## Number Theory: Lecture 10

In which we develop our understanding of binary quadratic forms.

• We remarked that while equivalent forms have the same discriminant, the converse is not true: there are forms that have the same discriminant but that are not equivalent.  Try to find your own examples, in addition to the one we saw in lectures.
• Lemma 30: Let $d$ be an integer.  Then there is a binary quadratic form with discriminant $d$ if and only if $d$ is congruent to 0 or 1 modulo 4.  One direction follows from the congruence $d \equiv b^2 \pmod{4}$.  To prove the other direction, for each case we simply found a binary quadratic form that does the job.
• Definition: Positive definite, negative definite, and indefinite.
• Lemma 31: Let $f(x,y) = ax^2 + bxy + cy^2$ be a binary quadratic form with discriminant $d = b^2 - 4ac$, and with $a \neq 0$.  If $d > 0$, then $f$ is indefinite.  If $d < 0$, then $f$ is definite.  Moreover, if $d < 0$ and $a > 0$ then $f$ is positive definite, and if $d < 0$ and $a < 0$ then $f$ is negative definite.  We proved this by writing $4a(ax^2 + bxy + cy^2) = (2ax + by)^2 + (4ac - b^2) y^2$.  In the case that $d < 0$, the results then follow nicely.  We had to do a tiny bit more work to deal with $d > 0$: we noted that the right-hand side is the difference of two squares, and argued from there.
• Definition of reduced forms.  A positive definite binary quadratic form $(a,b,c)$ is reduced if $-a < b \leq a < c$ or $0 \leq b \leq a = c$.
• Lemma 32: Let $f$ be a positive definite binary quadratic form.  Then $f$ is equivalent to a reduced form.  We saw that there were a couple of particularly useful transformations that we could use to reduce forms, namely $S:(a,b,c) \mapsto (c,-b,a)$ and $T_{\pm}:(a,b,c) \mapsto (a, b \pm 2a, a \pm b + c)$.  Each of these can be used to reduce one of $a$ and $|b|$ while leaving the other fixed, and by doing a suitable combination of these operations we see that we always obtain a form $(a,b,c)$ with $|b| \leq a \leq c$.  Finally, if $a = c$ then we can apply $S$ if necessary to ensure that $b \geq 0$, and if $b = -a$ then we can apply $T_+$ to change the middle coefficient to $a$ (while leaving the others unchanged).

I gave out the second examples sheet today; you can find it on the course page.

Davenport (The Higher Arithmetic) discusses this material, as does Baker (A concise introduction to the theory of numbers).

#### Preparation for Lecture 11

We are making good progress with our task of understanding binary quadratic forms.  Here are some questions for you to consider before next time.

• For a fixed discriminant $d < 0$, how many reduced forms are there?  (Finitely many, or infinitely many?)
• Can you find any discriminants $d < 0$ for which there is just one reduced form?  Can you find any for which there are exactly two reduced forms?
• Can two reduced forms be equivalent to each other?