days26/Pries Project

Title: Computation of p-torsion of Jacobians of hyperelliptic curves

Abstract: An elliptic curve defined over a finite field of characteristic p can be ordinary or supersingular; this distinction measures certain properties of its p-torsion. The p-torsion of the Jacobian of a curve of higher genus can also be studied and classified by interesting combinatorial invariants, such as the p-rank, a-number, and Ekedahl-Oort type. Algorithms to compute these invariants exist but have not been implemented. In this talk, I will explain how to compute these invariants and describe the lag in producing explicit curves with given p-torsion invariants.

Project

Fq, q=pa, then E/Fq can be ordinary or supersingular. Some ways to determine this implemented in Sage: ap, newton_slopes of Frobenius_polynomial, Hasse_invariant.

Suppose C/Fq is a curve of genus g. The easiest type of curve to look at are hyperelliptic curves y2=f(x) where f(x) has degree 2g+1. The p-torsion of its Jacobian has invariants generalizing the ordinary/supersingular distinction. These are called p-rank, a-number, Ekedahl-Oort type, etc. Its Jacobian also has a Newton polygon (the length of slope 0 portion equals the p-rank). The Newton polygon has been implemented for hyperelliptic curves in Sage for large p. The easiest type of curve to look at is y2=f(x) where f(x) has degree 2g+1.

To compute some of these: set up y2=f(x), raise f(x)(p−1)2=∑cixi. Create the (g×g) matrix M=(cp*i−j) (the ijth entry is the coefficient of x^{pi-j}). Look at the g by g matrix,

M(pi)=(cpip*i−j)

(take the pith power of each coefficient and create N=MM(p)M(p2)...M(pg−1).

The matrix M is the matrix for the Cartier operator on the 1-forms. The p-rank is the rank of N. The a-number equals g-rank(M).

For the Ekedahl-Oort type you need the action of F and V on the deRham cohomology (more difficult).

Test cases: y2=xp−x (p-rank 0, and (if I remember correctly) a-number (p−1)/2).

Some questions: for genus 4 (or higher), and given prime - is there a curve of p-rank 0 and a-number 1.

I will describe more motivation and questions on Thursday.

References: Yui, Voloch,

Possible reference http://www.math.colostate.edu/~pries/Preprints/00DecPreprints/08groupschemeconm1007.pdf

Cartier matrix and Hasse-Witt Matrix(this version uses caching): Best and most up to date (12/14)

Code for Documentation Reference: Doc Ref

Alternative exponentiation f^((p-1)/2). So far not faster. alternative exponentiation of f.sws Intermediate worksheet intermediate worksheet for exponentiation.sws

See this published version.