Diff for "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

\mathbb{F}_q, q = p^a, then E/\mathbb{F}_q can be ordinary or supersingular. Some ways to determine this implemented in Sage: a_p, newton_slopes of Frobenius_polynomial, Hasse_invariant.

Suppose C/\mathbb{F}_q is a curve of genus g. The easiest type of curve to look at are hyperelliptic curves y^2=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 y^2 = f(x) where f(x) has degree 2g+1.

To compute some of these: set up y^2 = f(x), raise f(x)^{(p-1)}{2} = \sum c_i x^i. Create the (g\times g) matrix M = (c_{p*i-j}) (the ijth entry is the coefficient of x^{pi-j}). Look at the g by g matrix,

M^{(p^i)} = (c_{p*i-j}^{p^i})

(take the p^ith power of each coefficient and create N = M M^{(p)} M^{(p^2)} ... M^{(p^{g-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: y^2=x^p-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/07g3smallphyper907.pdf

Computing M and a-rank Computing Hasse Vitt M

Corrected, Edited version: Computing N Edit (sws) (After fixing indexing problem)

Separate Commands for N and M: Separate N and M (sws)

Slightly Cleaner Code for N, with some time tests: Newest N Code (sws)

First way to speed up the exponentiation.fast exponentiation of f-1.sws

Removed p < 2g-1 test, insert zeros instead Final Sage Code for N - MG (sws)

See this published version.