The Cartier–Miller evaluation of coefficient sums

Truncated powers of a polynomial of degree three or four over a prime field and the curve v^2 = f(w): the coefficient that gives the trace of the curve, the truncated sum as a sum over the points of the curve and as a character sum, the Cartier operator on the function field of the curve with its values on the two differentials that carry the sums, the points of the curve with orders and residues, functions and differentials without poles, the regular identity that evaluates the sums through a function on the curve, and the statement of D. Jordan's theorem (2026) that evaluates these sums with about log p operations through the arithmetic of the curve.

Four lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Truncated powers and points of a curve
2. The Cartier operator
3. Points, orders and residues
4. The regular identity

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Finite fields, polynomials, derivations and formal power series (core course Abstract Algebra II); the lessons prove the other facts they use.