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The complex AGM, periods of elliptic curves over and complex elliptic logarithms
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Cremona, J. E. and Thongjunthug, Thotsaphon (2013) The complex AGM, periods of elliptic curves over and complex elliptic logarithms. Journal of Number Theory, Volume 133 (Number 8). pp. 2813-2841. doi:10.1016/j.jnt.2013.02.002 ISSN 0022-314X.
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Official URL: http://dx.doi.org/10.1016/j.jnt.2013.02.002
Abstract
We give an account of the complex Arithmetic–Geometric Mean (AGM), as first studied by Gauss, together with details of its relationship with the theory of elliptic curves over C, their period lattices and complex parametrisation. As an application, we present efficient methods for computing bases for the period lattices and elliptic logarithms of points, for arbitrary elliptic curves defined over C. Earlier authors have only treated the case of elliptic curves defined over the real numbers; here, the multi-valued nature of the complex AGM plays an important role. Our method, which we have implemented in both MAGMA and Sage, is illustrated with several examples using elliptic curves defined over number fields with real and complex embeddings.
| Item Type: | Journal Article | ||||
|---|---|---|---|---|---|
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
| Journal or Publication Title: | Journal of Number Theory | ||||
| Publisher: | Academic Press | ||||
| ISSN: | 0022-314X | ||||
| Official Date: | 2013 | ||||
| Dates: |
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| Volume: | Volume 133 | ||||
| Number: | Number 8 | ||||
| Page Range: | pp. 2813-2841 | ||||
| DOI: | 10.1016/j.jnt.2013.02.002 | ||||
| Status: | Peer Reviewed | ||||
| Publication Status: | Published | ||||
| Access rights to Published version: | Restricted or Subscription Access |
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