The reason that the p-rank is lower is that multiplication by p on A is an inseparable isogeny: the differential is p which is 0 in K. By looking at the kernel as a group scheme one can get the more complete structure (reference David Mumford Abelian Varieties pp. 146–7); but if for example one looks at reduction mod p of a division equation, the number of solutions must drop. The rank of the Cartier–Manin operator, or Hasse–Witt matrix, therefore gives an upper bound for the p-rank. The p-rank is the rank of the Frobenius operator composed with itself g times. In the original paper of Hasse and Witt the problem is phrased in terms intrinsic to C, not relying on J. It is there a question of classifying the possible Artin–Schreier extensions of the function field F(C) (the analogue in this case of Kummer theory). Case of genus 1 The case of elliptic curves was worked out by Hasse in 1934. Since the genus is 1, the only possibilities for the matrix H are: H is zero, Hasse invariant 0, p-rank 0, the supersingular case; or H non-zero, Hasse invariant 1, p-rank 1, the ordinary case. Here there is a congruence formula saying that H is congruent modulo p to the number N of points on C over F, at least when q = p. Because of Hasse's theorem on elliptic curves, knowing N modulo p determines N for p ≥ 5. This connection with local zeta-functions has been investigated in depth. For a plane curve defined by a cubic f(X,Y,Z) = 0, the Hasse invariant is zero if and only if the coefficient of (XYZ)p−1 in fp−1 is zero. Notes References (English translation of a Russian original) Algebraic curves Finite fields Matrices Complex manifolds