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The Formal Group of an Elliptic Curve

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The Arithmetic of Elliptic Curves

Part of the book series: Graduate Texts in Mathematics ((GTM,volume 106))

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Abstract

Let E be an elliptic curve. In this chapter we study an “infinitesimal” neighborhood of E centered at the origin O.

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Notes

  1. 1.

    More precisely, they imply that P F(z, z′) = P z + P z′ for distinct \(z,z' \in \mathcal{M}\). For z = z′, we can let z′ ↦ z and use the fact that the map z ↦ P z and the addition law on E(K) are continuous for the topology induced from K. Alternatively, we could do an explicit, albeit messy, calculation with power series and the duplication formula.

  2. 2.

    The assumption that R has no torsion elements means that if \(n \in \mathbb{Z}\) and α ∈ R satisfy n α = 0, then either n = 0 or α = 0. Equivalently, the natural map \(R \rightarrow K = R \otimes \mathbb{Q}\) is an injection.

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    Joseph H. Silverman

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Silverman, J.H. (2009). The Formal Group of an Elliptic Curve. In: The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol 106. Springer, New York, NY. https://doi.org/10.1007/978-0-387-09494-6_4

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