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== 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 $p^i$th 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, Initial code by Alyson [[attachment:Pries Aly code.sws|Initial code by Alyson (sws)]] Version 2: Anja, Gagan [[attachment:Pries project1 code.sws|version 2 (sws)]] Computing M and a-rank [[attachment:compute Hasse Vitt M.sws|Computing Hasse Vitt M]] Computing N [[attachment:Computing N for Hyperelliptic curve.sws|Computing N]] Corrected, Edited version: [[attachment:Computing N MG Edit(3).sws|Computing N Edit (sws)]] (After fixing indexing problem) See [[http://demo.sagenb.org/home/pub/64/|this published version]]. |
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,
Initial code by Alyson Initial code by Alyson (sws) Version 2: Anja, Gagan version 2 (sws) Computing M and a-rank Computing Hasse Vitt M
Computing N Computing N
Corrected, Edited version: Computing N Edit (sws) (After fixing indexing problem)