@inproceedings{a5bbc3fb8f17477dbfed0ab6b438df11,

title = "Rational points on elliptic and hyperelliptic curves",

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.",

keywords = "Birch-Swinnerton-Dyer Conjecture, Elliptic curve, Hasse principle, Hyperelliptic curve, Rank, Rational points",

author = "Manjul Bhargava",

note = "Funding Information: Acknowledgements. The author was partially funded by NSF grant DMS-1001828 and a Simons Investigator Grant. I thank Benedict Gross, Wei Ho, Bjorn Poonen, Arul Shankar, Christopher Skinner, Michael Stoll, Xiaoheng Wang, and Wei Zhang for their help.; 2014 International Congress of Mathematicans, ICM 2014 ; Conference date: 13-08-2014 Through 21-08-2014",

year = "2014",

language = "English (US)",

series = "Proceeding of the International Congress of Mathematicans, ICM 2014",

publisher = "KYUNG MOON SA Co. Ltd.",

pages = "657--684",

editor = "Jang, {Sun Young} and Kim, {Young Rock} and Dae-Woong Lee and Ikkwon Yie",

booktitle = "Plenary Lectures and Ceremonies",

}