Number Theory: Lecture 9

In which we prove quadratic reciprocity for the Jacobi symbol, and meet binary quadratic forms.

  • Theorem 28 (Quadratic reciprocity for the Jacobi symbol): Let m and n be odd natural numbers.  Then \genfrac{(}{)}{}{}{m}{n} = (-1)^{\frac{m-1}{2} \frac{n-1}{2}} \genfrac{(}{)}{}{}{n}{m}.  We saw that we could prove this using just the definition of the Jacobi symbol and the law of quadratic reciprocity for the Legendre symbol (and a bit of book-keeping).  So the law of quadratic reciprocity for the Legendre symbol is the really interesting result; the version for the Jacobi symbol was a pretty straightforward consequence.
  • Definition of binary quadratic forms.
  • Definition of unimodular substitutions, and equivalence of forms.
  • Exercise: Check that equivalence of forms is an equivalence relation.
  • Definition of the discriminant of a form.
  • Lemma 29: Equivalent forms have the same discriminant.  We saw that we could prove this by direct calculation, or by using the matrix interpretation.  Is the converse true?

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

You are now well placed to tackle some questions about binary quadratic forms that we shall address in the next lecture or two.

  • If two binary quadratic forms have the same discriminant, must they be equivalent?
  • Which numbers can be the discriminant of a binary quadratic form?
  • We have an equivalence relation on the collection of binary quadratic forms, and so we have equivalence classes of forms.  It would be useful to be able to pick exactly one (hopefully quite nice) representative of each equivalence class.  How might we do that?  Can we somehow ‘reduce’ each form?
  • Our big aim for this section is to be able to decide which numbers are represented by which forms.  You could start (continue?) to try to make some conjectures about this, and of course to try to prove those conjectures.
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3 Responses to “Number Theory: Lecture 9”

  1. wapa Says:

    I just wonder about what we did in today’s lecture when we tried to figure out whether 4x^2+12xy+10y^2=(2x+3y)^2+x^2 and X^2+y^2 represent the same numbers. It seems (at least, to me) like we were trying to find a one-to-one correspondence between (x,y) and (X,Y), but I guess that would not be enough to answer our question. I mean, those two forms (or any two forms in general) might actually represent the same set of numbers even though there is no unimodular substitution relating the two forms together. Or is what we did actually sufficient to justify that the two forms do not represent the same numbers?

    Sorry for quite a long and complicated comment. I might be so confused that I even make my message confusing!

  2. theoremoftheweek Says:

    I gave the example (4,12,10), because it looks moderately complicated, and we can imagine that if we want to know which numbers it represents, then it might be easier to work with a simpler form that represents the same numbers.

    Since (4,12,10)~(2,0,2), we could look at (2,0,2) instead, and that looks potentially simpler.

    I agree that the fact that (4,12,10) is not related to (1,0,1) by a unimodular substitution does not in itself show that they do not represent the same sets of numbers, but it does show that we can’t immediately tackle the problem for (4,12,10) by instead studying (1,0,1).

    Does that clarify things?

  3. wapa Says:

    Yes, it does. Thank you very much.

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