main_project/semistable_reduction/picard/Picard

main_project/semistable_reduction/picard/Picard_example1.sagews

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 = QQR.<x> = K[]f = x^4 +14*x^2 + 72*x -41f.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 = pAdicValuation(K, 2)vK.montes_factorization(f)      # shows that f is irreducible over QQ_2L0.<alpha0> = K.extension(f)L0.<alpha>= L0.galois_closure()L0vL0 = 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)v1v2 = 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")vL0 = padic_splitting_field(vK, f)L0 = vL0.domain()L0P = 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 = padic_ramification_filtration(vL, compute_subfields=True)filt

F = x^23 + 24*pi*x^22 + 276*pi^2*x^21 + (2024*pi^3 + 20)*x^20 + (10626*pi^4 + 420*pi)*x^19 + (42504*pi^5 + 4200*pi^2)*x^18 + (134596*pi^6 + 26600*pi^3 + 8)*x^17 + (346104*pi^7 + 119700*pi^4 + 144*pi)*x^16 + (735471*pi^8 + 406980*pi^5 + 1224*pi^2)*x^15 + (1307504*pi^9 + 1085280*pi^6 + 6528*pi^3 + 72)*x^14 + (1961256*pi^10 + 2325600*pi^7 + 24480*pi^4 + 1080*pi)*x^13 + (2496144*pi^11 + 4069800*pi^8 + 68544*pi^5 + 7560*pi^2)*x^12 + (2704156*pi^12 + 5878600*pi^9 + 148512*pi^6 + 32760*pi^3 + 70)*x^11 + (2496144*pi^13 + 7054320*pi^10 + 254592*pi^7 + 98280*pi^4 + 840*pi)*x^10 + (1961256*pi^14 + 7054320*pi^11 + 350064*pi^8 + 216216*pi^5 + 4620*pi^2)*x^9 + (1307504*pi^15 + 5878600*pi^12 + 388960*pi^9 + 360360*pi^6 + 15400*pi^3 + 48)*x^8 + (735471*pi^16 + 4069800*pi^13 + 350064*pi^10 + 463320*pi^7 + 34650*pi^4 + 432*pi)*x^7 + (346104*pi^17 + 2325600*pi^14 + 254592*pi^11 + 463320*pi^8 + 55440*pi^5 + 1728*pi^2)*x^6 + (134596*pi^18 + 1085280*pi^15 + 148512*pi^12 + 360360*pi^9 + 64680*pi^6 + 4032*pi^3 + 76)*x^5 + (42504*pi^19 + 406980*pi^16 + 68544*pi^13 + 216216*pi^10 + 55440*pi^7 + 6048*pi^4 + 456*pi)*x^4 + (10626*pi^20 + 119700*pi^17 + 24480*pi^14 + 98280*pi^11 + 34650*pi^8 + 6048*pi^5 + 1140*pi^2)*x^3 + (2024*pi^21 + 26600*pi^18 + 6528*pi^15 + 32760*pi^12 + 15400*pi^9 + 4032*pi^6 + 1520*pi^3 + 16)*x^2 + (276*pi^22 + 4200*pi^19 + 1224*pi^16 + 7560*pi^13 + 4620*pi^10 + 1728*pi^7 + 1140*pi^4 + 48*pi)*x + 24*pi^23 + 420*pi^20 + 144*pi^17 + 1080*pi^14 + 840*pi^11 + 432*pi^8 + 456*pi^5 + 48*pi^2 np = Newton Polygon with vertices [(0, 68), (1, 52), (3, 32), (7, 16), (23, 0)] .... calling padic_approximate_factor with vk= pi-adic valuation on Number Field in pi with defining polynomial x^24 + 20*x^21 + 8*x^18 + 72*x^15 + 70*x^12 + 48*x^9 + 76*x^6 + 16*x^3 + 82 , f= x^23 + 24*pi*x^22 + 276*pi^2*x^21 + (2024*pi^3 + 20)*x^20 + (10626*pi^4 + 420*pi)*x^19 + (42504*pi^5 + 4200*pi^2)*x^18 + (134596*pi^6 + 26600*pi^3 + 8)*x^17 + (346104*pi^7 + 119700*pi^4 + 