Abstract
On some extensions of Gauss’ work and applications
Highlights
For a negative integer D such that D ≡ 0 or 1, let (D) be the set of primitive positive definite binary quadratic forms Q(x, y) = ax2 + bxy + cy2 ∈ [x, y] of discriminant b2 − 4ac = D
The modular group SL2( ) (or PSL2( )) acts on the set (D) from the right and defines the proper equivalence ∼ as for some γ. In his celebrated work Disquisitiones Arithmeticae of 1801 [1], Gauss introduced the beautiful law of composition of integral binary quadratic forms
Let N be a positive integer, n = N K and P be a subgroup of IK(n) satisfying PK,1(n) ⊆ P ⊆ PK(n)
Summary
The modular group SL2( ) (or PSL2( )) acts on the set (D) from the right and defines the proper equivalence ∼ as In his celebrated work Disquisitiones Arithmeticae of 1801 [1], Gauss introduced the beautiful law of composition of integral binary quadratic forms. Given the order of discriminant D in the imaginary quadratic field K = ( D ), let I( ) be the group of proper fractional -ideals and P( ) be its subgroup of nonzero principal -ideals. We shall develop an algorithm of finding distinct form classes in N(dK)/∼Γ and give a concrete example (Proposition 6.2 and Example 6.3) To this end, we shall apply Shimura’s theory which links the class field theory for imaginary quadratic fields and the theory of modular functions ([7, Chapter 6]).
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
