CoCalc Public Filesmain_project / semistable_reduction / picard / Picard_example1.sagews
Authors: Jeroen Sijsling, Stefan Wewers
Views : 33
Description: Computation of conductor exponent for some Picard curves
Compute Environment: Ubuntu 18.04 (Deprecated)

Computing the conductor of Picard curves

1. Example

We consider the Picard curve $Y:\; y^3 = f(x) = x^4 + 14x^2 + 72x -41.$ The discriminant of $f$ is $\Delta(f)=-2^{10}3^45^6$.

Our goal is to compute the conductor $N_Y = 2^{f_2}3^{f_3}5^{f_5}$.

K = QQ
R.<x> = K[]
f = x^4 +14*x^2 + 72*x -41
f.discriminant().factor()

-1 * 2^10 * 3^4 * 5^6

We check that $p=5$ is an exceptional prime. It follows that $f_5=0$. (Note that $v_5(\Delta(f))=6$.)

f.change_ring(GF(5)).factor()
(f(-58+5^3*x)/5^6).change_ring(GF(5))

(x + 3)^2 * (x^2 + 4*x + 1) 3*x^2 + 4*x + 2
vK = pAdicValuation(K, 5)
vK.mac_lane_approximants(f)

[[ Gauss valuation induced by 5-adic valuation, v(x + 58) = 3, v(x^2 + 491*x + 87614) = 7 ], [ Gauss valuation induced by 5-adic valuation, v(x^2 + 134*x + 356) = 4 ]]

Now we want to compute $f_2$.

First we compute the splitting field of $f$. It is a totally ramified extension of $\QQ$ of degree $8$. The configuration of the roots of $f$ is as follows: they all lie in a disk of radius $3/4$, and inside in two smaller disks of radius $1$.

f.change_ring(GF(2)).factor()
vK.montes_factorization(f)      # shows that f is irreducible over QQ_2
L0.<alpha0> = K.extension(f)
L0.<alpha>= L0.galois_closure()
L0
vL0 = vK.extension(L0)
vL0(2)
fL0 = f.change_ring(L0)
a = fL0.roots(multiplicities=None)
[vL0(a[0]-a[i]) for i in range(4)]

(x + 1)^4 x^4 + 14*x^2 + 72*x - 41 Number Field in alpha with defining polynomial x^8 + 40*x^6 + 2400*x^4 + 81472*x^2 + 1000000 8 [+Infinity, 6, 6, 8]

We compute the MacLane valuations $v_1,v_2$ on $K[x]$ corresponding to the configuration of roots.



v0 = GaussValuation(R, vK)
v = v0.mac_lane_step(f)[0]
v1 = v.extension(f, 3)
v1
v2 = v.extension(f, 7/2)
v2

[ Gauss valuation induced by 2-adic valuation, v(x + 1) = 1/4, v(x^4 + 14*x^2 + 72*x - 41) = 3 ] [ Gauss valuation induced by 2-adic valuation, v(x + 1) = 1/4, v(x^4 + 14*x^2 + 72*x - 41) = 7/2 ]

From the configuration of roots of $f$ it is clear that the stable reduction of $Y$ has three components of genus $1$ (Case (c) in Theorem 4.1). The extension $L/K$ over which $Y$ aquires semistable reduction is a (tamely ramified) extension of $L_0$ of degree $3$.

v0 = GaussValuation(fL0.parent(), vL0)
v0(f((x-a[0])/(a[1]-a[0])))
v0(f((x-a[0])/(a[3]-a[0])))   # since this is not divisible by 3, we need L/L_0 to be ramified

-24 -32



We compute the jumps in the filtration of higher ramification groups of the extension $L$. It turns out that these are $0,3,9,15$. Here $\Gamma_3$ is a dihedral group of order $8$, $\Gamma_9$ is of order $4$, and $\Gamma_{15}$ is the center (of order $2$).

