{"title":"Local and global densities for Weierstrass models of elliptic curves","authors":"J. E. Cremona, M. Sadek","doi":"10.4310/mrl.2023.v30.n2.a5","DOIUrl":null,"url":null,"abstract":"We prove local results on the $p$-adic density of elliptic curves over $\\mathbb{Q}_p$ with different reduction types, together with global results on densities of elliptic curves over $\\mathbb{Q}$ 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 $\\mathbb{Q}$ (that is, elliptic curves with square-free conductor) is $1/\\zeta(2)\\approx60.79\\%$, the same as the density of square-free integers; the density of semistable elliptic curves over $\\mathbb{Q}$ is $\\zeta(10)/\\zeta(2)\\approx60.85\\%$; the density of integral Weierstrass equations which have square-free discriminant is $\\prod_p\\left(1-\\frac{2}{p^2}+\\frac{1}{p^3}\\right) \\approx 42.89\\%$, 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 previous result of Baier and Browning for short Weierstrass equations); and the density of elliptic curves over $\\mathbb{Q}$ with square-free minimal discriminant is $\\zeta(10)\\prod_p\\left(1-\\frac{2}{p^2}+\\frac{1}{p^3}\\right)\\approx42.93\\%$. 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.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"11 1","pages":"0"},"PeriodicalIF":0.6000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"25","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Research Letters","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4310/mrl.2023.v30.n2.a5","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 25
Abstract
We prove local results on the $p$-adic density of elliptic curves over $\mathbb{Q}_p$ with different reduction types, together with global results on densities of elliptic curves over $\mathbb{Q}$ 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 $\mathbb{Q}$ (that is, elliptic curves with square-free conductor) is $1/\zeta(2)\approx60.79\%$, the same as the density of square-free integers; the density of semistable elliptic curves over $\mathbb{Q}$ is $\zeta(10)/\zeta(2)\approx60.85\%$; the density of integral Weierstrass equations which have square-free discriminant is $\prod_p\left(1-\frac{2}{p^2}+\frac{1}{p^3}\right) \approx 42.89\%$, 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 previous result of Baier and Browning for short Weierstrass equations); and the density of elliptic curves over $\mathbb{Q}$ with square-free minimal discriminant is $\zeta(10)\prod_p\left(1-\frac{2}{p^2}+\frac{1}{p^3}\right)\approx42.93\%$. 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.
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