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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
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY
"... The book under review gives a comprehensive account of the RosserIwaniec method, the most important development in the construction of number sieves since the advent in 1947 of Selberg’s λmethod. Sieve literature has grown prodigiously since the publication of [HR], and Dr. Greaves has had to make ..."
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The book under review gives a comprehensive account of the RosserIwaniec method, the most important development in the construction of number sieves since the advent in 1947 of Selberg’s λmethod. Sieve literature has grown prodigiously since the publication of [HR], and Dr. Greaves has had to make some difficult decisions on what to include and what to omit. On the whole his choices have been wise; students and experts alike will have much to learn from his careful presentation. If the book does not always make for easy reading, that is due largely to the nature of the subject: not only does sieve architecture rest on complicated combinatorial foundations, but these culminate nowadays in none too easy linear differential delay boundary value problems and also link up with results and techniques from modern analytic number theory. We begin with a brief description of sieve methods by setting the stage: Let P be a finite set of primes—usually this is an infinite, increasing sequence of primes truncated at some number z>2—and refer to P as a ‘sieve’. Let P denote the product of all the primes in P. In Selberg’s terminology, P is said to ‘sift out ’ an integer n if n is divisible by some prime p in P. Then, writing (n, P)forthehighest common factor of n and P, P sifts out n if and only if (n, P)> 1; by the same token, the indicator function of all integers n that are not sifted out by P is ∑ 1 when (n, P)=1