RATIONAL POINTS ON SOME FAMILIES OF ELLIPTIC CURVES
| dc.contributor.advisor | Washington, Lawrence C | en_US |
| dc.contributor.author | Eikenberg, Edward Vincent | en_US |
| dc.contributor.department | Mathematics | en_US |
| dc.date.accessioned | 2004-06-04T05:28:28Z | |
| dc.date.available | 2004-06-04T05:28:28Z | |
| dc.date.issued | 2004-04-26 | en_US |
| dc.description.abstract | Let E_m be the family of elliptic curves given by y^2=x^3-x+m^2, which has rank 2 when regarded as an elliptic curve over Q(m). (Here Q represents the field of rational numbers.) Brown and Myers show that a certain quadratic polynomial m(t) has the property that E_m(t) contains an additional rational point that is independent from the two original generators. This implies that there are infinitely many rational numbers n such that E_n(Q) has rank at least 3. We generalize this result, showing that every nonzero rational number n has the property that E_n sits inside such a subfamily of rank 3. Moreover, given any rational point P in E_n, there exists a quadratic polynomial m(t) and a Q(t)-point R(t) in E_m(t) that is independent from the original generators, such that the specialization to t=0 gives m(0)=n and R(0)=P. Such subfamilies can be intersected to increase the rank, demonstrating the existence of a rational subfamily of rank 4 over Q(t), and infinitely many rational numbers n such that E_n(Q) has rank at least 5. Shioda's theory of Mordell-Weil lattices is used to find the generators of such E_m(t) over both Qbar(t) and Q(t) in these cases. (Here Qbar represents the algebraic closure of Q.) All quadratic polynomials m(t) are classified by whether or not E_m(t) contains an additional rational point of low degree. Results similar to these are also obtained for other families of elliptic curves. | en_US |
| dc.format.extent | 365252 bytes | |
| dc.format.mimetype | application/pdf | |
| dc.identifier.uri | http://hdl.handle.net/1903/1384 | |
| dc.language.iso | en_US | |
| dc.relation.isAvailableAt | Digital Repository at the University of Maryland | en_US |
| dc.relation.isAvailableAt | University of Maryland (College Park, Md.) | en_US |
| dc.subject.pqcontrolled | Mathematics | en_US |
| dc.subject.pquncontrolled | elliptic curve | en_US |
| dc.subject.pquncontrolled | rank | en_US |
| dc.subject.pquncontrolled | rational point | en_US |
| dc.subject.pquncontrolled | subfamily | en_US |
| dc.subject.pquncontrolled | elliptic surface | en_US |
| dc.title | RATIONAL POINTS ON SOME FAMILIES OF ELLIPTIC CURVES | en_US |
| dc.type | Dissertation | en_US |
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