USING ELLIPTIC CURVES TO PRODUCE QUADRATIC NUMBER FIELDS OF HIGH THREE-RANK
We use a connection between the arithmetic of elliptic curves of the form y² = x³ + k and the arithmetic of the quadratic number fields Q(√k) and Q(√-3k) to look for quadratic fields with high three-rank. We give a geometric proof of known results on polynomials that give rise to infinite families o...
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| Veröffentlicht in: | The Rocky Mountain journal of mathematics 2004-06, Vol.34 (2), p.599-618 |
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| container_title | The Rocky Mountain journal of mathematics |
| container_volume | 34 |
| creator | DELONG, MATT |
| description | We use a connection between the arithmetic of elliptic curves of the form y² = x³ + k and the arithmetic of the quadratic number fields Q(√k) and Q(√-3k) to look for quadratic fields with high three-rank. We give a geometric proof of known results on polynomials that give rise to infinite families of quadratic number fields possessing non-trivial lower bounds on their three-rank. We then generalize the method to produce infinitely many such polynomials. Finally, we produce specific examples of quadratic number fields with high three-rank. |
| doi_str_mv | 10.1216/rmjm/1181069870 |
| format | Article |
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| ispartof | The Rocky Mountain journal of mathematics, 2004-06, Vol.34 (2), p.599-618 |
| issn | 0035-7596 1945-3795 |
| language | eng |
| recordid | cdi_projecteuclid_primary_oai_CULeuclid_euclid_rmjm_1181069870 |
| source | Jstor Complete Legacy; Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals; Project Euclid Complete; JSTOR Mathematics & Statistics |
| subjects | 11G05 11R29 Arithmetic Curves Integers Mathematical lattices Mathematical theorems Number theory Numbers Polynomials Series convergence |
| title | USING ELLIPTIC CURVES TO PRODUCE QUADRATIC NUMBER FIELDS OF HIGH THREE-RANK |
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