Abstract
In this paper, we begin to consider the problem of computing p-adic periods of certain genus 2 curves with totally split reduction, using techniques of the arithmetic–geometric mean. For this, we synthesise the work of Henniart and Mestre on a p-adic arithmetic–geometric mean in genus 1 with the work of Bost and Mestre on a real arithmetic–geometric mean in genus 2 (via the so-called Richelot isogeny). We prove that, for a certain class of p-adic genus 2 curves, the Richelot isogeny plays the same role in the genus 2 theory as the maps appearing in Henniart–Mestre, in that the Richelot isogeny squares the p-adic periods, and leads to a quadratically converging sequence of genus 2 curves. This suggests that this may provide a quadratically convergent method to compute p-adic periods for these curves, once we have a suitably explicit p-adic Tate uniformisation in genus 2.
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Acknowledgements
We thank Tobias Berger, John Cremona, Victor Flynn, David Grant, Haluk Şengün and Michael Stoll for their interest in this project, and Jordi Pujolàs for useful discussions relating to Sect. 6 (and in particular, for providing an early draft of [16]). We also thank the anonymous referee for providing a number of useful suggestions which have improved the exposition of the paper.
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The first author thanks the University of Sheffield for its support as a graduate student via a Graduate Teaching Assistantship. This work formed part of his thesis ([5]), supervised by the second author. The authors received no further funding from any organisation for this work. The authors have no competing interests to declare that are relevant to the content of this article.
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Chow, R., Jarvis, F. A p-adic study of the Richelot isogeny with applications to periods of certain genus 2 curves. Ramanujan J 61, 935–956 (2023). https://doi.org/10.1007/s11139-022-00697-8
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DOI: https://doi.org/10.1007/s11139-022-00697-8