Cremona, John E. and Sadek, Mohammad
(2023)
*Local and global densities for Weierstrass models of elliptic curves.*
Mathematical Research Letters, 30
(2).
pp. 413-461.
ISSN 1073-2780 (Print) 1945-001X (Online)

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Official URL: http://dx.doi.org/10.4310/MRL.2023.v30.n2.a5

## Abstract

We prove local results on the p-adic density of elliptic curves over Qp with different reduction types, together with global results on densities of elliptic curves over Q with specified reduction types at one or more (including infinitely many) primes. These global results include: the density of integral Weierstrass equations which are minimal models of semistable elliptic curves over Q (that is, elliptic curves with square-free conductor) is 1/ζ(2) ≈ 60.79%, the same as the density of square-free integers; the density of semistable elliptic curves over Q is ζ(10)/ζ(2) ≈ 60.85%; the density of integral Weierstrass equations which have square-free discriminant is Qp (1 - p22 + p13) ≈ 42.89%, which is the same (except for a different factor at the prime 2) as the density of monic integral cubic polynomials with square-free discriminant (and agrees with a 2013 result of Baier and Browning for short Weierstrass equations); and the density of elliptic curves over Q with square-free minimal discriminant is ζ(10) Qp (1 - p22 + p13) ≈ 42.93%. The local results derive from a detailed analysis of Tate’s Algorithm, while the global ones are obtained through the use of the Ekedahl Sieve, as developed by Poonen, Stoll, and Bhargava.

Item Type: | Article |
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Subjects: | Q Science > QA Mathematics Q Science > QA Mathematics > QA150-272.5 Algebra |

Divisions: | Faculty of Engineering and Natural Sciences > Basic Sciences > Mathematics Faculty of Engineering and Natural Sciences |

Depositing User: | Mohammad Sadek |

Date Deposited: | 04 Oct 2023 16:07 |

Last Modified: | 03 Feb 2024 21:54 |

URI: | https://research.sabanciuniv.edu/id/eprint/48129 |