Results on truncated powers and points of curves over prime fields, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Power sums over a prime field — Lemma 2.1. The sum of the k-th powers of the elements of the field with p elements is -1 if k is a positive multiple of p - 1 and 0 otherwise.
• The middle coefficient is the trace — Theorem 3.1. For a polynomial f of degree at most four over the field with p elements, the coefficient of w^(p-1) in f^((p-1)/2) is the residue modulo p of p + 1 minus the number of points of the curve v^2 = f(w), points at infinity included.
• The truncated sum as a sum over points — Theorem 4.1. For a nonzero point of the field at which f does not vanish, the truncated sum equals an explicit term minus the sum of 1/(1 - lambda w(P)) over the affine points of the curve away from that point.
• The truncated sum as a character sum — Corollary 4.2. The same sum equals an explicit term plus a sum of quadratic characters of the values of f with weights 1/(w - mu).
• A congruence for iterated derivatives — Lemma 3.1. In a commutative ring of characteristic p with a derivation D, the p-th term of the recursion P_0 = 1, P_(n+1) = D(P_n) + a P_n is D^(p-1)(a) + a^p.
• Properties of the Cartier operator — Theorem 4.1. On the function field of a curve v^2 = f(w) over a perfect field of odd characteristic, the Cartier operator is additive, takes p-th powers out as their p-th roots, vanishes exactly on exact differentials, and fixes logarithmic differentials.
• The Cartier operator on expansions — Lemma 5.2. In the expansion at the point (0,1) the Cartier operator keeps the coefficients at the exponents pj + p - 1 and takes their p-th roots.
• The Cartier operator on the differentials of the curve — Theorem 6.1. The Cartier operator multiplies dw/v by the p-th root of the coefficient of w^(p-1) in f^((p-1)/2), and its value at (0,1) on dw/((1 - lambda w) v), relative to dw/v, is the p-th root of the truncated sum.
• Functions without poles are constant — Theorem 4.2. On a curve v^2 = f(w) with f of degree three or four without repeated root over an algebraically closed field of odd characteristic, a function that is regular at all points, the points at infinity included, is constant.
• The invariant differential has no zeros and no poles — Proposition 5.2. The differential dw/v has order zero at every point of the curve.
• Differentials without poles — Corollary 5.3. Every differential without poles on the curve is a constant multiple of dw/v.
• Residues of logarithmic differentials — Proposition 5.4. The logarithmic differential of a function has at most simple poles, and its residue at a point is the order of the function there, reduced modulo p.
• Poles of the differential that carries the sums — Lemma 2.1. The differential dw/((1 - lambda w) v) has simple poles at the two points over w = 1/lambda, with residues -r and r, r = 1/(lambda v_0), and no other pole.
• The regular identity — Theorem 3.1. If F has a zero of order M at one point over w = 1/lambda, a pole of order M at the other and no other zero or pole, and Psi is the value at (0,1) of (dF/F)/(dw/v), then M (S_f(lambda) - H) = r (H Psi - Psi^p).
• The Frobenius eigenvalue of the logarithmic derivative — Proposition 4.2. For lambda in the prime field, Psi^p = chi Psi with chi the quadratic character of f(1/lambda); the product r Psi lies in the prime field.
• Evaluation of the truncated sum in the regular case — Theorem 5.1. For lambda in the prime field, M (S_f(lambda) - H) = (H - chi) r Psi; if M is congruent to 1 - chi H modulo p and H is not chi, then S_f(lambda) = H - chi r Psi.

Reading order

Read the lessons in order. The first four have complete proofs; the fourth proves the evaluation formula in its regular case for a given function on the curve, and names what is still missing for the full theorem.