A Sage toolbox for computations with Models of Curves over Local Fields
Switch branches/tags
Clone or download
swewers Merge pull request #104 from swewers/fix_urgent_bug
This merge fixes several bugs and started a reorganization of `reduction_trees`

The key commit is 8844501:

In this new version of reduction_trees, 
the maps between different components
(inertial, lower and upper) are constructed
in a systematic way and such that they are 
compatible with the maps of generic fibers.

All components are created as curves over an
*absolute* finite fields, thus fixing issue #103
and adressing in part issue #66. This is done
by using the helper functions and  
`make_finite_fields` which turns a finite field into an absolute fielte, 
and `make_function_fields`
which turns a function field into a 'true' function
field over an absolute finite field. Once `trac:26103` is merged,
these functions can probably be removed. 

I think a further reorganization of `reduction_trees` and, related
to this, of `smooth_projective_curves` and `morphisms_of_smooth_projective_curves`
is necessary, but this will do for a while.
Latest commit 54591ab Aug 24, 2018

README.md

MCLF

Documentation Status CircleCI Coverage Status asv PyPI

A Sage toolbox for computing with Models of Curves over Local Fields

This is still a rather immature version of our toolbox. Nevertheless, you can use it to compute, for a large class of curves over the rationals, the stable reduction at primes of bad reduction.

Let Y be a smooth projective curve over a field K and let vK be a discrete valuation on K. The principal goal is to compute the semistable reduction of Y with respect to vK. This means that we want to know

  • a finite Galois extension L/K,
  • an extension vL of vK to L,
  • the special fiber of an integral semistable model of Y over the valuation ring of vL, and
  • the action of the decomposition group of vL on that special fiber.

At the moment we can do this only in certain special cases, which should nevertheless be useful.

If you have at least Sage 8.2 you can install the latest version of this package with sage -pip install --user --upgrade mclf.

If you can not install Sage on your local machine, you can also click Launch on mybinder.org to run an interactive Jupyter notebook with mclf preinstalled.

The package can be loaded with

sage: from mclf import *

We create a Picard curve over the rational number field.

sage: R.<x> = QQ[]
sage: Y = SuperellipticCurve(x^4-1, 3)
sage: Y
superelliptic curve y^3 = x^4 - 1 over Rational Field

In general, the class SuperellipticCurve allows you to create a superelliptic curve of the form yn = f(x), for a polynomial f over an arbitrary field K. But you can also define any smooth projective curve Y with given function field.

We define the 2-adic valuation on the rational field. Then we are able to create an object of the class SemistableModel which represents a semistable model of the curve Y with respect to the 2-adic valuation.

sage: v_2 = QQ.valuation(2)
sage: Y2 = SemistableModel(Y, v_2)
sage: Y2.is_semistable() # this may take a while
True

The stable reduction of Y at p=2 has four components, one of genus 0 and three of genus 1.

sage: [Z.genus() for Z in Y2.components()]
[0, 1, 1, 1]
sage: Y2.components_of_positive_genus()
[the smooth projective curve with Function field in y defined by y^3 + x^4 + x^2,
 the smooth projective curve with Function field in y defined by y^3 + x^2 + x,
 the smooth projective curve with Function field in y defined by y^3 + x^2 + x + 1]

We can also extract some arithmetic information on the curve Y from the stable reduction. For instance, we can compute the conductor exponent of Y at p=2:

sage: Y2.conductor_exponent()
6

Now let us compute the semistable reduction of Y at p=3:

sage: v_3 = QQ.valuation(3)
sage: Y3 = SemistableModel(Y, v_3)
sage: Y3.is_semistable()
True
sage: Y3.components_of_positive_genus()
[the smooth projective curve with Function field in y defined by y^3 + y + 2*x^4]

We see that Y has potentially good reduction at p=3. The conductor exponent is:

sage: Y3.conductor_exponent()
6

For more details on the functionality and the restrictions of the toolbox, see the Documentation. For the mathematical background see

Known bugs and issues

See our issues list, and tell us of any bugs or ommissions that are not covered there.

Experimental Changes

We also have an unstable experimental version with the latest experimental features and bugs that you can try out by clicking on Launch on mybinder.org, note that this version currently CircleCI our own test suite.

Development workflow

Most development happens on feature branches against the master branch. The master branch is considered stable and usually we create a new release and upload it to PyPI whenever there is something merged into master. We sometimes collect a number of experimental changes on the experimental branch.