Lecture 23: Quadratic Forms III
Reduction Theory

William Stein

Date: Math 124 $\quad$ HARVARD UNIVERSITY $\quad$ Fall 2001

Recall that a binary quadratic form is a function $f(x,y) = ax^2 + bxy+cy^2$. Our motivating problem is to decide which numbers are ``represented'' by $f$; i.e., for which integers $n$ do there exist integers $x, y$ such that $ax^2 + bxy + cy^2 = n$? If $g\in\SL_2(\mathbb{Z})$ then $f(x,y)$ and $f\vert _g(x,y) = f\left(g\vtwo{x}{y}\right)$ represent exactly the same set of integers. Also, $\disc(f)=\disc(f\vert _g)$, where $\disc(f) = b^2 - 4ac$, and $f$ is called positive definite if $\disc(f)<0$ and $a>0$.

In today's lecture, we will learn about reduction theory, which allows us to decide whether or not two positive definite binary quadratic forms are equivalent under the action of $\SL_2(\mathbb{Z})$.

If, in the future, you would like to pursue the theory of binary quadratic forms in either a more algebraic or algorithmic direction, I highly recommend that you look at Chapter 5 of Henri Cohen's book A Course in Computational Algebraic Number Theory (GTM 138).