Cartier Operator Research Articles

The a-number of yn = xm + x over finite fields

This paper presents a formula for the [Formula: see text]-number of certain maximal curves characterized by the equation [Formula: see text] over the finite field [Formula: see text]. [Formula: see text]-number serves as an invariant for the isomorphism class of the [Formula: see text]-torsion group scheme. By utilizing the action of the Cartier operator on [Formula: see text], we establish a closed formula for the [Formula: see text]-number of [Formula: see text].

Extensions representing Nori-Srinivas obstruction

Let (X,F) be a pair of a smooth variety X over an algebraically closed field k of characteristic p>0 and its Frobenius morphism F. Given a Frobenius Wn(k)-lifting (X¯,F¯) of the pair (X,F) for n≥1, Nori and Srinivas [9] determined the obstruction obsX¯,F¯∈Ext(ΩX/k1,BF⁎ΩX/k1) to Frobenius Wn+1(k)-lifting of (X¯,F¯) in terms of Čech cohomology. The extension representing obsX¯,F¯ has been only known for n=1, which uses the Cartier operator. In this paper, we interpret obsX¯,F¯ in terms of Kato's version of de Rham-Witt Cartier operator [8] and determine the extension representing obsX¯,F¯ for n≥2.

Powers of the Cartier operator on Artin-Schreier covers

Powers of the Cartier operator on Artin-Schreier covers

The rank of the Cartier operator on Picard curves

For an algebraic curve [Formula: see text] defined over an algebraically closed field of characteristic [Formula: see text], [Formula: see text]-number [Formula: see text] is the dimension of the space of exact holomorphic differentials on [Formula: see text]. We computed the [Formula: see text]-number for a family of certain Picard curves using the action of the Cartier operator on [Formula: see text].

Bloch’s cycle complex and coherent dualizing complexes in positive characteristic

Let X X be a separated scheme of dimension d d of finite type over a perfect field k k of positive characteristic p p . In this work, we show that Bloch’s cycle complex Z X c \mathbb {Z}^c_X of zero cycles mod p n p^n is quasi-isomorphic to the Cartier operator fixed part of a certain dualizing complex from coherent duality theory. From this we obtain new vanishing results for the higher Chow groups of zero cycles with mod p n p^n coefficients for singular varieties.

THE RANK OF THE CARTIER OPERATOR ON CERTAIN Fq2-MAXIMAL FUNCTION FIELDS

The a-number is an invariant of the isomorphism class of the p-torsion group scheme. In this paper, we compute a closed formula for the a-number of Xq+1+Yq+1+Zq+1=0 with q=ps over the finite field Fq2, using the action of the Cartier operator on H0(X,Ω1).

Twisted differential operators of negative level and prismatic crystals

We introduce twisted differential calculus of negative level and prove a descent theorem: Frobenius pullback provides an equivalence between finitely presented modules endowed with a topologically quasi-nilpotent twisted connection of level minus one and those of level zero. We explain how this is related to the existence of a Cartier operator on prismatic crystals. For the sake of readability, we limit ourselves to the case of dimension one.

On Vlasenko’s formal group laws

Given a Laurent polynomial over a flat \(\mathbf {Z}\)-algebra, Vlasenko defines a formal group law. We identify this formal group law with a coordinate system of a formal group functor. When the “Hasse–Witt matrix” of the Laurent polynomial is invertible, Vlasenko defines a matrix by taking a certain \(p\)-adic limit. We show that this matrix is the Frobenius of the Dieudonné module of this formal group modulo \(p\).

P-torsion étale sheaves on the Jacobian of a curve

Suppose X is a smooth, proper, geometrically connected curve over Fq with an Fq-rational point x0. For any Fq×-character σ of π1(X) trivial on x0, we construct a functor Lnσ from the derived category of coherent sheaves on the moduli space of deformations of σ over the Witt ring Wn(Fq) to the derived category of constructible Wn(Fq)-sheaves on the Jacobian of X. The functors Lnσ categorify the Artin reciprocity map for geometric class field theory with p-torsion coefficients. We then give a criterion for the fully faithfulness of (an enhanced version of) Lnσ in terms of the Hasse–Witt matrix of X.

A-hypergeometric series and a p-adic refinement of the Hasse-Witt matrix

We identify the p-adic unit roots of the zeta function of a projective hypersurface over a finite field of characteristic p as the eigenvalues of a product of special values of a certain matrix of p-adic series. That matrix is a product \(F(\varLambda ^p)^{-1}F(\varLambda )\), where the entries in the matrix \(F(\varLambda )\) are A-hypergeometric series with integral coefficients and \(F(\varLambda )\) is independent of p.

Hasse–Witt matrices for polynomials, and applications

In a classical paper, Manin gives a congruence [15, Theorem 1] for the characteristic polynomial of the action of Frobenius on the Jacobian of a curve C , defined over the finite field \mathbf{F}_{q} , q=p^m , in terms of its Hasse–Witt matrix. The aim of this article is to prove a congruence similar to Manin’s one, valid for any L -function L(f,T) associated to the exponential sums over affine space attached to an additive character of \mathbf{F}_q , and a polynomial f . In order to do this, we define a Hasse–Witt matrix \mathrm{HW}(f) , which depends on the characteristic p , the set D of exponents of f , and its coefficients. We also give some applications to the study of the Newton polygons of Artin–Schreier (hyperelliptic when p=2 ) curves, and zeta functions of varieties.

