This is a very preliminary version. At the moment it is only possible to compute the semistable reduction of superelliptic curves of degree p over Q_p and the exponent of conductor at p, under some restrictions (see an example below).
Everything is based on Julian Rüth's Sage package mac_lane, which allows us to work with quite general valuations. See
https://github.com/saraedum/mac_lane
The package can be loaded with
from mclf import *
However, the mac_lane package needs to be available -- at the moment it is necessary to
clone the branch modified available under https://github.com/swewers/mac_lane.
Hopefully, there will soon be an integrated version of mac_lane in Sage 8.1.
Let us do an example: a Picard curve over the rational number field, relative to the 3-adic valuation.
sage: K = QQ
sage: v_3 = pAdicValuation(K, 3)
sage: R.<x> = K[]
sage: Y = Superp(x^4+1, v_3, 3)
sage: Y
superelliptic curve Y: y^3 = x^4 + 1 over Rational Field, with respect to 3-adic valuation
In general, the class Superp allows you to create a superelliptic curve of the form y^p = f(x),
for a polynomial f over a number field K and a prime p. We also have to specify a discrete valuation on K with
residue characteristic p. The class should then provide access to a variety of functions which allows you to compute
the semistable reduction of the curve and analyze it. For instance, it is possible to compute the
conductor exponent at the chosen p-adic valuation.
The mathematical background will soon appear in
S. Wewers, Semistable reduction of superelliptic curves of degree p, preprint, 2017
A rough sketch and some examples can already be found in
Semistable reduction of curves and computation of bad Euler factors of L-functions, S. Wewers and I.I. Bouw, lecture notes for a minicourse at ICERM
For the time being, our implementation only works under certain restrictions on the superelliptic curve Y: y^p=f(x)
- the base field has to be the field of rational numbers
- the degree of the polynomial f has to prime to p (we can always bring any superelliptic curve of degree p into this form as long as there is at least one rational branch point),
- the Jacobian of Y has potentially good reduction at p.
These restrictions should disappear soon.
In the explicit example from above, we get:
sage: Y.compute_semistable_reduction()
We try to compute the semistable reduction of the
superelliptic curve Y: y^3 = x^4 + 1 over Rational Field, with respect to 3-adic valuation
which has genus 3
First we compute the etale locus:
Affinoid with 1 components:
Elementary affinoid defined by
v(x) >= 3/8
There is exactly one reduction component to consider:
Reduction component corresponding to
Elementary affinoid defined by
v(x) >= 3/8
It splits over 3-adic completion of Number Field in pi8 with defining polynomial x^8 - 3
into 1 lower components.
The upper components are:
The smooth projective curve over Finite Field of size 3 with Function field in y defined by y^3 + y + 2*x^4.
Contribution of this component to the reduction genus is 3
The curve has abelian reduction, since the total reduction genus
is equal to the genus of the generic fiber.
This result shows that the curve Y has potentially good reduction over a tame extension of the 3-adic numbers of ramification index 8. The reduction is the
curve over the finite field with 3 elements, given by the equation y^3 + y +2x^4 = 0.
Now we can also compute the conductor exponent at p=3:
sage: Y.conductor_exponent()
6
Updates with more functionality should appear regularly.