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23
GEOMETRY AND ARITHMETIC OF VERBAL DYNAMICAL SYSTEMS ON SIMPLE GROUPS
, 809
"... Abstract. We study dynamical systems arising from word maps on simple groups. We develop a geometric method based on the classical trace map for investigating periodic points of such systems. These results lead to a new approach to the search of Engellike sequences of words in two variables which c ..."
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Abstract. We study dynamical systems arising from word maps on simple groups. We develop a geometric method based on the classical trace map for investigating periodic points of such systems. These results lead to a new approach to the search of Engellike sequences of words in two variables which characterize finite solvable groups. They also give rise to some new phenomena and concepts in the arithmetic of dynamical systems. �1 Contents
Averages of elliptic curve constants
, 711
"... We compute the averages over elliptic curves of the constants occurring in the LangTrotter conjecture, the Koblitz conjecture, and the cyclicity conjecture. The results obtained confirm the consistency of these conjectures with the corresponding “theorems on average ” obtained recently by various a ..."
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We compute the averages over elliptic curves of the constants occurring in the LangTrotter conjecture, the Koblitz conjecture, and the cyclicity conjecture. The results obtained confirm the consistency of these conjectures with the corresponding “theorems on average ” obtained recently by various authors. 1
A REFINEMENT OF KOBLITZ’S CONJECTURE
, 909
"... Abstract. Let E be an elliptic curve over the number field Q. In 1988, Koblitz conjectured an asymptotic for the number of primes p for which the cardinality of the group of Fppoints of E is prime. However, the constant occurring in his asymptotic does not take into account that the distributions o ..."
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Abstract. Let E be an elliptic curve over the number field Q. In 1988, Koblitz conjectured an asymptotic for the number of primes p for which the cardinality of the group of Fppoints of E is prime. However, the constant occurring in his asymptotic does not take into account that the distributions of the E(Fp)  need not be independent modulo distinct primes. We shall describe a corrected constant. We also take the opportunity to extend the scope of the original conjecture to ask how often E(Fp)/t is prime for a fixed positive integer t, and to consider elliptic curves over arbitrary number fields. Several worked out examples are provided to supply numerical evidence for the new conjecture. 1.
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.
Reductions of an elliptic curve with almost prime orders
"... 1 Let E be an elliptic curve over Q. For a prime p of good reduction, let Ep be the reduction of E modulo p. We investigate Koblitz’s Conjecture about the number of primes p for which Ep(Fp) has prime order. More precisely, our main result is that if E is with Complex Multiplication, then there exis ..."
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1 Let E be an elliptic curve over Q. For a prime p of good reduction, let Ep be the reduction of E modulo p. We investigate Koblitz’s Conjecture about the number of primes p for which Ep(Fp) has prime order. More precisely, our main result is that if E is with Complex Multiplication, then there exist infinitely many primes p for which #Ep(Fp) has at most 5 prime factors. We also obtain upper bounds for the number of primes p ≤ x for which #Ep(Fp) is a prime. 1
Almost prime values of the order of elliptic curves over finite fields
, 2008
"... Abstract. Let E be an elliptic curve over Q without complex multiplication, and which is not isogenous to a curve with nontrivial rational torsion. For each prime p of good reduction, let E(Fp)  be the order of the group of points of the reduced curve over Fp. We prove in this paper that, under t ..."
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Abstract. Let E be an elliptic curve over Q without complex multiplication, and which is not isogenous to a curve with nontrivial rational torsion. For each prime p of good reduction, let E(Fp)  be the order of the group of points of the reduced curve over Fp. We prove in this paper that, under the GRH, there are at least 2.778Ctwin E x/(log x)2 primes p such that E(Fp)  has at most 8 prime factors, counted with multiplicity. This improves previous results of Steuding & Weng [18] and Murty & Miri [13]. This is also the first result where the dependence on the conjectural constant Ctwin E appearing in the twin prime conjecture for elliptic curves (also known as Koblitz’s conjecture) is made explicit. This is achieved by sieving a slightly different sequence than the one of [18] and [13]. By sieving the same sequence and using Selberg’s linear sieve, we can also improve the constant of Zywina [22] appearing in the upper bound for the number of primes p such that E(Fp)  is prime. Finally, we remark that our results still hold under an hypothesis weaker than the GRH. 1.
