A p-adic study of the Richelot isogeny with applications to periods of certain genus 2 curves

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.