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

Kasparian, Azniv; Holzapfel, Rolf-Peter

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].

MSC:

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