An explicit theory of heights

Flynn, E. V.

doi:10.1090/S0002-9947-1995-1297525-9

An explicit theory of heights
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Trans. Amer. Math. Soc. 347 (1995), 3003-3015 Request permission

Abstract:

We consider the problem of explicitly determining the naive height constants for Jacobians of hyperelliptic curves. For genus $> 1$, it is impractical to apply Hilbert’s Nullstellensatz directly to the defining equations of the duplication law; we indicate how this technical difficulty can be overcome by use of isogenies. The height constants are computed in detail for the Jacobian of an arbitrary curve of genus $2$, and we apply the technique to compute generators of $\mathcal {J}(\mathbb {Q})$, the Mordell-Weil group for a selection of rank $1$ examples.

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Moyenne arithmético-géometrique et périodes des courbes de genre

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