Holomorphic differentials of generalized Fermat curves

A non-singular complete irreducible algebraic curve Fk,n, defined over an algebraically closed field K, is called a generalized Fermat curve of type (k,n), where n,k≥2 are integers and k is relatively prime to the characteristic p of K, if it admits a group H≅Zkn of automorphisms such that Fk,n/H is isomorphic to PK1 and it has exactly (n+1) cone points, each one of order k. By the Riemann-Hurwitz-Hasse formula, Fk,n has genus at least one if and only if (k−1)(n−1)>1. In such a situation, we construct a basis, called a standard basis, of its space H1,0(Fk,n) of regular forms, containing a subset of cardinality n+1 that provides an embedding of Fk,n into PKn whose image is the fiber product of (n−1) classical Fermat curves of degree k. For p=2, we obtain a lower bound (which is sharp for n=2,3) for the dimension of the space of exact one-forms, that is, the kernel of the Cartier operator. We also do this for (p,k,n)=(3,2,4).

A-numbers of curves in Artin–Schreier covers

Let $\pi : Y \to X$ be a branched $\mathbf{Z}/p \mathbf{Z}$-cover of smooth, projective, geometrically connected curves over a perfect field of characteristic $p>0$. We investigate the relationship between the $a$-numbers of $Y$ and $X$ and the ramification of the map $\pi$. This is analogous to the relationship between the genus (respectively $p$-rank) of $Y$ and $X$ given the Riemann-Hurwitz (respectively Deuring--Shafarevich) formula. Except in special situations, the $a$-number of $Y$ is not determined by the $a$-number of $X$ and the ramification of the cover, so we instead give bounds on the $a$-number of $Y$. We provide examples showing our bounds are sharp. The bounds come from a detailed analysis of the kernel of the Cartier operator.

A-numbers of curves in Artin–Schreier covers

Let $\pi : Y \to X$ be a branched $\mathbf{Z}/p \mathbf{Z}$-cover of smooth, projective, geometrically connected curves over a perfect field of characteristic $p>0$. We investigate the relationship between the $a$-numbers of $Y$ and $X$ and the ramification of the map $\pi$. This is analogous to the relationship between the genus (respectively $p$-rank) of $Y$ and $X$ given the Riemann-Hurwitz (respectively Deuring--Shafarevich) formula. Except in special situations, the $a$-number of $Y$ is not determined by the $a$-number of $X$ and the ramification of the cover, so we instead give bounds on the $a$-number of $Y$. We provide examples showing our bounds are sharp. The bounds come from a detailed analysis of the kernel of the Cartier operator.

A bound on the genus of a curve with Cartier operator of small rank

Ekedahl showed that the genus of a curve in characteristic p>0 with zero Cartier operator is bounded by p(p-1)/2. We show the bound p+p(p-1)/2 in case the rank of the Cartier operator is 1, improving a result of Re.

A generalization of the Hasse–Witt matrix of a hypersurface

The Hasse–Witt matrix of a hypersurface in Pn over a finite field of characteristic p gives essentially complete mod p information about the zeta function of the hypersurface. But if the degree d of the hypersurface is ≤n, the zeta function is trivial mod p and the Hasse–Witt matrix is zero-by-zero. We generalize a classical formula for the Hasse–Witt matrix to obtain a matrix that gives a nontrivial congruence for the zeta function for all d. We also describe the differential equations satisfied by this matrix and prove that it is generically invertible.

On subadditivity of Kodaira dimension in positive characteristic over a general type base

We show that for a surjective, separable morphism f f of smooth projective varieties over a field of positive characteristic such that f ∗ O X ≅ O Y f_* \mathcal {O}_X \cong \mathcal {O}_Y subadditivity of Kodaira dimension holds, provided the base is of general type and the Hasse-Witt matrix of the geometric general fiber is not nilpotent.

A-hypergeometric series and the Hasse–Witt matrix of a hypersurface

We give a short combinatorial proof of the generic invertibility of the Hasse–Witt matrix of a projective hypersurface. We also examine the relationship between the Hasse–Witt matrix and certain A-hypergeometric series, which is what motivated the proof.

Stratified order one differential equations in positive characteristic

Stratification for nonlinear differential equations in positive characteristic is introduced. Testing this notion for first order equations is discussed, and related to the Cartier operator on curves. A variant of the Grothendieck–Katz conjecture is formulated, and proved in a special case.

Codes and the Cartier operator

In this article, we present a new construction of codes from algebraic curves. Given a curve over a non-prime finite field, the obtained codes are defined over a subfield. We call themCartier codessince their construction involves the Cartier operator. This new class of codes can be regarded as a natural geometric generalisation of classical Goppa codes. In particular, we prove that a well-known property satisfied by classical Goppa codes extends naturally to Cartier codes. We prove general lower bounds for the dimension and the minimum distance of these codes and compare our construction with a classical one: the subfield subcodes of Algebraic Geometry codes. We prove that every Cartier code is contained in a subfield subcode of an Algebraic Geometry code and that the two constructions have similar asymptotic performances.We also show that some known results on subfield subcodes of Algebraic Geometry codes can be proved nicely by using properties of the Cartier operator and that some known bounds on the dimension of subfield subcodes of Algebraic Geometry codes can be improved thanks to Cartier codes and the Cartier operator.