{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/quadratic-fields-admitting-elliptic-curves","title":"Quadratic Fields Admitting Elliptic Curves with Rational $j$-Invariant and Good Reduction Everywhere","arxiv_id":"2103.09814","date":"2021-03-17","proceeding":null,"authors":["Benjamin Matschke","Abhijit S. Mudigonda"],"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.","url_abs":"https://arxiv.org/abs/2103.09814v2","url_pdf":"https://arxiv.org/pdf/2103.09814v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"quadratic-fields-admitting-elliptic-curves","repo_url":"https://github.com/abhijit-mudigonda/everywhere-good-reduction","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}