Solving S-Unit, Mordell, Thue, Thue-Mahler and Generalized Ramanujan-Nagell Equations Via the Shimura-Taniyama Conjecture

Solving S-Unit, Mordell, Thue, Thue-Mahler and Generalized Ramanujan-Nagell Equations Via the Shimura-Taniyama Conjecture Synopsis

"In the first part we construct algorithms (over Q) which we apply to solve Sunit, Mordell, cubic Thue, cubic Thue-Mahler and generalized Ramanujan-Nagell equations. As a byproduct we obtain alternative practical approaches for various classical Diophantine problems, including the fundamental problem of finding all elliptic curves over Q with good reduction outside a given finite set of rational primes. The first type of our algorithms uses modular symbols, and the second type combines explicit height bounds with efficient sieves. In particular we construct a refined sieve for S-unit equations which combines Diophantine approximation techniques of de Weger with new geometric ideas. To illustrate the utility of our algorithms we determined the solutions of large classes of equations, containing many examples of interest which are out of reach for the known methods. In addition we used the resulting data to motivate various conjectures an