A Classification of Genus 0 Modular Curves with Rational Points
Let E be a non-CM elliptic curve defined over Q. Fix an algebraic closure Q of Q. We get a Galois representation ρE : Gal(Q/Q) →GL2( ˆZ) associated to E by choosing a compatible bases for the N-torsion subgroups of E(Q). Associated to an open subgroup G of GL2( ˆZ) satisfying −I ∈G and det(G) = ˆZ×, we have the modular curve (XG ,πG ) over Q which loosely parametrises elliptic curves E such that the image of ρE is conjugate to a subgroup of Gt. In this article we give a complete classification of all such genus 0 modular curves that have a rational point. This classification is given in finitely many families. Moreover, each such modular curve can be explicitly computed.
Elliptic curves; Galois representations; Modular curves; Number Theory
Zywina, David J.
Speh, Birgit Else Marie; Ramakrishna, Ravi Kumar
Ph. D., Mathematics
Doctor of Philosophy
dissertation or thesis