We also compute the subfields of $L$ corresponding to the jumps.

load("../../padic_extensions/padic_extensions.sage")
L0 = vL0.domain()
L0
P = L0.polynomial()
P = P(P.parent().gen()^3)
P

Number Field in pi8 with defining polynomial x^8 + 20*x^7 + 8*x^6 + 72*x^5 + 70*x^4 + 48*x^3 + 76*x^2 + 16*x + 82 x^24 + 20*x^21 + 8*x^18 + 72*x^15 + 70*x^12 + 48*x^9 + 76*x^6 + 16*x^3 + 82
load("../../padic_extensions/padic_extensions.sage")
L.<pi> = K.extension(P)
vL = vK.extension(L)
filt




For $m=0,\ldots,15$, we have to compute the genus of the curve $\bar{Y}_m:=\bar{Y}/\Gamma_m$. We do this by extending the valuations $v_1,v_2$ from $K(x)$ to the function field $L_m(Y)=L_m(x,y)$ and computing the genera of the residue fields of these extensions.







load("../../MacLane_valuations/maclane.sage")
def reduction_component(v, vLm, f):

p = vLm.residue_field().characteristic()
Lm = vLm.domain()
FXL.<z> = FunctionField(Lm)
h = FXL._ring(v.phi())
V = maclane_basefield_extensions(v, vLm, h)
S.<T> = FXL[]
FYL.<y> = FXL.extension(T^3-FXL(f))
for w in V:
w = RationalFunctionFieldValuation(FXL, w)
w = w.extension(FYL)
print "    Reduction component:"
print "        ", w.residue_field()
print "        ", w
print "         ground field: ", Lm
print "    ------------"

V = [v1, v2]
print "Computing stable reduction Yb:"
for v in V:
print "over ", v
reduction_component(v, vL, f)
for i in range(len(filt)):
m = filt[i][0]+1
vLm = filt[i][2]
print "Computing Y_m for m=%s:"%m
for v in V:
print "over ", v
reduction_component(v, vLm, f)


We see that $g(\bar{Y}_m)=0$ for $m=0,\ldots,9$, $=1$ for $m=10,\ldots,15$ and $=3$ for $m\geq 16$. It follows that $f_2 = \sum_{m=0}^\infty \frac{|\Gamma_m|}{|\Gamma_0|}(2g_y-2g_{\bar{Y}_m}) = 6(1+3\cdot\frac{1}{3}+5\cdot\frac{1}{6})+4(6\cdot\frac{1}{12}) = 19.$

Now we have to compute $f_3$.

p=3

Y = Superp(f, v_K, p)
Y.FY

Function field in y defined by y^3 - x^4 + 3*x^3 + 24*x^2 + x
Y.compute_semistable_reduction()

L0 = Rational Field L1 = Number Field in pi2 with defining polynomial x^2 + 3 Inertial reduction component: Function field in u2 defined by x*u2^3 + u2 + 2 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 1, v(x + 8) = 5/4 ] -------------- Reduction component: Function field in u2 defined by u2^3 + 2*u2 + 2*z^2 constant field: Finite Field of size 3 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 1, v(x + 8) = 5/4 ] splitting field: Number Field in pi4 with defining polynomial x^4 + 3 ------------ L0 = Rational Field L1 = Number Field in pi3 with defining polynomial x^3 + 2181*x^2 + 12*x + 2181 Inertial reduction component: Rational function field in x over Finite Field of size 3 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 1 ] -------------- Reduction component: Function field in u2 defined by u2^3 + 2*z^4 + 2*z^3 + 2*z^2 constant field: Finite Field of size 3 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 1 ] splitting field: Number Field in pi3 with defining polynomial x^3 + 2181*x^2 + 12*x + 2181 ------------ L0 = Rational Field L1 = Number Field in pi3 with defining polynomial x^3 + 2181*x^2 + 12*x + 2181 Inertial reduction component: Rational function field in x over Finite Field of size 3 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 5/4 ] -------------- Reduction component: Function field in u2 defined by u2^3 + 2*u2 + 2*z^2 constant field: Finite Field of size 3 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 5/4 ] splitting field: Number Field in pi12 with defining polynomial x^12 + 2181*x^8 + 12*x^4 + 2181 ------------ L0 = Number Field in pi6 with defining polynomial x^6 + 2184 L1 = Number Field in pi6 with defining polynomial x^6 + 2184 Inertial reduction component: Function field in u2 defined by 2*x^2*u2^3 + 2 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 1, v(x + 5) = 5/3, v(x^3 + 15*x^2 + 75*x + 368) = 21/4 ] -------------- Reduction component: Function field in u2 defined by u2^3 + u2 + 2*z^2 constant field: Finite Field of size 3 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 1, v(x + 5) = 5/3, v(x^3 + 15*x^2 + 75*x + 368) = 21/4 ] splitting field: Number Field in pi12 with defining polynomial x^12 + 2184 ------------

We see that the stable reduction of $Y$ is of type (c), i.e. with three tail components of genus one. One of these components has a splitting field which is a totally ramified Galois extension of $\mathbb{Q}_3$ of degree $12$. The two other components form an orbit under the Galois group of $\mathbb{F}_9/\mathbb{F}_3$. A splitting field is a Galois extension of degree $24$ of $\mathbb{Q}_3$ with ramification index $12$.

