Prof. Dr. Rolf-Peter Holzapfel
Cyclotomic curve Families over Elliptic Curves with Complete Picard-Einstein Metric
Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976)

MSC:

 11G15 Complex multiplication and moduli of abelian varieties, See also {14K22}

11G18 Arithmetic aspects of modular and Shimura varieties, See also {14G35}

ZDM: 14H52
PACS: 32L07
CR: 14J25
Abstract: According to a problem of Hirzebruch we look for models of
biproducts of elliptic CM-curves with Picard modular structure. We introduce
the singular mean value of crossing elliptic divisors on surfaces and determine
its maximum for all abelian surfaces. For any maximal crossing elliptic
divisor on an abelian surface A we construct infinite towers of coverings
of A whose members, inclusively A, are contracted compactified ball quo-
tients. On this way we find towers of Picard modular surfaces of the Gauss
number field including E × E blown up at six points (E \cong C/Z[i]),
the Kummer surface of the rational cuboid problem (3-dimensional extension
of congruence number problem) and some interesting rational surfaces
together with the corresponding congruence subgroups of U((2,1),Z[i]).
Keywords: algebraic curve, elliptic curve, algebraic surface, Shimura variety,
arithmetic group, Picard modular group, Gauss numbers, congruence numbers,
Kähler-Einstein metric, negative constant curvature, unit ball