
Lower bounds for the number of smooth values of a polynomial
, 1998
"... We investigate the problem of showing that the values of a given polynomial are smooth (i.e., have no large prime factors) a positive proportion of the time. Although some results exist that bound the number of smooth values of a polynomial from above, a corresponding lower bound of the correct ord ..."
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We investigate the problem of showing that the values of a given polynomial are smooth (i.e., have no large prime factors) a positive proportion of the time. Although some results exist that bound the number of smooth values of a polynomial from above, a corresponding lower bound of the correct order of magnitude has hitherto been established only in a few special cases. The purpose of this paper is to provide such a lower bound for an arbitrary polynomial. Various generalizations to subsets of the set of values taken by a polynomial are also obtained.
A COHENLENSTRA PHENOMENON FOR ELLIPTIC CURVES
"... Abstract. Given an elliptic curve E and a finite Abelian group G, we consider the problem of counting the number of primes p for which the group of points modulo p is isomorphic to G. Under a certain conjecture concerning the distribution of primes in short intervals, we obtain an asymptotic formula ..."
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Abstract. Given an elliptic curve E and a finite Abelian group G, we consider the problem of counting the number of primes p for which the group of points modulo p is isomorphic to G. Under a certain conjecture concerning the distribution of primes in short intervals, we obtain an asymptotic formula for this problem on average over a family of elliptic curves. 1. Introduction. Let E be an elliptic curve defined over the rational field Q. Given a prime p where E has good reduction, we consider the reduced curve, which we denote by Ep. In previous work [DS], we studied the arithmetic function ME(N): = #{p: #Ep(Fp) = N},
Abstracts of papers by Alessandro Zaccagnini
, 2013
"... This file contains full references and abstracts of my papers. Chronologically, these deal with the following topics: • Large gaps between primes in arithmetic progressions [1]. • An additive problem of Hardy and Littlewood and its variants [2, 3, 4]. • The Selberg integral [5, 6]. • An elementary t ..."
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This file contains full references and abstracts of my papers. Chronologically, these deal with the following topics: • Large gaps between primes in arithmetic progressions [1]. • An additive problem of Hardy and Littlewood and its variants [2, 3, 4]. • The Selberg integral [5, 6]. • An elementary talk on Goldbach’s problem [7]. • An “inverse ” problem related to the Selberg integral [8]. • Another additive problem of the Hardy–Littlewood type [9]. • An elementary talk on the continued fraction for √ 2 [10]. • An elementary talk on properties of prime numbers and their relations to cryptography [11]. • A review paper on the Selberg integral and related topics [12]. • An elementary paper that deals with various approximations to π [13]. • A paper on the Mertens product over primes in arithmetic progressions [14]. 1 • A paper dealing with an additive problem involving primes and powers of 2 [15]. • An elementary paper on the use of handheld nonprogrammable calculators
References
"... ∣∣∣∣ψ(x+h;q,a)−ψ(x;q,a) − hϕ(q) ∣∣∣∣2 and κ = 1+ γ+ log2pi+∑p(log p)/p(p − 1). Denote the expected main term by M(x,h,Q) = hQ log(xQ/h)+ (x+ h)Q log(1+ h/x) − κhQ. Let ε, A> 0 be arbitrary, x7/12+ε ≤ h ≤ x and Q ≤ h. There exists a positive constant c1 such that S(x,h,Q)−M(X,h,Q) h1/2Q3/2 exp − ..."
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∣∣∣∣ψ(x+h;q,a)−ψ(x;q,a) − hϕ(q) ∣∣∣∣2 and κ = 1+ γ+ log2pi+∑p(log p)/p(p − 1). Denote the expected main term by M(x,h,Q) = hQ log(xQ/h)+ (x+ h)Q log(1+ h/x) − κhQ. Let ε, A> 0 be arbitrary, x7/12+ε ≤ h ≤ x and Q ≤ h. There exists a positive constant c1 such that S(x,h,Q)−M(X,h,Q) h1/2Q3/2 exp −c1 (log2h/Q)