|
⇤ ← Revision 1 as of 2010-11-30 19:33:03
Size: 655
Comment:
|
Size: 1462
Comment:
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 4: | Line 4: |
== Project == $\mathbb{F}_q$, $q = p^a$, then $E/\mathbb{F}_q$ can be ordinary or supersingular. Some invariants, $a_p$, Frobenius polynomial, Newton slopes, Hasse invariants. Suppose $C/\mathbb{F}_q$ is a curve of genus g. It has invariants, p-rank, a-number, Ekedahl-Oort type, newton polygon. The easiest type of curve to look at is $y^2 = f(x)$ where $f(x)$ has degree $2g+1$. Only the Newton polygon has been implemented in Sage (for primes large enough). To compute some of these ?? which ??, set up $y^2 = f(x)$, raise $f(x)^{(p-1)}{2} = \sum c_i x^i$. Create the $(g\times g)$ matrix $M = (c_{p*i-j})$. For a hyper elliptic curve, $M^{(p)} = (c_{p*i-j}^p)$ and create $N = M M^{(p)} M^{(p^2)} ... M^{(p^{g-1})}$. For the Ekedahl-Oort type you need the deRham operator. |
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 invariants, a_p, Frobenius polynomial, Newton slopes, Hasse invariants.
Suppose C/\mathbb{F}_q is a curve of genus g. It has invariants, p-rank, a-number, Ekedahl-Oort type, newton polygon. The easiest type of curve to look at is y^2 = f(x) where f(x) has degree 2g+1. Only the Newton polygon has been implemented in Sage (for primes large enough).
To compute some of these ?? which ??, 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}). For a hyper elliptic curve, M^{(p)} = (c_{p*i-j}^p) and create N = M M^{(p)} M^{(p^2)} ... M^{(p^{g-1})}.
For the Ekedahl-Oort type you need the deRham operator.