Papers › Quadratic Fields Admitting Elliptic Curves with Rational j-Invariant and Good...
Quadratic Fields Admitting Elliptic Curves with Rational j-Invariant and Good Reduction Everywhere
Benjamin Matschke, Abhijit S. Mudigonda
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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^(-1/2)(x). In this paper, we assume the abc-conjecture to show the sharp asymptotic ∼cxlog^(-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/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 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-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.
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