144*pi)*x^16 + (735471*pi^8 + 406980*pi^5 + 1224*pi^2)*x^15 + (1307504*pi^9 + 1085280*pi^6 + 6528*pi^3 + 72)*x^14 + (1961256*pi^10 + 2325600*pi^7 + 24480*pi^4 + 1080*pi)*x^13 + (2496144*pi^11 + 4069800*pi^8 + 68544*pi^5 + 7560*pi^2)*x^12 + (2704156*pi^12 + 5878600*pi^9 + 148512*pi^6 + 32760*pi^3 + 70)*x^11 + (2496144*pi^13 + 7054320*pi^10 + 254592*pi^7 + 98280*pi^4 + 840*pi)*x^10 + (1961256*pi^14 + 7054320*pi^11 + 350064*pi^8 + 216216*pi^5 + 4620*pi^2)*x^9 + (1307504*pi^15 + 5878600*pi^12 + 388960*pi^9 + 360360*pi^6 + 15400*pi^3 + 48)*x^8 + (735471*pi^16 + 4069800*pi^13 + 350064*pi^10 + 463320*pi^7 + 34650*pi^4 + 432*pi)*x^7 + (346104*pi^17 + 2325600*pi^14 + 254592*pi^11 + 463320*pi^8 + 55440*pi^5 + 1728*pi^2)*x^6 + (134596*pi^18 + 1085280*pi^15 + 148512*pi^12 + 360360*pi^9 + 64680*pi^6 + 4032*pi^3 + 76)*x^5 + (42504*pi^19 + 406980*pi^16 + 68544*pi^13 + 216216*pi^10 + 55440*pi^7 + 6048*pi^4 + 456*pi)*x^4 + (10626*pi^20 + 119700*pi^17 + 24480*pi^14 + 98280*pi^11 + 34650*pi^8 + 6048*pi^5 + 1140*pi^2)*x^3 + (2024*pi^21 + 26600*pi^18 + 6528*pi^15 + 32760*pi^12 + 15400*pi^9 + 4032*pi^6 + 1520*pi^3 + 16)*x^2 + (276*pi^22 + 4200*pi^19 + 1224*pi^16 + 7560*pi^13 + 4620*pi^10 + 1728*pi^7 + 1140*pi^4 + 48*pi)*x + 24*pi^23 + 420*pi^20 + 144*pi^17 + 1080*pi^14 + 840*pi^11 + 432*pi^8 + 456*pi^5 + 48*pi^2 s= 16 and k= 1 g= y9 + pi^16 Li = Number Field in pi12 with defining polynomial x^12 + 4*x^9 + 6*x^6 + 4*x^3 + 2 .... calling padic_approximate_factor with vk= pi-adic valuation on Number Field in pi with defining polynomial x^24 + 20*x^21 + 8*x^18 + 72*x^15 + 70*x^12 + 48*x^9 + 76*x^6 + 16*x^3 + 82 , f= x^23 + 24*pi*x^22 + 276*pi^2*x^21 + (2024*pi^3 + 20)*x^20 + (10626*pi^4 + 420*pi)*x^19 + (42504*pi^5 + 4200*pi^2)*x^18 + (134596*pi^6 + 26600*pi^3 + 8)*x^17 + (346104*pi^7 + 119700*pi^4 + 144*pi)*x^16 + (735471*pi^8 + 406980*pi^5 + 1224*pi^2)*x^15 + (1307504*pi^9 + 1085280*pi^6 + 6528*pi^3 + 72)*x^14 + (1961256*pi^10 + 2325600*pi^7 + 24480*pi^4 + 1080*pi)*x^13 + (2496144*pi^11 + 4069800*pi^8 + 68544*pi^5 + 7560*pi^2)*x^12 + (2704156*pi^12 + 5878600*pi^9 + 148512*pi^6 + 32760*pi^3 + 70)*x^11 + (2496144*pi^13 + 7054320*pi^10 + 254592*pi^7 + 98280*pi^4 + 840*pi)*x^10 + (1961256*pi^14 + 7054320*pi^11 + 350064*pi^8 + 216216*pi^5 + 4620*pi^2)*x^9 + (1307504*pi^15 + 5878600*pi^12 + 388960*pi^9 + 360360*pi^6 + 15400*pi^3 + 48)*x^8 + (735471*pi^16 + 4069800*pi^13 + 350064*pi^10 + 463320*pi^7 + 34650*pi^4 + 432*pi)*x^7 + (346104*pi^17 + 2325600*pi^14 + 254592*pi^11 + 463320*pi^8 + 55440*pi^5 + 1728*pi^2)*x^6 + (134596*pi^18 + 1085280*pi^15 + 148512*pi^12 + 360360*pi^9 + 64680*pi^6 + 4032*pi^3 + 76)*x^5 + (42504*pi^19 + 406980*pi^16 + 68544*pi^13 + 216216*pi^10 + 55440*pi^7 + 6048*pi^4 + 456*pi)*x^4 + (10626*pi^20 + 119700*pi^17 + 24480*pi^14 + 98280*pi^11 + 34650*pi^8 + 6048*pi^5 + 1140*pi^2)*x^3 + (2024*pi^21 + 26600*pi^18 + 6528*pi^15 + 32760*pi^12 + 15400*pi^9 + 4032*pi^6 + 1520*pi^3 + 16)*x^2 + (276*pi^22 + 4200*pi^19 + 1224*pi^16 + 7560*pi^13 + 4620*pi^10 + 1728*pi^7 + 1140*pi^4 + 48*pi)*x + 24*pi^23 + 420*pi^20 + 144*pi^17 + 1080*pi^14 + 840*pi^11 + 432*pi^8 + 456*pi^5 + 48*pi^2 s= 10 and k= 2 g= y15^2 + 2*pi^10*y15 + pi^20 Li = Number Field in pi6 with defining polynomial x^6 + 2*x^3 + 2 .... calling padic_approximate_factor with vk= pi-adic valuation on Number Field in pi with defining polynomial x^24 + 20*x^21 + 8*x^18 + 72*x^15 + 70*x^12 + 48*x^9 + 76*x^6 + 16*x^3 + 82 , f= x^23 + 24*pi*x^22 + 276*pi^2*x^21 + (2024*pi^3 + 20)*x^20 + (10626*pi^4 + 420*pi)*x^19 + (42504*pi^5 + 4200*pi^2)*x^18 + (134596*pi^6 + 26600*pi^3 + 8)*x^17 + (346104*pi^7 + 119700*pi^4 + 144*pi)*x^16 + (735471*pi^8 + 406980*pi^5 + 1224*pi^2)*x^15 + (1307504*pi^9 + 1085280*pi^6 + 6528*pi^3 + 72)*x^14 + (1961256*pi^10 + 2325600*pi^7 + 24480*pi^4 + 1080*pi)*x^13 + (2496144*pi^11 + 4069800*pi^8 + 68544*pi^5 + 7560*pi^2)*x^12 + (2704156*pi^12 + 5878600*pi^9 + 148512*pi^6 + 32760*pi^3 + 70)*x^11 + (2496144*pi^13 + 7054320*pi^10 + 254592*pi^7 + 98280*pi^4 + 840*pi)*x^10 + (1961256*pi^14 + 7054320*pi^11 + 350064*pi^8 + 216216*pi^5 + 4620*pi^2)*x^9 + (1307504*pi^15 + 5878600*pi^12 + 388960*pi^9 + 360360*pi^6 + 15400*pi^3 + 48)*x^8 + (735471*pi^16 + 4069800*pi^13 + 350064*pi^10 + 463320*pi^7 + 34650*pi^4 + 432*pi)*x^7 + (346104*pi^17 + 2325600*pi^14 + 254592*pi^11 + 463320*pi^8 + 55440*pi^5 + 1728*pi^2)*x^6 + (134596*pi^18 + 1085280*pi^15 + 148512*pi^12 + 360360*pi^9 + 64680*pi^6 + 4032*pi^3 + 76)*x^5 + (42504*pi^19 + 406980*pi^16 + 68544*pi^13 + 216216*pi^10 + 55440*pi^7 + 6048*pi^4 + 456*pi)*x^4 + (10626*pi^20 + 119700*pi^17 + 24480*pi^14 + 98280*pi^11 + 34650*pi^8 + 6048*pi^5 + 1140*pi^2)*x^3 + (2024*pi^21 + 26600*pi^18 + 6528*pi^15 + 32760*pi^12 + 15400*pi^9 + 4032*pi^6 + 1520*pi^3 + 16)*x^2 + (276*pi^22 + 4200*pi^19 + 1224*pi^16 + 7560*pi^13 + 4620*pi^10 + 1728*pi^7 + 1140*pi^4 + 48*pi)*x + 24*pi^23 + 420*pi^20 + 144*pi^17 + 1080*pi^14 + 840*pi^11 + 432*pi^8 + 456*pi^5 + 48*pi^2 s= 4 and k= 4 g= y17^4 + (2*pi^10 + 4*pi^4)*y17^3 + (pi^20 + 6*pi^14 + 6*pi^8)*y17^2 + (-40*pi^21 - 10*pi^18 - 144*pi^15 - 136*pi^12 - 96*pi^9 - 152*pi^6 - 32*pi^3 - 164)*y17 + 394*pi^22 + 88*pi^19 + 1371*pi^16 + 1352*pi^13 + 884*pi^10 + 1504*pi^7 + 238*pi^4 + 1640*pi Li = Number Field in pi3 with defining polynomial x^3 + 2 [(15, 2, pi12-adic valuation), (9, 4, pi6-adic valuation), (3, 8, pi3-adic valuation), (0, 24, 2-adic valuation)]

