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.
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Supported in part by NSF DMS-8501724
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Stiller, P.F. On the classification of elliptic surfaces withq=1. Manuscripta Math 60, 299–321 (1988). https://doi.org/10.1007/BF01169342
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DOI: https://doi.org/10.1007/BF01169342