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oai:drum.lib.umd.edu:1903/13842016-03-29T07:28:21Zcom_1903_2261com_1903_12com_1903_2col_1903_2793col_1903_3

Washington, Lawrence C Eikenberg, Edward Vincent Mathematics 2004-06-04T05:28:28Z 2004-06-04T05:28:28Z 2004-04-26 http://hdl.handle.net/1903/1384 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. 365252 bytes application/pdf en_US RATIONAL POINTS ON SOME FAMILIES OF ELLIPTIC CURVES Dissertation Digital Repository at the University of Maryland University of Maryland (College Park, Md.) Mathematics elliptic curve rank rational point subfamily elliptic surface open.access