We study the interplay between canonical heights and endomorphisms of an abelian variety $A$ over a number field $k$. In particular we show that whenever the ring of endomorphisms defined over $k$ is strictly larger than $\Z$ there will be $\Q$-linear relations among the values of a canonical height pairing evaluated at a basis modulo torsion of $A(k)$.

MSC Classifications:

11G10, 14K15 show english descriptions Abelian varieties of dimension $> 1$ [See also 14Kxx]
Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx] 11G10 - Abelian varieties of dimension $> 1$ [See also 14Kxx]
14K15 - Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx]

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