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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
Distribution of Farey Fractions in Residue Classes and Lang–Trotter Conjectures on Average
"... 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 trac ..."
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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.
Drinfeld Modules With No Supersingular Primes
"... We give examples of Drinfeld modules # of rank 2 and higher over F q (T ) that have no primes of supersingular reduction. The idea is to construct # so that the associated mod # representations are incompatible with the existence of supersingular primes. We also answer a question of Elkies by pro ..."
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We give examples of Drinfeld modules # of rank 2 and higher over F q (T ) that have no primes of supersingular reduction. The idea is to construct # so that the associated mod # representations are incompatible with the existence of supersingular primes. We also answer a question of Elkies by proving that such obstructions cannot exist for elliptic curves over number fields.