Now we compute the quotients $W_m=W/\Gamma_m$ of these three components.

for c in Y.components:
c.compute_ramification_filtration_reduction()



The splitting field of the first component has ramification breaks $1,3$, with $|\Gamma_0|=12$ and $|\Gamma_1|=3$. We have $g(W_3)=0$, hence the contribution to the conductor is $2(1+2\cdot\frac{1}{4}) = 3.$ The splitting field of the each of the other two components has ramification breaks $1,7$, with $|\Gamma_0|=12$ and $|\Gamma_6|=3$. So the contribution of one of these components is $2(1+6\cdot \frac{1}{4}) = 5.$ So the conductor exponents at $p=3$ is $f_3 = 3 + 2\cdot 5 = 13.$

All in all, the conductor of $Y$ is $N_Y = 2^{19}3^{13} = 835884417024.$

2^19*3^13

835884417024

2. Example

A similar example is the curve $Y:; y^3 = x^4 - 24x^2 - 76x +24.$ Here $\Delta(f) = 2^83^55^6$.

f = x^4-24*x^2-76*x+24
f.discriminant().factor()

-1 * 2^8 * 3^5 * 5^6

Let us first see what happens for $p=2$.

f.change_ring(GF(2)).factor()
p = 2
V = vK.mac_lane_approximants(f)
V[0].mac_lane_step(f)
V[1].mac_lane_step(f)

x^4 [[ Gauss valuation induced by 2-adic valuation, v(x + 2) = 3 ]] [[ Gauss valuation induced by 2-adic valuation, v(x) = 2/3, v(x^3 + 4) = 7/3 ]]
L0.<alpha> = f.splitting_field()
L0
vL0 = vK.extension(L0)
vL0.residue_field()
vL0(2)
fL0 = f.change_ring(L0)
a = fL0.roots(multiplicities=None)
[vL0(a[0]-a[i]) for i in range(4)]

Number Field in alpha with defining polynomial x^6 + 15*x^5 + 96*x^4 + 311*x^3 + 492*x^2 + 339*x + 247 Finite Field in u1 of size 2^2 3 [+Infinity, 2, 2, 2]
vL = padic_sufficiently_ramified_extension(vL0, 9)
vL
v0 = GaussValuation(R, vK)
v1 = v0.extension(x, 2/3)
reduction_component(v1, vL, f)

pi18-adic valuation Reduction component: Function field in u2 defined by u2^3 + z^4 + z Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi18-adic valuation, v(z) = 6 ], v(y) = 8, v(y^3 - z^4 + 24*z^2 + 76*z - 24) = +Infinity ] ground field: Number Field in pi18 with defining polynomial x^18 + 20*x^15 + 28*x^12 + 76*x^9 + 104*x^6 + 112*x^3 + 68 ------------

We see that $Y$ has potentially good reduction over a tamely ramified extension of $\mathbb{Q}_2$ (of degree $18$). However, $\bar{Y}_0=\bar{Y}/\Gamma_0$ has genus $0$. It follows that $f_2 = 6.$

Now we look at $p=3$.

p=3
Y = Superp(f, v_K, p)
Y.FY

Function field in y defined by y^3 - x^4 + 24*x^2 + 76*x - 24
Y.compute_semistable_reduction()

Inertial reduction component: Function field in u2 defined by x^2*u2^3 + 1 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 1/3, v(x^3 + 6*x^2 + 12*x + 14) = 5/3, v(x^3 + 9*x^2 + 24*x + 26) = 2, v(x^3 + 9*x^2 + 24*x + 44) = 19/9, v(x^9 + 27*x^8 + 315*x^7 + 2157*x^6 + 9936*x^5 + 32580*x^4 + 76656*x^3 + 128304*x^2 + 140850*x + 88100) = 20/3, v(x^9 + 27*x^8 + 315*x^7 + 2157*x^6 + 9936*x^5 + 32580*x^4 + 76656*x^3 + 129033*x^2 + 143766*x + 91016) = 27/4 ] --------------

Unfortunately, the computation of the splitting field leads to an error...

