International audienc

International audienceWe prove that, with "obvious" exceptions, a CM-point (j(\tau),j(\tau')) cannot belong to a straight line in C^2 defined over Q. This generalizes a result of K\"uhne, who proved this for the line x+y=1

Abstract. In this paper we improve on Odlyzko's lower bounds for the Artin conductor.

We estimate from below the lower density of the set of prime numbers p such that p − 1 has a prime factor of size at least p c , where 1/4 ≤ c ≤ 1/2. We also establish upper and lower bounds on the counting function of the set of positive integers n ≤ x with exactly k prime factors, counted with or without multiplicity, such that the largest prime factor of gcd(p − 1 : p | n) exceeds n 1/2k .

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