Hatcher-Topology-of-Numbers · Problem 7.2.5
The Class Group for Quadratic Forms
Mathematics — Algebraic Topology · symmetric-forms-and-products-of-primes
Problem
For a fixed discriminant consider the set of primes that do not divide the conductor and are represented by primitive forms with mirror symmetry. Show that numbers that are products of primes in are represented by at most one form of discriminant , up to equivalence, and this form has mirror symmetry.Intuitive explanation
The key insight is that mirror symmetry in binary quadratic forms is preserved under multiplication of represented integers, and the conductor's role is to eliminate "ramified" primes that would create ambiguity. When you multiply two numbers each represented by a symmetric form of discriminant , their product is represented by a form with the same symmetry property—and the restriction to primes outside the conductor ensures this representation is unique up to equivalence. Think of it as a rigidity principle: once you fix the symmetry constraint and exclude problematic primes, the form representing a product is forced to be the same form that represents its factors.
Intuitive explanation
The key insight is that mirror symmetry in binary quadratic forms is preserved under multiplication of represented integers, and the conductor's role is to eliminate "ramified" primes that would create ambiguity. When you multiply two numbers each represented by a symmetric form of discriminant , their product is represented by a form with the same symmetry property—and the restriction to primes outside the conductor ensures this representation is unique up to equivalence. Think of it as a rigidity principle: once you fix the symmetry constraint and exclude problematic primes, the form representing a product is forced to be the same form that represents its factors.