# Heights and the specialization map for families of abelian varieties.

@article{Silverman1983HeightsAT, title={Heights and the specialization map for families of abelian varieties.}, author={Joseph H. Silverman}, journal={Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)}, year={1983}, volume={1983}, pages={197 - 211} }

Let C be a non-singular projective curve, and let A — > C be a (flat) family of abelian varieties, all defmed over a global field K. There are three natural height functions associated to such a family, namely a Weil height hc on C(AT), a canonical height hAti on the generic fiber An(K(C}\ and for each point teC(K) for which the special fiber At is non-singular, a canonical height hAt on At(K). Extending ideas of Dem'janenko and Manin ([2], [10]), we prove the following theorem comparing these…

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