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23
On Kummer's Conjecture
, 2001
"... Kummer conjectured the asymptotic behavior of the first factor of the class number of a cyclotomic field. If we only ask for upper and lower bounds of the order of growth predicted by Kummer, then this modified Kummer conjecture is true for almost all primes. ..."
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Cited by 67 (8 self)
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Kummer conjectured the asymptotic behavior of the first factor of the class number of a cyclotomic field. If we only ask for upper and lower bounds of the order of growth predicted by Kummer, then this modified Kummer conjecture is true for almost all primes.
Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem
, 2001
"... 1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are ..."
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Cited by 20 (3 self)
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1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are substantial improvements of earlier work of J.P. Serre and M. Ram Murty. We also consider the problem of finding the size of the smallest prime p = pE for which the group E(Fp) is cyclic and we show that, under the generalized Riemann hypothesis, pE = O � (log N) 4+ε � if E is without complex multiplication, and pE = O � (log N) 2+ε � if E is with complex multiplication, for any 0 < ε < 1. 1
Implementation Of The AtkinGoldwasserKilian Primality Testing Algorithm
 RAPPORT DE RECHERCHE 911, INRIA, OCTOBRE
, 1988
"... We describe a primality testing algorithm, due essentially to Atkin, that uses elliptic curves over finite fields and the theory of complex multiplication. In particular, we explain how the use of class fields and genus fields can speed up certain phases of the algorithm. We sketch the actual implem ..."
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Cited by 9 (7 self)
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We describe a primality testing algorithm, due essentially to Atkin, that uses elliptic curves over finite fields and the theory of complex multiplication. In particular, we explain how the use of class fields and genus fields can speed up certain phases of the algorithm. We sketch the actual implementation of this test and its use on testing large primes, the records being two numbers of more than 550 decimal digits. Finally, we give a precise answer to the question of the reliability of our computations, providing a certificate of primality for a prime number.
On the rrank Artin Conjecture
 Math. Comp
, 1999
"... Abstract. We assume the generalized Riemann hypothesis and prove an asymptotic formula for the number of primes for which F ∗ p can be generated by r given multiplicatively independent numbers. In the case when the r given numbers are primes, we express the density as an Euler product and apply this ..."
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Cited by 7 (3 self)
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Abstract. We assume the generalized Riemann hypothesis and prove an asymptotic formula for the number of primes for which F ∗ p can be generated by r given multiplicatively independent numbers. In the case when the r given numbers are primes, we express the density as an Euler product and apply this to a conjecture of Brown–Zassenhaus (J. Number Theory 3 (1971), 306–309). Finally, in some examples, we compare the densities approximated with the natural densities calculated with primes up to 9 · 10 4. 1.
Reduction mod p of subgroups of the MordellWeil group of an elliptic curve
 INT J. OF NUMBER THEORY
"... Let E be an elliptic curve defined over Q. Let Γ be a free subgroup of rank r of E(Q). For any prime p of good reduction, let Γp be the reduction of Γ modulo p and Ep be the reduction of E modulo p. We prove that if E has CM then for all but o(x / log x) of primes p ≤ x, Γp  ≥ p r r+2 +ɛ(p), wher ..."
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Cited by 4 (3 self)
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Let E be an elliptic curve defined over Q. Let Γ be a free subgroup of rank r of E(Q). For any prime p of good reduction, let Γp be the reduction of Γ modulo p and Ep be the reduction of E modulo p. We prove that if E has CM then for all but o(x / log x) of primes p ≤ x, Γp  ≥ p r r+2 +ɛ(p), where ɛ(p) is any function of p such that ɛ(p) → 0 as p → ∞. This is a consequence of two other results. Denote by Np the cardinality of Ep(Fp), where Fp is a finite field of p elements. Then for any δ> 0, the set of primes p for which Np has a divisor in the range (pδ−ɛ(p), pδ+ɛ(p) ) has density zero. Moreover, the set of primes p for which Γp  < p r r+2 −ɛ(p) has density zero.
On Arithmetic of Hyperelliptic Curves
 Aspects of Mathematics, HKU
"... In this exposé, Pell’s equation is put in a geometric perspective, and a version of Artin’s primitive roots conjecture is formulated for hyperelliptic jacobians. Also explained are some recent results which throw new lights, having to do with AnkenyArtinChowla’s conjecture, class number relation ..."
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Cited by 4 (2 self)
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In this exposé, Pell’s equation is put in a geometric perspective, and a version of Artin’s primitive roots conjecture is formulated for hyperelliptic jacobians. Also explained are some recent results which throw new lights, having to do with AnkenyArtinChowla’s conjecture, class number relations, and CohenLenstra heuristics.
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
On the greatest prime divisor of Np
 J. Ramanujan Math. Soc
"... Let E be an elliptic curve defined over Q. For any prime p of good reduction, let Ep be the reduction of E mod p. Denote by Np the cardinality of Ep(Fp), where Fp is the finite field of p elements. Let P (Np) be the greatest prime divisor of Np. We prove that if E has CM then for all but o(x / log x ..."
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Cited by 2 (1 self)
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Let E be an elliptic curve defined over Q. For any prime p of good reduction, let Ep be the reduction of E mod p. Denote by Np the cardinality of Ep(Fp), where Fp is the finite field of p elements. Let P (Np) be the greatest prime divisor of Np. We prove that if E has CM then for all but o(x / log x) of primes p ≤ x, P (Np)> p ϑ(p), where ϑ(p) is any function of p such that ϑ(p) → 0 as p → ∞. Moreover we show that for such E there is a positive proportion of primes p ≤ x for which P (Np)> p ϑ, where ϑ is any number less than ϑ0 = 1 − 1 2 prove the following. Let Γ be a free subgroup of rank r ≥ 2 of the group of rational points E(Q), and Γp be the reduction of Γ mod p, then for a positive proportion of primes p ≤ x, we have where ɛ> 0. e− 1 4 = 0.6105 · · ·. As an application of this result we Γp > p ϑ0−ɛ Keywords: Reduction mod p of elliptic curves, Elliptic curves over finite fields, BrunTitchmarsh inequality in number fields, BombieriVinogradov theorem in number fields, Abelian extensions of imaginary quadratic number fields. 2000 Mathematics Subject Classification. Primary 11G20, Secondary 11N37. 1
Towards LangTrotter for Elliptic Curves over Function Fields
"... Introduction Let K be a global field of char p and let F q K denote the algebraic closure of F p in K. We fix an elliptic curve E/K with nonconstant jinvariant and a torsionfree subgroup E(K) of rank r > 0. We write V for the open set of places v of K such that the special fiber E v is a ..."
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Introduction Let K be a global field of char p and let F q K denote the algebraic closure of F p in K. We fix an elliptic curve E/K with nonconstant jinvariant and a torsionfree subgroup E(K) of rank r > 0. We write V for the open set of places v of K such that the special fiber E v is an elliptic curve and, for v in V , we let # v E v (k v ) be the image of # under reduction modulo v, where k v is the residue field of K at v. We fix a finite set of (rational) prime numbers S which is large enough to include the exceptional primes which we will define explicitly in section 2.4 and section 3), and we let G(#, S) denote the subset of v V such that # v contains the primetoS part of E v (k v ). For every n > 0, we write V n for the subset of v V such that deg(v) = n and let G n (#, S) = V n G(#, S). Theorem 1. Suppose r 6. There exists constants a, b satisfying 0 1 and depending only on r and S such that, for each n 1 there exists # n (#, S), depending on r, S an