Multiple supersingular elliptic fibers on elliptic surfaces

Elliptic surfaces over an algebraically closed field in characteristic p>0p>0 with multiple supersingular elliptic fibers, that is, multiple fibers of a supersingular elliptic curve, are investigated. In particular, it is shown that for an elliptic surface with q=g+1q=g+1 and a supersingular elliptic curve as a general fiber, where q is the dimension of an Albanese variety of the surface and g   is the genus of the base curve, the multiplicities of the multiple supersingular elliptic fibers are not divisible by p2p2. As an application of this result, the structure of false hyperelliptic surfaces is discussed on this basis.