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Arithmetic and GeometryTen Years in Alpbach (AMS-202)$
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Gisbert Wüstholz and Clemens Fuchs

Print publication date: 2019

Print ISBN-13: 9780691193779

Published to Princeton Scholarship Online: May 2020

DOI: 10.23943/princeton/9780691193779.001.0001

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PRINTED FROM PRINCETON SCHOLARSHIP ONLINE (www.princeton.universitypressscholarship.com). (c) Copyright Princeton University Press, 2022. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in PRSO for personal use.date: 30 June 2022

Hyperelliptic Continued Fractions and Generalized Jacobians: Minicourse Given by Umberto Zannier

Hyperelliptic Continued Fractions and Generalized Jacobians: Minicourse Given by Umberto Zannier

Chapter:
(p.56) Chapter Three Hyperelliptic Continued Fractions and Generalized Jacobians: Minicourse Given by Umberto Zannier
Source:
Arithmetic and Geometry
Author(s):

Laura Capuano

Peter Jossen

Christina Karolus

Francesco Veneziano

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691193779.003.0003

This chapter details Umberto Zannier's minicourse on hyperelliptic continued fractions and generalized Jacobians. It begins by presenting the Pell equation, which was studied by Indian, and later by Arabic and Greek, mathematicians. The chapter then addresses two questions about continued fractions of algebraic functions. The first concerns the behavior of the solvability of the polynomial Pell equation for families of polynomials. It must be noted that these questions are related to problems of unlikely intersections in families of Jacobians of hyperelliptic curves (or generalized Jacobians). The chapter also reviews several classical definitions and results related to the continued fraction expansion of real numbers and illustrates them by examples.

Keywords:   Umberto Zannier, hyperelliptic continued fractions, generalized Jacobians, Pell equation, continued fractions, algebraic functions, polynomials, hyperelliptic curves, real numbers

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