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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 $\mathbb{Z}$ there will be $\mathbb{Q}$ -linear relations among the values of a canonical height pairing evaluated at a basis modulo torsion of $A(k)$ .

Linear Relations Among the Values of Canonical Heights from the Existence of Non-Trivial Endomorphisms