Number Theory: Lecture 11

In which we explore reduced binary quadratic forms.

  • Lemma 33: Let f(x,y) = ax^2 + bxy + cy^2 be a reduced form.  Then the smallest values represented by f(x,y) for coprime x and y are 0, a, c and a - |b| + c, in that order.  We showed that if x and y are non-zero integers then f(x,y) \geq a - |b| + c.  (In the lecture, I forgot to include the condition that x and y are coprime.  This doesn’t matter if they’re both non-zero, but it does matter when one variable is zero, because it means that the only pairs (x,y) that we need to consider are (0,0) and (\pm 1,0) and (0,\pm 1).  It also doesn’t affect our application of Lemma 33 in Theorem 34.  I’ll say something about this at the start of the next lecture.)
  • Theorem 34: Each positive definite binary quadratic form is equivalent to a unique reduced form.  We have already seen that each positive definite binary quadratic form is equivalent to some reduced form, so it suffices to check that no two reduced forms are equivalent.  We did this using Lemma 33.
  • Proposition 35: Let d be a fixed negative integer.  Then there are finitely many reduced forms of discriminant d.  We used the condition b^2 - 4ac = d together with the criterion for being a reduced form to obtain bounds on the coefficients, from which the result follows.
  • Definition of the class number.

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 34, but discusses many of the other ideas in today’s lecture.  There are some online notes by Andrew Granville that cover this material.

I mentioned the class number problem.  There are many places where you can read about this, including this paper by Stark, and this paper by Goldfeld, to give just two examples.  Either of these would give you a sense of the breadth of mathematical ideas that go into the study of the class number.

We’ve only discussed the reduction of definite binary quadratic forms.  There is also a very interesting theory of indefinite forms.  The bad news is that we don’t have time to explore it in lectures.  The good news is that there is a CATAM project that invites you to do exactly that, and you should now be well placed to tackle this project.

Preparation for Lecture 12

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

One thing to consider is whether we need to study representations by all pairs (x,y), or whether we can concentrate on pairs (x,y) where the variables are coprime.  If we start with coprime x and y and apply a unimodular substitution, do we still get coprime values of the variables?

And here’s another question to consider.  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?


2 Responses to “Number Theory: Lecture 11”

  1. Vaidotas Says:

    I have a question about theorem 34: we deduced that a=a’ using minimality of values attained by f and f’ ; but I can’t see why c=c’ : you claimed that f and f’ attain the same values at (+-1,0),(0,+-1) by claiming that these are the smallest values non-zero represented by f and f’ ;
    What if I take two reduced forms (1,0,1) and (1,1,2) – the minimal values (0,a,c,a-/b/+c) for these are (0,1,1,2) and (0,1,2,2) respectively. So two smallest values attained are the same.
    (of course discriminants are different, but I want explanation for the first argument )

  2. theoremoftheweek Says:

    The only way to get a using coprime variables in your first example is by using (\pm 1, 0) or (0, \pm 1) (because we know that if both variables are non-zero then we get at least a - |b| + c > a). This gives four pairs of variables that all give a (which is, if you like, the two smallest non-zero values taken by f).

    These four pairs of coprime variables for f must correspond to four pairs of coprime variables for f', via whatever unimodular substitution makes the two forms equivalent.

    Only two of those pairs give f'(x,y) = a' (you might need to think for a moment to convince yourself of this). So the other two must give c', which must therefore be the same as c.

    Does that clarify matters at all?

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