This page hosts information on Fritz Hörmann's talk "What is.. the Birch-Swinnerton-Dyer conjecture?" at the

WhatIsSeminar.

## Abstract

Elliptic curves - which can be given by equations of degree 3 (e.g.
X^3+Y^3=1) - are the most interesting among all algebraic curves. It is an
old question in number theory, called a Diophantine problem, to determine
the set of rational points on such a curve. The elliptic case, again, is
the most interesting and mysterious of all Diophantine problems of
dimension 1. For example, there may or may not be infinitely many rational
solutions. At present no known algorithm can determine this.
However, already in the 60's, Birch and Swinnerton-Dyer experimentally
found a deep and mysterious relation of this question to analytic
properties of the zeta-function of the curve, which encodes the easily
determined solutions of the congruences (e.g. X^3+Y^3 == 1 modulo N). It
later became one of the most famous conjectures of mathematics, and is one
of the millenium prize problems, for whose solution the Clay Mathematical
Institute offers a reward of $1,000,000.

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