Number Theory: Lecture 9

In which we meet binary quadratic forms.

  • Definition of a binary quadratic form.
  • Definition of a unimodular substitution.
  • Definition of what it means for two binary quadratic forms to be equivalent.
  • Definition of the discriminant of a binary quadratic form.
  • Lemma 29: If f and f' are equivalent binary quadratic forms, then \mathrm{disc}(f) = \mathrm{disc}(f').  Our first proof was a slog using algebra and no thought.  Our second used the matrix representation of binary quadratic forms: then the discriminant is -4 times the determinant.

I gave out the second examples sheet.

Understanding today’s lecture

You could pick your own examples of binary quadratic forms to explore.

How many forms can you find that are equivalent to (4, 12, 9) or (4, 12, 10)?  Which are the ‘simplest’?

Can you find another example of two binary quadratic forms that have the same discriminant but that are not equivalent?

I gave a couple of exercises during the lecture too.

Further reading

Davenport (The Higher Arithmetic) and Baker (A concise introduction to the theory of numbers) both have chapters on binary quadratic forms that cover the material in this section of the course.

Preparation for Lecture 10

If d is congruent to 0 or 1 modulo 4, is there necessarily a binary quadratic form with discriminant d?

Can you find conditions (perhaps involving the coefficients and/or the discriminant) for a form to represent only non-negative numbers, or only non-positive numbers?  Are there any forms that represent both positive and negative numbers?

Can you find any unimodular substitutions that seem particularly helpful for ‘simplifying’ forms?

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One Response to “Number Theory: Lecture 9”

  1. theoremoftheweek Says:

    A student wrote to me to ask why we rule out determinant -1 in our definition of equivalence of binary quadratic forms. Here’s my response, in case it’s puzzling anyone else (and my remark about it during the lecture was insufficient).

    There is no fundamental reason why we should rule out determinant -1, except that it doesn’t make a big difference to the theory and it’s slightly cleaner if we do rule it out. We’re still able to address the question of which numbers can be represented by which forms this way (with determinant -1 not allowed). If you look at books or notes by other people, you’ll find that it’s very standard to focus on determinant +1 only, but it’s not strictly necessary for thinking about equivalence of forms if we just care about being able to invert the matrices.

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