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| 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 $2g+1$. | 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 $2g+1$. |
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| 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 $N = M M^{(p)} M^{(p^2)} ... M^{(p^{g-1})}$. |
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 $N = M M^{(p)} M^{(p^2)} ... M^{(p^{g-1})}$. |
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| Test cases: y^2=x^p-x (p-rank 0, and (if I remember correctly) a-number (p-1)/2. | Test cases: $y^2=x^p-x$ (p-rank 0, and (if I remember correctly) a-number $(p-1)/2$). |
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| References: Yui, Voloch, | References: [[attachment:yui-on_the_jacobian_of_hyperelliptic_curves_over_fields_of_characteristic_p_gt_2.pdf|Yui]], Voloch, Possible reference http://www.math.colostate.edu/~pries/Preprints/00DecPreprints/07g3smallphyper907.pdf Computing M and a-rank [[attachment:compute Hasse Vitt M.sws|Computing Hasse Vitt M]] Corrected, Edited version: [[attachment:Computing N MG Edit(3).sws|Computing N Edit (sws)]] (After fixing indexing problem) Separate Commands for N and M: [[attachment:Separate N and M.sws|Separate N and M (sws)]] Slightly Cleaner Code for N, with some time tests: [[attachment:Computing N - Time Tests for Large p.sws|Newest N Code (sws)]] First way to speed up the exponentiation. Buggy!!! [[attachment:fast exponentiation of f.sws]] Removed p < 2g-1 test, insert zeros instead [[attachment:Fixed Indexing.sws|Fixed Indexing (sws)]] 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,
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. Buggy!!! fast exponentiation of f.sws
Removed p < 2g-1 test, insert zeros instead Fixed Indexing (sws)