ROOT NUMBERS OF FIBERS OF ELLIPTIC-SURFACES

The variation of the root number on fibers of elliptic surfaces over the rationals with base the projective line is studied. It is proved that for a large class of such surfaces the sets of rational t's such that the fiber over t is an elliptic curve with root number 1 and -1 respectively are both dense in the set of real numbers. This result provides some evidence for a recent conjecture of B. Mazur. A similar result and some applications are also discussed.