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The Faltings height of an elliptic curve $E$ defined over $\Q$ is the quantity $$ -\frac{1}{2}\log(A), $$ where $A$ is the covolume (that is, the area of a fundamental period parallelogram) of the Néron lattice of $E$.

The stable Faltings height of $E$ is $$ \frac{1}{12}(\log\mathrm{denom}(j)-\log(|\Delta|)) - \frac{1}{2}\log(A), $$ where $j$ is the $j$-invariant of $E$, $\Delta$ the discriminant of any model of $E$ and $A$ the covolume of the period lattice of that model. The stable height is independent of the model of $E$, and the unstable and stable heights are equal for semistable curves, for which $\mathrm{denom}(j)=|\Delta|$.

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  • Review status: beta
  • Last edited by John Cremona on 2021-03-22 09:53:05
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