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On Arithmetic Of Hyperelliptic Curves
Jing Yu
Abstract
In this expos´e, Pell's equation is put in a geometric perspective, and
a version of Artin's primitive roots conjecture is formulated for hyperel-
liptic jacobians. Also explained are some recent results which throw new
lights, having to do with Ankeny-Artin-Chowla's conjecture, class number
relations, and Cohen-Lenstra heuristics.
Introduction
It is well known that there are close connections between the arithmetic
behavior of algebraic number fields and that of the algebraic function fields
in one variable over finite fields. This connection has been a constant source
for exciting developments of number theory in 20th century. In this article
we shall further explore the subject by examining some arithmetic questions
about hyperelliptic function fields viewed as analogues of quadratic number
1
2 Jing Yu
fields.
Let C be a hyperelliptic curve over an arbitrary base field k, i.e. a
double cover of the projective line P=k. Its function f...

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