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15
The square sieve and the Lang–Trotter conjecture
 Canadian Journal of Mathematics
, 2001
"... 1 Let E be an elliptic curve defined over Q and without complex multiplication. Let K be a fixed imaginary quadratic field. We find nontrivial upper bounds for the number of ordinary primes p ≤ x for which Q(πp) = K, where πp denotes the Frobenius endomorphism of E at p. More precisely, �under a ce ..."
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1 Let E be an elliptic curve defined over Q and without complex multiplication. Let K be a fixed imaginary quadratic field. We find nontrivial upper bounds for the number of ordinary primes p ≤ x for which Q(πp) = K, where πp denotes the Frobenius endomorphism of E at p. More precisely, �under a certain generalized Riemann hypothesis we show that this number is OE x 17 18 log x, and unconditionally. We also prove that the number we show that this number is OE,K � 13 x(log log x) 12 (log x) 25 24 of imaginary quadratic fields K, with − disc K ≤ x and of the form K = Q(πp), is ≫E log log log x for x ≥ x0(E). These results represent progress towards a 1976 LangTrotter conjecture. 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
Irregularities in the distribution of primes in function fields, J. Number Theory, accepted for publication
"... Abstract. We adapt the Maier matrix method to the polynomial ring Fq[t], and prove analogues of results of Maier [4] and Shiu [10] concerning the distribution of primes in short intervals. 1. ..."
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Cited by 3 (2 self)
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Abstract. We adapt the Maier matrix method to the polynomial ring Fq[t], and prove analogues of results of Maier [4] and Shiu [10] concerning the distribution of primes in short intervals. 1.
7 A PROBLEM OF FOMENKO’S RELATED TO ARTIN’S CONJECTURE
, 2012
"... Let a be a natural number greater than 1. For each prime p, letia(p) denote the index of the group generated by a in F ∗ p. Assuming the generalized Riemann hypothesis and Conjecture A of Hooley, Fomenko proved in 2004 that the average value of ia(p) is constant. We prove that the average value of i ..."
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Let a be a natural number greater than 1. For each prime p, letia(p) denote the index of the group generated by a in F ∗ p. Assuming the generalized Riemann hypothesis and Conjecture A of Hooley, Fomenko proved in 2004 that the average value of ia(p) is constant. We prove that the average value of ia(p) is constant without using Conjecture A of Hooley. More precisely, we show upon GRH that for any α with 0 ≤ α<1, there is a positive constant cα> 0 such that X (log ia(p)) α ∼ cαπ(x), p≤x where π(x) is the number of primes p ≤ x. We also study related questions.
WALKS ON GRAPHS AND LATTICES – EFFECTIVE BOUNDS AND APPLICATIONS
, 2008
"... Abstract. We continue the investigations started in [7, 8]. We consider the following situation: G is a finite directed graph, where to each vertex of G is assigned an element of a finite groupΓ. We consider all walks of length N on G, starting from vi and ending at vj. To each such walk w we assign ..."
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Abstract. We continue the investigations started in [7, 8]. We consider the following situation: G is a finite directed graph, where to each vertex of G is assigned an element of a finite groupΓ. We consider all walks of length N on G, starting from vi and ending at vj. To each such walk w we assign the element ofΓequal to the product of the elements along the walk. The set of all walks of length N from vi to vj thus induces a probability distribution FN,i,j onΓ. In [7] we give necessary and sufficient conditions for the limit as N goes to infinity of FN,i,j to exist and to be the uniform density onΓ(a detailed argument is presented in [8]). The convergence speed is then exponential in N. In this paper we consider (G,Γ), whereΓis a group possessing Kazhdan’s property T (or, less restrictively, propertyτwith respect to representations with finite image), and a family of homomorphismsψk:Γ→Γk with finite image. Each FN,i,j induces a distribution Fk N,i,j onΓk (by pushforward underψk). Our main result is that, under mild technical assumptions, the exponential rate of convergence of Fk N,i,k to the uniform distribution onΓk does not depend on k. As an application, we prove effective versions of the results of [8] on the probability that a random (in a suitable sence) element of SL(n,Z) or Sp(n,Z) has irreducible characteristic polynomial, generic Galois group, etc.
SPARSE POLYNOMIAL INTERPOLATION AND THE FAST EUCLIDEAN ALGORITHM
, 2012
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THE EUCLIDEAN ALGORITHM FOR NUMBER FIELDS AND PRIMITIVE ROOTS
"... In 1927 Artin formulated his famous conjecture about primitive roots. Artin’s Primitive Root Conjecture. If a is not1 or a square then there are infinitely many primes p such that a is a primitive root modulo p. In fact, for an explicit constant A(a) Artin conjectured that the number of primes p ≤ ..."
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In 1927 Artin formulated his famous conjecture about primitive roots. Artin’s Primitive Root Conjecture. If a is not1 or a square then there are infinitely many primes p such that a is a primitive root modulo p. In fact, for an explicit constant A(a) Artin conjectured that the number of primes p ≤ x such that a is a
A SHORT PROOF OF LEGENDRE’S CONJECTURE
"... Abstract. It was proved for every natural number n, there is always a prime number between the integers n 2 and (n + 1) 2. Therefore, it was demonstrated the Legendre’s conjecture as a consequence of using a modification to the sieve of Legendre. 1. ..."
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Abstract. It was proved for every natural number n, there is always a prime number between the integers n 2 and (n + 1) 2. Therefore, it was demonstrated the Legendre’s conjecture as a consequence of using a modification to the sieve of Legendre. 1.
A SHORT PROOF OF GOLDBACH’S CONJECTURE
"... Abstract. It was proved for every even number greater than or equal to 4, there is always a prime number which is the result of subtracting the even number with some prime. Therefore, it was demonstrated the strong Goldbach’s conjecture as a consequence of applying a modification to the sieve of Leg ..."
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Abstract. It was proved for every even number greater than or equal to 4, there is always a prime number which is the result of subtracting the even number with some prime. Therefore, it was demonstrated the strong Goldbach’s conjecture as a consequence of applying a modification to the sieve of Legendre. 1.