Eisenstein Discriminant Representations

Abstract

Forms in V at discriminant 4k are in bijection with the Eisenstein representations of k.

Theorem 1.1 (Forms at discriminant 4k biject with Eisenstein representations).

$$\left\{(A,B,C)\in {\mathbb{Z}}^3\mid B=-2(A+C),B^2-4AC=4k\right\}\to \left\{(A,C)\in {\mathbb{Z}}^2\mid A^2+AC+C^2=k\right\},(A,B,C)\mapsto (A,C)\text{ is bijective}.$$

Proof. Machine-checked in Lean as D5/S3/PrimeForms/EisensteinDiscriminant.forms_biject_eisenstein_representations (✓ std3). ∎

Source. Repository-derived.

Commentary.

For each integer k, a binary quadratic form with coefficients A, B, C lies in V when B = -2(A + C). Under that constraint its discriminant equals 4k exactly when A^2 + AC + C^2 = k. The coefficient projection sending the form to (A, C) is bijective, with inverse (A, C) |-> (A, -2(A + C), C). Thus the V-form incidence total and the Eisenstein representation number are identified by an explicit bijection, rather than only by a numerical equality.

References

• Truth anchor: D5/S3/PrimeForms/EisensteinDiscriminant.forms_biject_eisenstein_representations