Quadratic fields admitting elliptic curves with rational j-invariant and good reduction everywhere

Abstract

Clemm and Trebat-Leder (2014) proved that the number of quadratic number fields with absolute discriminant bounded by x over which there exist elliptic curves with good reduction everywhere and rational j-invariant is ≫xlog−12⁡(x). In this paper, we assume the abc-conjecture to show the sharp asymptotic ∼cxlog−12⁡(x) for this number, obtaining formulae for c in both the real and imaginary cases. Our method has three ingredients:(1)We make progress towards a conjecture of Granville: Given a fixed elliptic curve E/Q with short Weierstrass equation y2=f(x) for reducible f∈Z[x], we show that the number of integers d, |d|≤D, for which the quadratic twist dy2=f(x) has an integral non-2-torsion point is at most D23+o(1), assuming the abc-conjecture.(2)We apply the Selberg–Delange method to obtain a Tauberian theorem which allows us to count integers satisfying certain congruences while also being divisible only by certain primes.(3)We show that for a polynomially sparse subset of the natural numbers, the number of pairs of elements with least common multiple at most x is O(x1−ϵ) for some ϵ>0. We also exhibit a matching lower bound. If instead of the abc-conjecture we assume a particular tail bound, we can prove all the aforementioned results and that the coefficient c above is greater in the real quadratic case than in the imaginary quadratic case, in agreement with an experimentally observed bias.