### Abstract

A hyperelliptic curve C over Q is the graph of an equation of the form y^{2} = 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: y^{2} = f(x), with x and y rational numbers, is called a rational point on C. Given a random elliptic or hyperelliptic curve C: y^{2} = 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 language | English (US) |
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Title of host publication | Plenary Lectures and Ceremonies |

Editors | Sun Young Jang, Young Rock Kim, Dae-Woong Lee, Ikkwon Yie |

Publisher | KYUNG MOON SA Co. Ltd. |

Pages | 657-684 |

Number of pages | 28 |

ISBN (Electronic) | 9788961058049 |

State | Published - Jan 1 2014 |

Event | 2014 International Congress of Mathematicans, ICM 2014 - Seoul, Korea, Republic of Duration: Aug 13 2014 → Aug 21 2014 |

### Publication series

Name | Proceeding of the International Congress of Mathematicans, ICM 2014 |
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Volume | 1 |

### Conference

Conference | 2014 International Congress of Mathematicans, ICM 2014 |
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Country | Korea, Republic of |

City | Seoul |

Period | 8/13/14 → 8/21/14 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Keywords

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

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## Cite this

*Plenary Lectures and Ceremonies*(pp. 657-684). (Proceeding of the International Congress of Mathematicans, ICM 2014; Vol. 1). KYUNG MOON SA Co. Ltd..