# Hasse–Witt matrix

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*. The rank of the Hasse–Witt matrix is the **Hasse ** or **Hasse–Witt invariant**.

## 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*; the *p*-rank is bounded by the rank of *H*, specifically it is the rank of the Frobenius mapping composed with itself *g* times. 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**/*p***Z** in *F*. Therefore the matrix can be written down but does not represent a linear mapping in the straightforward sense.

## Cohomology

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

*H*^{1}(*C*,*O*_{C}),

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

*H*^{0}(*C*, Ω_{C})

where Ω_{C} = Ω^{1}_{C} is the sheaf of Kähler differentials on *C*.

## 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 *p*^{k} points. It may take any value from 0 to *d*, the dimension of *A*; by contrast for any other prime number *l* there are *l*^{2d} 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 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.^{[1]} 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 *f*^{p−1} is zero.^{[1]}

## Notes

- 1 2 Hartshorne, Robin (1977).
*Algebraic Geometry*. Graduate Texts in Mathematics.**52**. Springer-Verlag. p. 332. ISBN 0-387-90244-9. MR 0463157. Zbl 0367.14001.

## References

- Hasse, Helmut (1934). "Existenz separabler zyklischer unverzweigter Erweiterungskörper vom Primzahlgrad
*p*über elliptischen Funktionenkörpern der Charakteristik*p*".*Journal f. d. reine u. angew. Math.***172**: 77–85. doi:10.1515/crll.1935.172.77. JFM 60.0910.02. Zbl 0010.14803. - Hasse, Helmut; Witt, Ernst (1936). "Zyklische unverzweigte Erweiterungskörper vom Primzahlgrad
*p*über einem algebraischen Funktionenkörper der Charakteristik*p*".*Monatshefte f. Math. und Phys*.**43**: 477–492. doi:10.1515/9783110835007.202. JFM 62.0112.01. Zbl 0013.34102. - Manin, Ju. I. (1965). "The Hasse–Witt matrix of an algebraic curve".
*Transl., Ser. 2, Am. Math. Soc*.**45**: 245–246. ISSN 0065-9290. Zbl 0148.28002. (English translation of a Russian original)