Arithmetic proportional elliptic configurations with comparatively large number of irreducible components. (English) Zbl 1158.14311

Mladenov, Ivaïlo M.(ed.) et al., Proceedings of the 6th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 3–10, 2004. Sofia: Bulgarian Academy of Sciences (ISBN 954-84952-9-5/pbk). 252-261 (2005).
Summary: Let \(T\) be an arithmetic proportional elliptic configuration on a bielliptic surface \(A_{\sqrt{-d}}\) with complex multiplication by an imaginary quadratic number field \(\mathbb{Q}(\sqrt{-d})\). The present note establishes that if T has s singular points and \(4s -5 \leq h \leq 4s\) irreducible smooth elliptic components, then \(d = 3\) and \(T\) is \(\text{Aut}(A_{\sqrt{-3}}\)-equivalent to Hirzebruch’s example \(T^{(1,4)}_{\sqrt{-d}}\) with a unique singular point and 4 irreducible components.
For the entire collection see [Zbl 1066.53003].


14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
11G15 Complex multiplication and moduli of abelian varieties