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)

Computing stable reduction Yb: over [ Gauss valuation induced by 2-adic valuation, v(x + 1) = 1/4, v(x^4 + 14*x^2 + 72*x - 41) = 3 ] Reduction component: Function field in u2 defined by u2^3 + z^4 + z^2 Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi-adic valuation, v(z + pi^9 + pi^6 + 1) = 18 ], v(y) = 24, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] ground field: Number Field in pi with defining polynomial x^24 + 20*x^21 + 8*x^18 + 72*x^15 + 70*x^12 + 48*x^9 + 76*x^6 + 16*x^3 + 82 ------------ over [ Gauss valuation induced by 2-adic valuation, v(x + 1) = 1/4, v(x^4 + 14*x^2 + 72*x - 41) = 7/2 ] Reduction component: Function field in u2 defined by u2^3 + z^2 + z Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi-adic valuation, v(z + pi^21 + pi^18 + pi^9 + pi^6 + 1) = 24 ], v(y) = 28, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] ground field: Number Field in pi with defining polynomial x^24 + 20*x^21 + 8*x^18 + 72*x^15 + 70*x^12 + 48*x^9 + 76*x^6 + 16*x^3 + 82 ------------ Reduction component: Function field in u2 defined by u2^3 + z^2 + z Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi-adic valuation, v(z + pi^21 + pi^9 + pi^6 + 1) = 24 ], v(y) = 28, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] ground field: Number Field in pi with defining polynomial x^24 + 20*x^21 + 8*x^18 + 72*x^15 + 70*x^12 + 48*x^9 + 76*x^6 + 16*x^3 + 82 ------------ Computing Y_m for m=16: over [ Gauss valuation induced by 2-adic valuation, v(x + 1) = 1/4, v(x^4 + 14*x^2 + 72*x - 41) = 3 ] Reduction component: Function field in u2 defined by u2^3 + z^2 + z Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi12-adic valuation, v(z + pi12^3 + 1) = 9/2, v(z^2 + (-4*pi12^9 - 6*pi12^6 - 2*pi12^3)*z + 7*pi12^9 + 15*pi12^6 + 12*pi12^3 + 7) = 18 ], v(y) = 12, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] ground field: Number Field in pi12 with defining polynomial x^12 + 4*x^9 + 6*x^6 + 4*x^3 + 2 ------------ over [ Gauss valuation induced by 2-adic valuation, v(x + 1) = 1/4, v(x^4 + 14*x^2 + 72*x - 41) = 7/2 ] Reduction component: Function field in u2 defined by u2^3 + z + 1 Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi12-adic valuation, v(z + pi12^3 + 1) = 9/2, v(z^2 + (6*pi12^9 + 14*pi12^6 + 12*pi12^3 + 8)*z - 23*pi12^9 - 57*pi12^6 - 38*pi12^3 - 25) = 24 ], v(y) = 14, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] ground field: Number Field in pi12 with defining polynomial x^12 + 4*x^9 + 6*x^6 + 4*x^3 + 2 ------------ Reduction component: Function field in u2 defined by u2^3 + z Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi12-adic valuation, v(z + pi12^3 + 1) = 9/2, v(z^2 + (6*pi12^9 + 14*pi12^6 + 12*pi12^3 + 8)*z + 31*pi12^9 + 77*pi12^6 + 54*pi12^3 + 35) = 24 ], v(y) = 14, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] ground field: Number Field in pi12 with defining polynomial x^12 + 4*x^9 + 6*x^6 + 4*x^3 + 2 ------------ Computing Y_m for m=10: over [ Gauss valuation induced by 2-adic valuation, v(x + 1) = 1/4, v(x^4 + 14*x^2 + 72*x - 41) = 3 ] Reduction component: Function field in u2 defined by u2^3 + z Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi6-adic valuation, v(z + 1) = 3/2, v(z^2 + (pi6^3 + 2)*z + 2*pi6^3 + 1) = 21/4, v(z^4 + 14*z^2 + 72*z - 41) = 18 ], v(y) = 6, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] ground field: Number Field in pi6 with defining polynomial x^6 + 2*x^3 + 2 ------------ over [ Gauss valuation induced by 2-adic valuation, v(x + 1) = 1/4, v(x^4 + 14*x^2 + 72*x - 41) = 7/2 ] Reduction component: Function field in u2 defined by u2^3 + z Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi6-adic valuation, v(z + 1) = 3/2, v(z^2 + (pi6^3 + 2)*z + 2*pi6^3 + 1) = 21/4, v(z^4 + 14*z^2 + 72*z - 41) = 21 ], v(y) = 7, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] ground field: Number Field in pi6 with defining polynomial x^6 + 2*x^3 + 2 ------------ Computing Y_m for m=4: over [ Gauss valuation induced by 2-adic valuation, v(x + 1) = 1/4, v(x^4 + 14*x^2 + 72*x - 41) = 3 ] Reduction component: Function field in u2 defined by u2^3 + z Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi3-adic valuation, v(z + 1) = 3/4, v(z^4 + 14*z^2 + 72*z - 41) = 9 ], v(y) = 3, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] ground field: Number Field in pi3 with defining polynomial x^3 + 2 ------------ over [ Gauss valuation induced by 2-adic valuation, v(x + 1) = 1/4, v(x^4 + 14*x^2 + 72*x - 41) = 7/2 ] Reduction component: Function field in u2 defined by u2^3 + z Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi3-adic valuation, v(z + 1) = 3/4, v(z^4 + 14*z^2 + 72*z - 41) = 21/2 ], v(y) = 7/2, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] ground field: Number Field in pi3 with defining polynomial x^3 + 2 ------------ Computing Y_m for m=1: over [ Gauss valuation induced by 2-adic valuation, v(x + 1) = 1/4, v(x^4 + 14*x^2 + 72*x - 41) = 3 ] Reduction component: Function field in u2 defined by u2^3 + z Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by 2-adic valuation, v(z + 1) = 1/4, v(z^4 + 14*z^2 + 72*z - 41) = 3 ], v(y) = 1, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] ground field: Rational Field ------------ over [ Gauss valuation induced by 2-adic valuation, v(x + 1) = 1/4, v(x^4 + 14*x^2 + 72*x - 41) = 7/2 ] Reduction component: Rational function field in z over Finite Field of size 2 Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by 2-adic valuation, v(z + 1) = 1/4, v(z^4 + 14*z^2 + 72*z - 41) = 7/2 ], v(y) = 7/6, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] ground field: Rational Field ------------

