Elliptic curve defined over completion

amatar
13 ●1 ●1 ●5

I have an elliptic curve E defined over the rationals and K is an imaginary quadratic field. I have a Heegner point P for E over K. I also have a rational prime p. Let q be a prime of K above p. I would like to use Sage to check whether the point P is divisible by p in E(K) and also in E(Kq) where Kq is the completion of K at the prime q. To check this in E(K) is easy; one can use the heegner_index() function or one can use the division_points() function. I am wondering if there is a way in Sage to do my required check in E(Kq). It seems to me that completions of number fields at finite primes are not defined in Sage.

answered 4 years ago

John Cremona
701 ●35 ●22

Sage does have p-adic fields, but no-one has yet implemented completions of number fields at finite places (see K.completeion? where K is a number field). You should still be able to manually construct the apprpriate p-adic field and base-change the elliptic curve to it.

Here's an example with a split prime, which is easier:

sage: K.<a> = NumberField(x^2 - x + 1)
sage: Q7 = Qp(7, prec=20)
sage: e1, e2 = K.embeddings(Q7)
sage: E = EllipticCurve('389a1').change_ring(K)
sage: Q = E.gens()[0]; Q
(-1 : 1 : 1)
sage: Q.division_points(5)
[]
sage: E.change_ring(e1)(Q).division_points(5)
[(2*7 + 5*7^2 + 3*7^4 + 6*7^5 + 3*7^6 + 4*7^7 + 7^8 + 7^9 + 4*7^10 + 3*7^11 + 4*7^12 + 3*7^13 + 7^14 + 5*7^16 + 4*7^18 + 5*7^19 + O(7^20) : 6 + 3*7 + 7^2 + 4*7^3 + 5*7^5 + 5*7^7 + 2*7^8 + 6*7^9 + 6*7^10 + 3*7^11 + 5*7^12 + 6*7^14 + 5*7^15 + 4*7^16 + 7^17 + 6*7^18 + O(7^20) : 1 + O(7^20))]

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