Example 3:

One more example: $Y:\; y^3 = f(x) = x^4-3x^3-24x^2-x.$ We have $\Delta(f) = 3^{10}$, so we only have to look at $p=3$.

f = x^4-3*x^3-24*x^2-x
f.discriminant().factor()

3^10
load("../../semistable_reduction/superp.sage")
Y = Superp(f, v_K, p)
Y.FY
Y.compute_semistable_reduction()

Function field in y defined by y^3 - x^4 + 3*x^3 + 24*x^2 + x Inertial reduction component: Function field in u2 defined by x*u2^3 + u2 + 2 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 1, v(x + 8) = 5/4 ] -------------- Inertial reduction component: Rational function field in x over Finite Field of size 3 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 1 ] -------------- Inertial reduction component: Rational function field in x over Finite Field of size 3 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 5/4 ] -------------- Inertial reduction component: Function field in u2 defined by 2*x^2*u2^3 + 2 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 1, v(x + 5) = 5/3, v(x^3 + 15*x^2 + 75*x + 368) = 21/4 ] --------------

We see that the stable reduction $\bar{Y}$ of $Y$ consists of $4$ components, three of genus $1$ and one of genus $0$ (type (c)).

The first component $W_1$ of genus $1$ has a tamely ramified splitting field and its inertial reduction has genus $0$. So $g(W_0)=0$ and $g(W_m)=1$ for $m\geq 1$. The contribution of $W_1$ to the conductor exponent is therefore $2$.

To compute the contribution of the second component $W_2$ fo genus $1$, we analyse the ramificaton filtration of its splitting field.



︠f01f02b5-0b5c-4fa7-b987-9c43995a7f09︠





c = Y.components[2]
v = c.v
vL, w = c.reduction[0]
vL.domain()
w.residue_field()

Error in lines 1-1 Traceback (most recent call last): File "/usr/local/lib/python2.7/dist-packages/smc_sagews/sage_server.py", line 995, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> IndexError: list index out of range
filt = padic_ramification_filtration(vL, compute_subfields=True)
filt

vK2 = filt[0][2]
vK2.domain()
reduction_component(v, vK2, f)

Number Field in pi4 with defining polynomial x^4 + 3 Reduction component: Rational function field in z over Finite Field of size 3 Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi4-adic valuation, v(z + 2) = 5 ], v(y + 2*z + 1) = 16/3, v(y^3 - z^4 + 3*z^3 + 24*z^2 + z) = +Infinity ] ground field: Number Field in pi4 with defining polynomial x^4 + 3 ------------

We see that $|\Gamma_0|=12$, $|\Gamma_m|=3$ for $m=1,2$ and $\Gamma_m=1$ for $m\geq 3$. Also, $g(W_2/\Gamma_2)=0$ and $g(W_2/\Gamma_3)=1$. It follows that the contribution is $2(1 + 2\frac{1}{4}) = 3.$

c = Y.components[3]
v = c.v
vL, w = c.reduction[0]
vL.domain()
w.residue_field()
filt

Number Field in pi12 with defining polynomial x^12 + 2184 Function field in u2 defined by u2^3 + u2 + 2*z^2 [(6, 3, pi4-adic valuation), (0, 12, 3-adic valuation)]
vK2 = filt[0][2]
vK2.domain()
reduction_component(v, vK2, f)

Number Field in pi4 with defining polynomial x^4 + 1419 Reduction component: Function field in u2 defined by 2*u2^3 + 2*z Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi4-adic valuation, v(z - 1417) = 20/3, v(z^3 + 15*z^2 + 75*z + 368) = 21 ], v(y + 1/2013561*z^2 - 2834/2013561*z + 2007889/2013561) = 17/3, v(y^3 - z^4 + 3*z^3 + 24*z^2 + z) = +Infinity ] ground field: Number Field in pi4 with defining polynomial x^4 + 1419 ------------

For the splitting field of the last component $W_3$ we have $|\Gamma_0|=12$, $|\Gamma_m|=3$ for $m=1,\ldots,6$ and $\Gamma_m=1$ for $m\geq 7$. Also, $g(W_3/\Gamma_m)=0$ for $m=0,\ldots,6$. Hence the contribution of $W_3$ is $2(1 + 6\cdot\frac{1}{4}) = 5.$
Altogether, we see that the conductor exponent is $f_3 = 2 + 3 + 5 = 10.$