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

%mdNow we have to compute $f_3$.

Now we have to compute $f_3$.

p=3v_K = pAdicValuation(K,p)load("../../semistable_reduction/superp.sage")

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

Function field in y defined by y^3 - x^4 - 14*x^2 - 72*x + 41

Y.compute_semistable_reduction()

Inertial reduction component: Rational function field in x over Finite Field of size 3 [ Gauss valuation induced by 3-adic valuation, v(x) = 3/4 ] -------------- Reduction component: Function field in u2 defined by u2^3 + u2 + z^2 constant field: Finite Field of size 3 [ Gauss valuation induced by 3-adic valuation, v(x) = 3/4 ] splitting field: Number Field in pi12 with defining polynomial x^12 + 2181*x^8 + 12*x^4 + 33 ------------ Inertial reduction component: Rational function field in x over Finite Field in u1 of size 3^2 [ Gauss valuation induced by 3-adic valuation, v(x^2 + 1) = 1, v(x^2 + 7) = 7/4 ] -------------- G = t^2 + 7 F = t^4 + 14*t^2 + 72*t - x^3 - 41 Reduction component: Function field in u2 defined by u2^3 + 2*u2 + z^2 constant field: Finite Field in u2 of size 3^2 [ Gauss valuation induced by 3-adic valuation, v(x^2 + 7) = 7/4 ] splitting field: Number Field in pi24 with defining polynomial x^24 + 2088*x^12 + 1791 ------------

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()

components over [ Gauss valuation induced by 3-adic valuation, v(x) = 3/4 ] the splitting field is Number Field in pi12 with defining polynomial x^12 + 2181*x^8 + 12*x^4 + 33 [(2, 3, pi4-adic valuation), (0, 12, 3-adic valuation)] The ramification breaks are: [3, 1] Computing Y_m for m=2 and |Gm|=3: intermediate field: Number Field in pi4 with defining polynomial x^4 + 6 with ramification index 4 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) = 3 ], v(y + 2) = 4/3, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] Computing Y_m for m=0 and |Gm|=12: intermediate field: Rational Field with ramification index 1 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 3-adic valuation, v(z) = 3/4 ], v(y + 2) = 1/3, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] components over [ Gauss valuation induced by 3-adic valuation, v(x^2 + 1) = 1, v(x^2 + 7) = 7/4 ] the splitting field is Number Field in pi24 with defining polynomial x^24 + 2088*x^12 + 1791 [(6, 3, pi8-adic valuation), (0, 12, 3-adic valuation)] The ramification breaks are: [7, 1] Computing Y_m for m=6 and |Gm|=3: intermediate field: Number Field in pi8 with defining polynomial x^8 + 576*x^6 + 2007*x^4 + 513*x^2 + 1359 with ramification index 4 Reduction component: Rational function field in z over Finite Field in u2 of size 3^2 Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi8-adic valuation, v(z + 219464*pi8^6 + 2307032/3*pi8^4 + 195744*pi8^2 + 520950) = 7 ], v(y) = 8/3, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] Reduction component: Rational function field in z over Finite Field in u2 of size 3^2 Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi8-adic valuation, v(z + 109732*pi8^6 + 1153513/3*pi8^4 + 97872*pi8^2 + 260475) = 7 ], v(y) = 8/3, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] Computing Y_m for m=0 and |Gm|=12: intermediate field: Rational Field with ramification index 1 Reduction component: Rational function field in z over Finite Field in u1 of size 3^2 Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by 3-adic valuation, v(z^2 + 7) = 7/4 ], v(y) = 2/3, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ]

