Rational points on elliptic and hyperelliptic curves

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

A hyperelliptic curve C over Q is the graph of an equation of the form y2 = f(x), where f is a polynomial having coefficients in the rational numbers Q and distinct roots in C. The special case where the degree of f is 3 is called an elliptic curve E over Q which, as we will discuss, has many special properties not shared by general hyperelliptic curves C. A solution (x, y) to C: y2 = f(x), with x and y rational numbers, is called a rational point on C. Given a random elliptic or hyperelliptic curve C: y2 = f(x) over Q with f(x) of a given degree n, how many rational points do we expect on the curve C? Equivalently, how often do we expect a random polynomial f(x) of degree n to take a square value over the rational numbers? In this article, we give an overview of a number of recent conjectures and theorems giving some answers and partial answers to this question.

Original languageEnglish (US)
Title of host publicationPlenary Lectures and Ceremonies
EditorsSun Young Jang, Young Rock Kim, Dae-Woong Lee, Ikkwon Yie
PublisherKYUNG MOON SA Co. Ltd.
Pages657-684
Number of pages28
ISBN (Electronic)9788961058049
StatePublished - Jan 1 2014
Event2014 International Congress of Mathematicans, ICM 2014 - Seoul, Korea, Republic of
Duration: Aug 13 2014Aug 21 2014

Publication series

NameProceeding of the International Congress of Mathematicans, ICM 2014
Volume1

Conference

Conference2014 International Congress of Mathematicans, ICM 2014
CountryKorea, Republic of
CitySeoul
Period8/13/148/21/14

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Keywords

  • Birch-Swinnerton-Dyer Conjecture
  • Elliptic curve
  • Hasse principle
  • Hyperelliptic curve
  • Rank
  • Rational points

Fingerprint Dive into the research topics of 'Rational points on elliptic and hyperelliptic curves'. Together they form a unique fingerprint.

  • Cite this

    Bhargava, M. (2014). Rational points on elliptic and hyperelliptic curves. In S. Y. Jang, Y. R. Kim, D-W. Lee, & I. Yie (Eds.), Plenary Lectures and Ceremonies (pp. 657-684). (Proceeding of the International Congress of Mathematicans, ICM 2014; Vol. 1). KYUNG MOON SA Co. Ltd..