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 and are equivalent binary quadratic forms, then . 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 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 or ? 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.
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 is congruent to or modulo , is there necessarily a binary quadratic form with discriminant ?
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?