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

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+24f.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 = 2vK = pAdicValuation(K, 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()L0vL0 = 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(vL, 9)vLv0 = 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=3v_K = pAdicValuation(K,p)load("../../semistable_reduction/superp.sage")Y = Superp(f, v_K, p)Y.FY

Function field in y defined by y^3 - x^4 - 14*x^2 - 72*x + 41

Y.compute_semistable_reduction()

Gauss valuation induced by 3-adic valuation is_in= False holes = [] [ Gauss valuation induced by 3-adic valuation, v(x) = 1/2 ] is_in= False holes = [] [ Gauss valuation induced by 3-adic valuation, v(x) = 3/4 ] is_in= True holes = [] [ Gauss valuation induced by 3-adic valuation, v(x^2 + 1) = 1 ] is_in= False holes = [] [ Gauss valuation induced by 3-adic valuation, v(x^2 + 1) = 1, v(x^2 + 7) = 3/2 ] is_in= False holes = [] [ Gauss valuation induced by 3-adic valuation, v(x^2 + 1) = 1, v(x^2 + 7) = 7/4 ] is_in= True holes = [] Inertial reduction component: Rational function field in x over Finite Field of size 3 [ Gauss valuation induced by 3-adic valuation, v(x) = 3/4 ] -------------- h = x L0 = Rational Field V = [[ Gauss valuation induced by 3-adic valuation, v(x) = 3/4 ]] L1 = Number Field in pi3 with defining polynomial x^3 + 2181*x^2 + 12*x + 33 L = Number Field in pi12 with defining polynomial x^12 + 2181*x^8 + 12*x^4 + 33 V = [[ Gauss valuation induced by 3-adic valuation, v(x) = 3/4 ]] Reduction component: Function field in u2 defined by u2^3 + u2 + z^2 constant field: Finite Field of size 3 [ Gauss valuation induced by 3-adic valuation, v(x) = 3/4 ] splitting field: Number Field in pi12 with defining polynomial x^12 + 2181*x^8 + 12*x^4 + 33 ------------ Inertial reduction component: Rational function field in x over Finite Field in u1 of size 3^2 [ Gauss valuation induced by 3-adic valuation, v(x^2 + 1) = 1, v(x^2 + 7) = 7/4 ] -------------- h = x^2 + 7 L0 = Rational Field V = [[ Gauss valuation induced by 3-adic valuation, v(x^2 + 7) = 7/4 ]] G = t^2 + 7 F = t^4 + 14*t^2 + 72*t - x^3 - 41 L1 = Number Field in pi6 with defining polynomial x^6 + 2088*x^3 + 1791 L = Number Field in pi24 with defining polynomial x^24 + 2088*x^12 + 1791 V = [[ Gauss valuation induced by 3-adic valuation, v(x^2 + 7) = 7/4 ]] Reduction component: Function field in u2 defined by u2^3 + 2*u2 + z^2 constant field: Finite Field in u2 of size 3^2 [ Gauss valuation induced by 3-adic valuation, v(x^2 + 7) = 7/4 ] splitting field: Number Field in pi24 with defining polynomial x^24 + 2088*x^12 + 1791 ------------

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-xf.discriminant().factor()

3^10

load("../../semistable_reduction/superp.sage")Y = Superp(f, v_K, p)Y.FYY.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 ] -------------- 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 ------------ 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 ------------ 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 ------------ 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 $\bar{Y}$ 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.

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

Number Field in pi12 with defining polynomial x^12 + 2181*x^8 + 12*x^4 + 2181 Function field in u2 defined by u2^3 + 2*u2 + 2*z^2

filt = padic_ramification_filtration(vL, compute_subfields=True)filt

[(2, 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 + 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 ------------