## Hodge cycles on abelian varieties associated to the complete binary trees.(English)Zbl 1117.14013

This paper is an investigation of the structure of the ring of Hodge cycles on a complex abelian variety $$A$$ of CM-type, where the CM-field $$K$$ is Galois over $$\mathbb Q$$ with Galois group isomorphic to $$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2^n\mathbb{Z}$$. The author constructs all degenerate CM-types for $$K$$, and enumerates the Hodge cycles on the associated abelian varieties. These results are obtained as applications of combinatorial results on complete binary trees.

### MSC:

 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 11G10 Abelian varieties of dimension $$> 1$$ 06A11 Algebraic aspects of posets

### Keywords:

Hodge cycle; abelian variety; binary tree
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