Let C be an elliptic curve over Q_p with Weierstrass form

y^2 = x^3 + ax^2 + bx + c:

The level of a point (x, y) is n if v_p(x) = -2n and v_p(y) = -3n,

and we let C^{(n)}(\mathbb{Q}_p) denote the set of all points of level at least n. Define the functions t = x/y and z = 1/y. Then there is another affine part C that is given by z = t^3 + at^2z + btz^2 + cz^3.

(1) Show that the point 0 corresponds to (t; z) = (0; 0) in this affine part.

(2) Show that on this new affine part negation is given by -(t; z) = (-t,-z).

(3) Show that if n > 0, then C^{(n)}(\mathbb{Q}_p) corresponds with

{(t, z) : v_p(t) \geq n and v_p(z) > 0}

(4) Show that the level of a point P \in C^(1)(\mathbb{Q}_p) equals v_p(t(P)).

(5) Show that for (t, z) \in C^(1)(\math{Q}_p) of level n we have v_p(z(P)) = 3n.

The coefficients a,b,c are supposed to be p-adic integers, or equivalently: $v_p(a),v_p(b),v_p(c) \geq 0$ (this is a standard convention for p-adic Weierstrass equations);

And by definition, v_p(0) = infinity.