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 $\gg x\log^{-1/2}(x)$. In this paper, we assume the $abc$-conjecture to show the sharp asymptotic $\sim cx\log^{-1/2}(x)$ 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 $E/\mathbb{Q}$ with short Weierstrass equation $y^2 = f(x)$ for reducible $f \in \mathbb{Z}[x]$, we show that the number of integers $d$, $|d| \leq D$, for which the quadratic twist $dy^2 = f(x)$ has an integral non-$2$-torsion point is at most $D^{2/3+o(1)}$, 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 $O(x^{1-\epsilon})$ for some $\epsilon > 0$. 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.