Inert Eisenstein Quotients and Frobenius
Abstract
Reduction modulo an inert rational prime identifies the Eisenstein quotient with a quadratic finite field, where Frobenius acts by conjugation.
Theorem 1.1 (An explicit quadratic quotient and its Frobenius action).
Proof. Machine-checked in Lean as D5/S3/Factorization/QuadraticIdeals/InertEisensteinFrobenius.inert_eisenstein_quotient_frobenius (✓ std3). ∎
Source. Repository-derived.
Commentary.
Let E be the Eisenstein order Z[omega], with omega squared plus omega plus one equal to zero. Let ell be a rational prime congruent to two modulo three, and let I be the ideal generated by ell. The ideal I is maximal. The quotient Q = E/I has ell squared elements and characteristic ell.
Write K for the quadratic algebra over Z/ellZ with a distinguished element satisfying the same relation as omega. There is a ring equivalence e from Q to K which sends the real and imaginary integer coordinates of every z in E to their residues modulo ell. This specifies the equivalence on all classes represented by Eisenstein integers.
For every z in E, raising its class in Q to the ell-th power gives the class of its quadratic conjugate. This Frobenius identity includes ell equal to two. The statement applies to the actual ideal generated by ell.