Hasse-Witt matrix
Encyclopedia
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.
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/pZ in F. Therefore the matrix can be written down but does not represent a linear mapping in the straightforward sense.
is this: the p-power map acts on
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
where ΩC = Ω1C is the sheaf of Kähler differential
s on C.
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 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
).
s 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-function
s has been investigated in depth. (Supersingular curves are related to supersingular prime
s, but by means of modular curve
s and particular points on them.)
to a product Eg where E is a supersingular elliptic curve.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Hasse–Witt matrix H of a non-singular algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
C over a finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
F is the matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
of the Frobenius mapping (p-th power mapping where F has q elements, q a power of the prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
p) with respect to a basis for the differentials of the first kind. It is a g × g matrix where C has genus
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
g.
Approach to the definition
This definition, as given in the introduction, is natural in classical terms, and is due to Helmut HasseHelmut Hasse
Helmut Hasse was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of p-adic numbers to local classfield theory and diophantine geometry , and to local zeta functions.-Life:He was born in Kassel, and died in...
and Ernst Witt
Ernst Witt
Ernst Witt was a German mathematician born on the island of Als . Shortly after his birth, he and his parents moved to China, and he did not return to Europe until he was nine....
(1936). It provides a solution to the question of the p-rank of the Jacobian variety
Jacobian variety
In mathematics, the Jacobian variety J of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles...
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
Cryptography
Cryptography is the practice and study of techniques for secure communication in the presence of third parties...
, 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
Arithmetic and geometric Frobenius
In mathematics, the Frobenius endomorphism is defined in any commutative ring R that has characteristic p, where p is a prime number. Namely, the mapping φ that takes r in R to rp is a ring endomorphism of R....
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.
Cohomology
The interpretation for sheaf cohomologySheaf cohomology
In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...
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
Pierre Cartier (mathematician)
Pierre Cartier is a mathematician. An associate of the Bourbaki group and at one time a colleague of Alexander Grothendieck, his interests have ranged over algebraic geometry, representation theory, mathematical physics, and category theory....
and Yuri Manin. The connection with the Hasse–Witt definition is by means of Serre duality
Serre duality
In algebraic geometry, a branch of mathematics, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n . It shows that a cohomology group Hi is the dual space of another one, Hn−i...
, which for a curve relates that group to
- H0(C, ΩC)
where ΩC = Ω1C is the sheaf of Kähler differential
Kähler differential
In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes.-Presentation:The idea was introduced by Erich Kähler in the 1930s...
s on C.
Abelian varieties and their p-rank
The p-rank of an abelian varietyAbelian variety
In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
A over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
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
Group scheme
In mathematics, a group scheme is a type of algebro-geometric object equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not...
one can get the more complete structure (reference David Mumford
David Mumford
David Bryant Mumford is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science...
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
Function field
Function field may refer to:*Function field of an algebraic variety*Function field...
F(C) (the analogue in this case of Kummer theory
Kummer theory
In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer around the 1840s in his pioneering work on Fermat's last...
).
Case of genus 1
The case of elliptic curveElliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
s 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-function
Local zeta-function
In number theory, a local zeta-functionis a function whose logarithmic derivative is a generating functionfor the number of solutions of a set of equations defined over a finite field F, in extension fields Fk of F.-Formulation:...
s has been investigated in depth. (Supersingular curves are related to supersingular prime
Supersingular prime
In mathematics, a supersingular prime is a prime number satisfying one of the following concepts:*Supersingular prime , in algebraic number theory*Supersingular prime...
s, but by means of modular curve
Modular curve
In number theory and algebraic geometry, a modular curve Y is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL...
s and particular points on them.)
Abelian variety case
A supersingular abelian variety A of dimension g is, by one definition, an abelian variety isogenousIsogenous
Isogenous is a term occasionally used in spectroscopy that denotes when two or more spectroscopic series of transition from a common state. The word was first used in this context in T.J. Stone & R.F. Barrow Laser excited fluorescence spectra of gaseous Te_2, Canadian Journal of Physics. Vol. 53,...
to a product Eg where E is a supersingular elliptic curve.