A naive p-adic height function on the Jacobians of curves of genus 2

Trip, Manoy (2022) A naive p-adic height function on the Jacobians of curves of genus 2. Master's Thesis / Essay, Mathematics.

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Abstract

In this thesis, we introduce a naive p-adic height on the Jacobians of smooth projective curves of genus 2. More generally, we explore both real and p-adic height functions on elliptic curves and on the Jacobians of genus 2 curves. Starting from the more established topic of real-valued height functions on elliptic curves, we discuss how methods can be adapted to the construction of p-adic height functions. A naive p-adic height was defined by Bernadette Perrin-Riou, and it can be used to obtain a quadratic p-adic height using a limit process. We give more details on her arguments. We then move on to the topic of height functions on the Jacobians of genus 2 curves. We discuss existing real height functions in this setting, and a quadratic p-adic height defined using local height functions. Then we turn to the main result of the thesis, which is the existence of a naive p-adic height on the Jacobian of a genus 2 curve that can be used to define a quadratic p-adic height. This height is defined analogously to Perrin-Riou’s height. We show that the resulting quadratic p-adic height is equal to the quadratic height obtained using local heights.

Item Type:	Thesis (Master's Thesis / Essay)
Supervisor name:	Muller, J.S. and Bianchi, F. and Lorscheid, O.
Degree programme:	Mathematics
Thesis type:	Master's Thesis / Essay
Language:	English
Date Deposited:	27 May 2022 07:45
Last Modified:	27 May 2022 07:45
URI:	https://fse.studenttheses.ub.rug.nl/id/eprint/27097

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