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Zbl 1158.14311
Kasparian, Azniv; Holzapfel, Rolf-Peter
Arithmetic proportional elliptic configurations with comparatively large number of irreducible components.
(English)
[A] Mladenov, Iva\"ilo 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. 252-261 (2005). ISBN 954-84952-9-5/pbk

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 $\bbfQ(\sqrt{-d})$. The present note establishes that if T has s singular points and $4s -5 \le h \le 4s$ 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.
MSC 2000:
*14J27 Elliptic surfaces
11G15 Complex multiplication and moduli of abelian varieties

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