Lecture Notes for MATH 426 - FUND CONCEPTS ANLYS at Washington (UW)

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University:University of Washington - Seattle
  • Number Fields
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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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