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.

Further reading

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?

One Response to “Number Theory: Lecture 10”

  1. Number Theory: Lecture 11 « Theorem of the week Says:

    […] Theorem of the week Expositions of interesting mathematical results « Number Theory: Lecture 10 […]

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