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Sato–Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height
, 2008
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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
A remark on the conjectures of Lang–Trotter and Sato–Tate on average
, 2007
"... We obtain new average results on the conjectures of LangTrotter and SatoTate about elliptic curves. ..."
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We obtain new average results on the conjectures of LangTrotter and SatoTate about elliptic curves.
Distribution of Farey fractions in residue classes and Lang–Trotter conjectures on average
 Proc. Amer. Math. Soc. 136, Number 6 (2008), 19771986. MR2383504 (2009a:11035
"... We prove that the set of Farey fractions of order T, that is, the set {α/β ∈ Q: gcd(α,β) = 1, 1 � α,β � T}, is uniformly distributed in residue classes modulo a prime p provided T � p 1/2+ε for any fixed ε> 0. We apply this to obtain upper bounds for the Lang–Trotter conjectures on Frobenius tra ..."
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We prove that the set of Farey fractions of order T, that is, the set {α/β ∈ Q: gcd(α,β) = 1, 1 � α,β � T}, is uniformly distributed in residue classes modulo a prime p provided T � p 1/2+ε for any fixed ε> 0. We apply this to obtain upper bounds for the Lang–Trotter conjectures on Frobenius traces and Frobenius fields “on average ” over a oneparametric family of elliptic curves. 2000 Mathematics Subject Classification: 11B57, 11G07, 14H52 1 1
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
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"... positive integers n dividing the nth term of an elliptic divisibility sequence ..."
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positive integers n dividing the nth term of an elliptic divisibility sequence
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"... positive integers n dividing the nth term of an elliptic divisibility sequence ..."
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positive integers n dividing the nth term of an elliptic divisibility sequence
BIRS Workshop 11w5075: WIN2 – Women in Numbers 2, C. David (Concordia University), M. Lalín (Université de Montréal),
, 2011
"... This workshop was a unique effort to combine strong, broad impact with a top level technical research program. In order to help raise the profile of active female researchers in number theory and increase their participation in research activities in the field, this event brought together female sen ..."
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This workshop was a unique effort to combine strong, broad impact with a top level technical research program. In order to help raise the profile of active female researchers in number theory and increase their participation in research activities in the field, this event brought together female senior and junior researchers in the field for collaboration. Emphasis was placed on onsite collaboration on open research problems as well as student training. Collaborative group projects introducing students to areas of active research were a key component of this workshop. We would like to thank the following organizations for their support of this workshop: BIRS, PIMS, Microsoft Research, and the Number Theory Foundation. 1 Rationale and Goals Number theory is a fundamental subject with connections to a broad spectrum of mathematical areas including algebra, arithmetic, analysis, topology, cryptography, and geometry. This very active area naturally attracts many female mathematicians. Although the number of female number theorists is steadly growing, there are still relatively few women reaching high profile positions and visibility at international workshops and conferences. The lack of female leaders in the area is an issue that tends to perpetuate itself, since it has repercussions in attracting and training the next generation of female mathematicians.