reduced integral binary quadratic forms
A positive binary form is one where .
If is positive, then . If and either or , then is positive.
If is positive, then , so .
If , then . Thus if , then . The proof for the case is identical. ∎
It turns out that each proper equivalence class of primitive positive forms contains a single reduced form, and thus we can understand how many classes there are of a given discriminant by studying only the reduced forms.
If is a primitive positive reduced form with discriminant , , then the minimum value assumed by if are not both zero is . If , then this value is assumed only for ; if ; it is assumed for and .
Since , it follows that
Thus, whenever , while if or is zero, then . So is clearly the smallest nonzero value of .
If , then if , and for , so achieves its minimum only at .
If , then since otherwise is not reduced (we cannot have else we have a form of discriminant ). Thus again and thus if , so in this case the result follows as well. ∎
Note that the reduced form of discriminant , , also achieves its minimum value at .
If are primitive positive reduced forms with , then .
We take the cases and separately.
First assume so we can apply the above theorem.
Since , we can write with . Suppose . Now, and have the same minimum value, so .
If , then achieves its minimum only at , so and thus . So and thus . Since is also reduced, and thus and .
If instead , then instead of concluding that we can only conclude that or . If , the argument carries through as above. If , then , so and thus . Thus . But then since the discriminants are equal, and thus both . So and we are done.
Finally, in the case , we see that for any such reduced form, , so . Thus since otherwise the form is not reduced. So the only reduced form of discriminant is in fact . ∎
Every primitive positive form is properly equivalent to a unique reduced form.
We just proved uniqueness, so we must show existence. Note that I used a different method of proof in class that relied on “infinite descent” to get the result in the first paragraph below; the method here is just as good but provides less insight into how to actually reduce a form.
We first show that any such form is properly equivalent to some form satisfying . Among all forms properly equivalent to the given one, choose such that is as small as possible (there may be multiple such forms; choose one of them). If , then
is properly equivalent to , and we can choose so that , contradicting our choice of minimal . So ; similarly, . Finally, if , simply interchange and (by applying the proper equivalence ) to get the required form.
To finish the proof, we show that if , where , then is properly equivalent to a reduced form. The form is already reduced unless and either or . But in these cases, the form is reduced, so it suffices to show that and are properly equivalent. If , then takes to , while if , then takes to . ∎
Let’s see how to reduce to :
If is a positive reduced form with , then .
since the form is reduced. So , and the result follows. ∎
If , then is the number of classes of primitive positive forms of discriminant .
If , then is finite, and is equal to the number of primitive positive reduced forms of discriminant .
Essentially obvious. Since every positive form is properly equivalent to a (unique) reduced form, is clearly equal to the number of positive reduced forms of discriminant . But given a reduced form of discriminant , there are only finitely many choices for , by the proposition. This constrains us to finitely many choices for , since . and determine since is fixed. ∎
Examples: : even, . So , corresponding to , is the only reduced form of discriminant . Note that this provides another proof that primes are representable as the sum of two squares, since all such primes have and thus are representable by this quadratic form.
: odd, . So . This gives us
|not reduced since ; properly equivalent to via|
There are three equivalence classes of positive reduced forms with discriminant .
: , so is odd, . So . So the forms are
|not reduced since , properly equivalent to via|
|not reduced since , equivalent to via|
There are four classes of forms of discriminant .
: odd, . So , and , so . Since , we must have ; thus and thus we get only . But is properly equivalent to via , so there is only one equivalence class of positive reduced forms with discriminant .
|Title||reduced integral binary quadratic forms|
|Date of creation||2013-03-22 19:18:52|
|Last modified on||2013-03-22 19:18:52|
|Last modified by||rm50 (10146)|