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Unformatted text preview: 8430 HANDOUT 4: IDENTIFICATION OF C ( D ) AND Pic( O ( D )) PETE L. CLARK 1. Statement of the Fundamental Bijection Let D be a negative quadratic discriminant. If f = ax 2 + bxy + c 2 is primitive posi- tive definite of discriminant D , then it is easy to check that I ( f ) := Z a + Z (- b + √ D 2 ) is an integral ideal of O ( D ). (The formula consolidates the cases D ≡ 0 (mod 4) and D ≡ 1 (mod 4); in the former case, O ( D ) = Z [ √ D 2 ] and in the latter case O ( D ) = Z [ 1+ √ D 2 ]. For an ideal I of O ( D ), we will denote its image in the ideal class monoid by [ I ]. In particular, composing the map f 7→ I ( f ) with the map which sends an ideal to its class, we get a map which sends a primitive, positive definite quadratic form f of discriminant D to the ideal class [ I ( f )]. 1 This has remarkable proper- ties. Indeed, we have the following important and beautiful result (unlike some of the other results presented in this course, this exploits particular properties of imaginary quadratic fields): Theorem 1. (Fundamental Bijection) Consider the function which maps a prim- itive, positive definite quadratic form f = ax 2 + bxy + cy 2 to the O ( D )-ideal I ( f ) = Z a + Z (- b + √ D 2 ) . a) The ideal I ( f ) is invertible. b) If f and f are properly equivalent, I ( f ) and I ( f ) have equal images in Pic( O ( D )) , thus I descends to a map [ I ] : C ( D ) → Pic( O ( D )) . The map [ I ] is a bijection . c) We have [ I ( ι ( f ))] = [ I ( ax 2- bxy + cy 2 )] = [ I ( f )] = [ I ( f )]- 1 . d) f represents m ∈ Z iff there exists some ideal J ∈ [ I ( f )] such that N ( J ) = m . Let us stop and record some immediate consequences. Part c) tells us that the subset of ambiguous classes corresponds precisely to the 2-torsion subgroup Pic( O ( D )), hence explains why the number of ambiguous classes is always a power of 2. Part d) gives further perspective on our problem: we saw first that if m is an integer prime to D such that D is a square modulo m , then there exists an ideal J of O ( D ) of norm m , and that the principal form q D represents m iff J is principal. Now we see the other part of this: if the ideal is nonprincipal, then m is represented instead by some non-properly equivalent form f . 1 In the first draft of the notes I wrote the ideal class as I ( f ), and got all the way into the second lecture before realizing that, since in part c) of the Theorem below we want to take complex conjugates of ideals, this is unacceptably poor notation. I have tried to change all the overbars to [ ]’s except for the ones that really are complex conjugation. Please let me know if I missed any! 1 2 PETE L. CLARK Moreover, if we further assume that m = p is prime, we know that there exists exactly two ideals p and p of norm p . Since for any inverible ideal J in O ( D ) we have J J = ( N ( J )), this means that in the Picard group [ J ][ J ] = 1, i.e., the complex conjugate of J represents the inverse ideal class. We conclude:represents the inverse ideal class....
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This note was uploaded on 12/03/2011 for the course MATH 8430 taught by Professor Clark during the Fall '11 term at UGA.
- Fall '11