2213.sagews

This function tests local conditions. The inputs are: a number field A, a finite place p, a uniformizer u for p, an element a of A, and a prime integer l. The output is a list of two booleans: the first is true when a is Mordell-Weil, and the second is true when a is transverse.

def local_conditions(A, p, u, a, l):
    F=A.residue_field(p)
    vp=a.valuation(p)
    b=F(a/u^vp)
    unramified=(vp%l==0)
    transverse=(b.nth_root(l, all=True)!=[])
    return [unramified, transverse]

One random example.

A.<i>=NumberField(x^2+1); A

Number Field in i with defining polynomial x^2 + 1

p=A.ideal(3*i+8); p

Fractional ideal (3*i + 8)

F=A.residue_field(p); F

Residue field of Fractional ideal (3*i + 8)

a=A(18); a

18

u=A.uniformizer(p); u

i + 27

local_conditions(A, p, u, a, 3)

[True, False]

LJ.<u>=NumberField( x^8 - 1134*x^4 - 18225*x^2 - 107163 )

LJ

Number Field in u with defining polynomial x^8 - 1134*x^4 - 18225*x^2 - 107163

a = 1/3402*(-78194096813332063817*u^7 - 226120053042379143270*u^6 +
   370356230483142417591*u^5 + 3146670160662562282161*u^4 +
   91124428396873011166719*u^3 + 228045575974541486007357*u^2 +
   921614535402170900230413*u + 684196453833522406049448)

a.norm().factor()

-1 * 7^6 * 11^27

proof.number_field(False)

LJ.ideal(11).factor()[0]

(Fractional ideal (4/1701*u^7 - 1/54*u^6 + 10/81*u^5 - 41/54*u^4 + 37/18*u^3 - 55/6*u^2 + 149/14*u - 73/2), 1)

P11=LJ.ideal(11).factor()[0][0]

local_conditions(LJ, P11, 11, a, 3)

[True, True]