Number Theory: Lecture 11

In which we explore reduced binary quadratic forms.

  • Lemma 32: Let f be a positive definite binary quadratic form.  Then f is equivalent to a reduced form.  We used three unimodular substitutions (S : (a,b,c) \mapsto (c,-b,a) and T_{\pm}:(a,b,c) \mapsto (a,b\pm 2a, a \pm b + c)) and gave an algorithm for reducing a form.
  • Lemma 33: Let f(x,y) = ax^2 + bxy + cy^2 be a reduced form.  The smallest integers represented by f(x,y) for coprime x and y (or x = y = 0) are 0, a, c and a - |b| + c, in that order.  We noted that f(0,0) = 0 and f(\pm 1, 0) = a and f(0, \pm 1) = c, and then checked what happens if |x| \geq |y| > 0 or |y| \geq |x| > 0.
  • Lemma 34: Let \displaystyle \left(\begin{array}{c} x' \\ y' \end{array}\right) = \left(\begin{array}{cc}p & q \\ r & s \end{array}\right) \left( \begin{array}{c} x \\ y \end{array}\right) where \displaystyle \left(\begin{array}{cc}p & q \\ r & s \end{array}\right) \in SL_2(\mathbb{Z}).  The integers x and y are coprime if and only if x' and y' are coprime.  This was a straightforward check.
  • Theorem 35: Each positive definite binary quadratic form is equivalent to a unique reduced form.  We used Lemma 32, and then showed that if two reduced forms are equivalent then they are the same, using Lemmas 33 and 34.

Understanding today’s lecture

Pick some positive definite binary quadratic forms and reduce them.

Further reading

Baker (A concise introduction to the theory numbers) covers all of this material.  Davenport (The Higher Arithmetic) doesn’t go into the details of the proof of Theorem 35, but discusses many of the other ideas in today’s lecture.  There are some online notes by Andrew Granville that cover this material.

Preparation for Lecture 12

How many reduced forms are there with a given discriminant?  (E.g. are there infinitely many or finitely many, or does it depend on the discriminant?)

Next time we’ll be tackling the question of which numbers can be represented by binary quadratic forms.

If f(x,y) = ax^2 + bxy + cy^2 is a binary quadratic form, then a can certainly be represented (because f(1,0) = a).  It would be interesting to know whether the converse is true, at least once we’re clear what the converse might be.  Let f be a form that represents a.  We know that f might not have first coefficient a, not least because there are many forms equivalent to f that represent the same set of numbers, and they don’t all have the same first coefficient.  But is it the case that if f represents a then f is equivalent to a form with first coefficient a?

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3 Responses to “Number Theory: Lecture 11”

  1. Number Theorist Says:

    Regarding lemma 33, isn’t it still true even if you don’t mention that x and y are coprime? If HCF(x,y)=d, say, then f(x/d,y/d) will be an integer and f(x/d,y/d)=1/d^2 f(x,y), so if we are trying to minimise f(x,y), then WLOG x and y are coprime.

    With this in mind, I don’t see why, in theorem 35, we couldn’t have just said ‘The second smallest positive integer represented by f and f’ respectively are c and c’, so c=c’.’

    Is there anything wrong with that argument?

    PS: Loving the blog! It’s a great idea 🙂

  2. theoremoftheweek Says:

    Good questions!

    In Lemma 33, the tiny wrinkle is that for example f(2,0) might be smaller than f(x,y) where x and y are both non-zero. Perhaps you or someone else would like to suggest an example of a form where this happens?

    The second smallest positive integers represented are indeed c and c’. But the hypothetical problem is something like this: maybe the smallest positive integers represented by f are a and c where c is different from a, but the smallest positive integers represented by f are a and c’ where c’ is actually the same as a.

    It turns out that can’t happen, because the four pairs (x,y) that represent a and c for f must correspond to four pairs that represent a and c’ for f’, but we do have to check carefully.

    Does that help? Please do post back if you have further/follow-up questions!

  3. Number Theory: Lecture 12 | Theorem of the week Says:

    […] Expositions of interesting mathematical results « Number Theory: Lecture 11 […]

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