On the classification of elliptic surfaces withq=1

Abstract

We classify, up to isomorphism, elliptic surfaces with irregularity one having exactly one singular fiber (necessarily of typeI 6 * ). All of them turn out to be elliptic modular surfaces (Shioda [11]), so that the problem is indirectly equivalent to classifying certain subgroups ofSL 2(Z). These surfaces are then used to produce examples of (elliptic) surfaces withq=1, anyp g ≥1, which have maximal Picard number (see Persson [7] for the caseq=0). Finally, the classification yields some interesting relationships between hypergeometric functions, theta functions, and certain automorphic forms.