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 them Cartier Codes since 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-kn...

in this note we review a simple criterion, due to ekedahl, for superspecial curves defined over finite fields.using this we generalize and give some simple proofs for some well-known superspecial curves.

is an isomorphism. Here F : X −→ X denotes the Frobenius morphism on X and H denotes the a cohomology sheaf of F∗Ω•X . If the variety is not smooth, not much is known about the properties of the Cartier operator and the poor behaviour of the deRham complex in this case makes its study difficult. If one substitutes the deRham complex with the Zariski-deRham complex the situation is better. For e...

We give a lower bound on the rank of the Cartier operator of Jacobian varieties of hyperelliptic and superelliptic curves in terms of their genus.

Throughout this paper Y will denote a complete, nonsingular, and irreducible algebraic curve of genus g > 0 over the algebraically closed field k of characteristic p. Eventually we will assume that g ≥ 2. Denote by W the g-dimensional k-vector space of regular differentials of Y/k, which is just the vector space H(Y,ΩY/k). Also let K(Y ) denote the function field of Y , and let ΩK(Y )/k be the ...

We study the complexity of computing one or several terms (not necessarily consecutive) in a recurrence with polynomial coefficients. As applications, we improve the best currently known upper bounds for factoring integers deterministically, and for computing the Cartier-Manin operator of hyperelliptic curves.

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