Hasse-Witt matrix

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In mathematics, the Hasse-Witt matrix H of a non-singular algebraic curve C over a finite field F is the matrix of the Frobenius mapping (p-th power mapping where F has q elements, q a power of the prime number p) with respect to a basis for the differentials of the first kind. It is a g × g matrix where C has genus g.

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[edit] Approach to the definition

This definition, as given in the introduction, is natural in classical terms, and is due to Helmut Hasse and Ernst Witt (1936). It provides a solution to the question of the p-rank of the Jacobian variety J of C; namely, that the p-rank is the same as the rank of H. It is also a definition that is in principle algorithmic. There has been substantial recent interest in this as of practical application to cryptography, in the case of C a hyperelliptic curve. The curve C is superspecial if H = 0.

That definition needs a couple of caveats, at least. Firstly, there is a convention about Frobenius mappings, and under the modern understanding what is required for H is the transpose of Frobenius (see arithmetic and geometric Frobenius for more discussion). Secondly, the Frobenius mapping is not F-linear; it is linear over the prime field Z/pZ in F. Therefore the matrix can be written down but does not represent a linear mapping in the straightforward sense.

[edit] Cohomology

The interpretation for sheaf cohomology is this: the p-power map acts on

H1(C,OC),

or in other words the first cohomology of C with coefficients in its structure sheaf. This is now called the Cartier-Manin operator (sometimes just Cartier operator), for Pierre Cartier and Yuri Manin. The connection with the Hasse-Witt definition is by means of Serre duality, which for a curve relates that group to

H0(C, ΩC)

where ΩC = Ω1C is the sheaf of Kähler differentials on C.

[edit] Abelian varieties and their p-rank

The p-rank of an abelian variety A over a field K of characteristic p is the integer k for which the kernel A[p] of multiplication by p has pk points. It may take any value from 0 to d, the dimension of A; by contrast for any other prime number l there are l2d points in A[l]. 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 a formula for the p-rank. 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).

[edit] Case of genus 1

The case of elliptic curves was worked out by Hasse in 1934. There are two cases: p-rank 0, or supersingular elliptic curve, with H = 0; and p-rank 1, ordinary elliptic curve, with H ≠ 0. 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. (Supersingular curves are related to supersingular primes, but by means of modular curves and particular points on them.)

[edit] References

  • Hasse, Helmut, Existenz separabler zyklischer unverzweigter Erweiterungskörper vom Primzahlgrad p über elliptischen Funktionenkörpern der Charakteristik p, Journal f. d. reine u. angew. Math. 172 (1934), 77-85.
  • Hasse, Helmut & Witt, Ernst, Zyklische unverzweigte Erweiterungskörper vom Primzahlgrad p über einem algebraischen Funktionenkörper der Charakteristik p, Monatshefte f. Math. und Phys. 43 (1936), 477-492
  • Ju. I. Manin. The Hasse-Witt matrix of an algebraic curve. Trans. Amer. Math. Soc., 45:245-246, 1965 (English translation of a Russian original)