Squarefree discriminants of Frobenius rings
, 2008
"... Let E be an elliptic curve over Q. It is well known that the ring of endomorphisms of Ep, the reduction of E modulo a prime p of ordinary reduction, is an order of the quadratic imaginary field Q(πp) generated by the Frobenius element πp. When the curve has complex multiplication (CM), this is alway ..."
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Let E be an elliptic curve over Q. It is well known that the ring of endomorphisms of Ep, the reduction of E modulo a prime p of ordinary reduction, is an order of the quadratic imaginary field Q(πp) generated by the Frobenius element πp. When the curve has complex multiplication (CM), this is always a fixed field as the prime varies. However, when the curve has no CM, very little is known, not only about the order, but about the fields that might appear as algebra of endomorphisms varying the prime. The ring of endomorphisms is obviously related with the arithmetic of a 2 p −4p, the discriminant of the characteristic polynomial of the Frobenius element. In this paper, we are interested in the function πE,r,h(x) counting the number of primes p up to x such that a 2 p − 4p is squarefree and in the congruence class r modulo h. We give in this paper the precise asymptotic for πE,r,h(x) when averaging over elliptic curves defined over the rationals, and we discuss the relation of this result with the LangTrotter conjecture, and with some other problems related to the curve modulo p. 1 Introduction and statement of results
Divisibility, Smoothness and Cryptographic Applications
, 2008
"... This paper deals with products of moderatesize primes, familiarly known as smooth numbers. Smooth numbers play an crucial role in information theory, signal processing and cryptography. We present various properties of smooth numbers relating to their enumeration, distribution and occurrence in var ..."
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This paper deals with products of moderatesize primes, familiarly known as smooth numbers. Smooth numbers play an crucial role in information theory, signal processing and cryptography. We present various properties of smooth numbers relating to their enumeration, distribution and occurrence in various integer sequences. We then turn our attention to cryptographic applications in which smooth numbers play a pivotal role. 1 1
AVERAGES OF THE NUMBER OF POINTS ON ELLIPTIC CURVES
"... ABSTRACT. If E is an elliptic curve defined over Q and p is a prime of good reduction for E, let E(Fp) denote the set of points on the reduced curve modulo p. Define an arithmetic function ME(N) by setting ME(N): = #{p: #E(Fp) = N}. Recently, David and the third author studied the average of ME(N) ..."
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ABSTRACT. If E is an elliptic curve defined over Q and p is a prime of good reduction for E, let E(Fp) denote the set of points on the reduced curve modulo p. Define an arithmetic function ME(N) by setting ME(N): = #{p: #E(Fp) = N}. Recently, David and the third author studied the average of ME(N) over certain “boxes ” of elliptic curves E. Assuming a plausible conjecture about primes in short intervals, they showed the following: for odd N, the average of ME(N) over a box with sufficiently large sides is ∼ K ∗ (N) log N for an explicitlygiven function K ∗ (N). The function K ∗ (N) is somewhat peculiar: defined as a product over the primes dividing N, it resembles a multiplicative function at first glance. But further inspection reveals that it is not, and so one cannot directly investigate its properties by the usual tools of multiplicative number theory. In this paper, we overcome these difficulties and prove a number of statistical results about K ∗ (N). For example, we determine the mean value of K ∗ (N) over odd N and over prime N, and we show that K ∗ (N) has a distribution function. We also explain how our results relate to existing theorems and conjectures on the multiplicative properties of #E(Fp), such as Koblitz’s conjecture. 1.