{"id":{"repo_id":"maryland","oai_identifier":"oai:drum.lib.umd.edu:1903/1384"},"canonical_url":"https://search.dev.ndltd.org/etd/maryland/oai:drum.lib.umd.edu:1903/1384","repository":{"repo_id":"maryland","name":"University of Maryland","base_url":"https://api.drum.lib.umd.edu/server/oai/request"},"display":{"title":"RATIONAL POINTS ON SOME FAMILIES OF ELLIPTIC CURVES","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Eikenberg, Edward Vincent"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Mathematics","school":null,"contributors":[],"advisors":["Washington, Lawrence C"],"committee_chairs":[],"committee_members":[],"year":2004,"date_issued":"2004-04-26","date_published":"2004-04-26","updated_at":"2026-07-24T03:02:25Z","subjects":[],"languages":["en_US"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1903/1384","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Washington, Lawrence C"]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Eikenberg, Edward Vincent"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2004-06-04T05:28:28Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2004-06-04T05:28:28Z"]},{"key":"dc:date.issued","label":"Date","values":["2004-04-26"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1903/1384"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["RATIONAL POINTS ON SOME FAMILIES OF ELLIPTIC CURVES"]}]}],"canonical_facts":{"dc:contributor.advisor":["Washington, Lawrence C"],"dc:contributor.department":["Mathematics"],"dc:creator":["Eikenberg, Edward Vincent"],"dc:date.accessioned":["2004-06-04T05:28:28Z"],"dc:date.available":["2004-06-04T05:28:28Z"],"dc:date.issued":["2004-04-26"],"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."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/1903/1384"],"dc:language.iso":["en_US"],"dc:title":["RATIONAL POINTS ON SOME FAMILIES OF ELLIPTIC CURVES"],"dc:type":["Dissertation"]},"updated_at":"2026-07-24T03:02:25Z"}