p-adic heights on Jacobians of genus 2 curves and applications

Abstract: We describe an algorithmic construction of a p-adic height on the Jacobian of a genus 2 curve over the rationals (here p is not necessarily of good reduction). In particular, the focus will be on the local component at p of the height, which is defined in terms of some p-adic sigma/theta functions.

These local heights differ from those in the Coleman--Gross construction in a crucial way; nevertheless, in some cases we can prove a suitable comparison theorem. Thus, we can use our heights as an alternative to the Coleman--Gross heights in some instances of the quadratic Chabauty method. The application given in this talk concerns the rational points on some quite special genus 4 hyperelliptic curves.

This talk is partly based on joint work with Enis Kaya and Steffen Müller.

number theory

Audience: researchers in the topic

( slides | video )

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Rational Points and Galois Representations

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