Elliptic curves of almost-prime conductor

Details Elliptic curves of almost-prime conductor Elliptic curves of almost-prime conductor

2012-01-22

arxiv.org/abs/1201.4566v1

Mathematics

Number Theory

10 pages (including 2 pages of tables)

Scientific paper

We use elementary techniques to study elliptic curves over Q of conductor N equal to the product of two distinct odd primes. Using a result of Iwaniec on the representation of almost-primes by quadratic forms, we show that infinitely many N equal to a prime or the product of two distinct odd primes arise as the conductors of elliptic curves over Q. Inspired by earlier work in the case of prime conductor, we also provide a non-existence result: We show that there are no elliptic curves over Q of conductor N when N is the product of two primes satisfying certain congruency and quadratic class number conditions. We use a computer to find all 67 values of N<10^7 satisfying these conditions.

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