Well-definedness of a map on an elliptic curve

Question

I am making my way through Lawrence Washington's book "Elliptic Curves - Number Theory and Cryptography" (2nd edition) and got stuck on page 200 on the following problem:

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ by the equation

$y^{2}= x^{3}+Ax+B$,

where A, B are both integers and let $p$ be a prime number. Now define the set

$E_{k}:=\{(x,y)\in E\; \vert\; v_p(x)\leq -2k, v_{p}(y)\leq -3k \}\cup\{\infty\}$,

where $\infty$ denotes the neutral element of the group structure on $E$ and $v_{p}(z)$ is the $p$-adic valuation of $z$.

He defines the map

$$ \lambda: E_{1}\rightarrow \mathbb{Z}/p^{4}\mathbb{Z}$$

$$ (x,y)\mapsto p^{-1}\frac{x}{y}\;\text{ mod } p\quad \text{ if }(x,y)\in E\backslash\{\infty\}$$

$$\infty\mapsto 0.$$

I interpret this assignment as: $\lambda$ factorizes over $\mathbb{Z}$ in the canonical way.

However, Washington does not lose a word about the question, why $p^{-1}\frac{x}{y}=: t$ is an integer, and it does not seem clear to me.

Does anyone see whether or why this is the case?

Two remarks:

It is equivalent to see that $t$ is annulated by a monic polynomial with integer coefficients. One can show that in this case and if $t\neq 0$, a such polynomial needs to have degree at least 4.

I understand that the p-adic valuation of $t$ must be nonnegative.

Answer 1

In fact, $\lambda$ factorizes to the ring of $p$-adic integers $\mathbb{Z}_p$ (since the $p$-adic valuation of $t$ is non-negative). Then reduce mod $p^4$ and use the isomorphism

$$\mathbb{Z}_p/p^4\mathbb{Z}_p\leftarrow \mathbb{Z}/p^4\mathbb{Z}.$$

This answer may seem very elliptic so feel free to ask if you have any question.

Answer 2

You'll probably need this commonly used result:

If $(x,y)\neq \infty$ is a rational point on $E(\mathbb Q)$ then $x$ and $y$ have the form $$ x=\frac{m}{e^2},\;\;y=\frac{n}{e^3} $$ for some integers $m,n,e$ satisfying $e>0$ and $$ \gcd(m,e)=\gcd(n,e)=1 $$

I imagine this should be given in that book too. Another reference is chapter III.2 of "Rational points on Elliptic Curves".

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Now $E_1$ says $$ v_p(x)\leq -2,\;\;v_p(y)\leq -3, $$ so in particular $p$ divides $e$ but not $m$ or $n$. Therefore $v_p(p^{-1}e)\geq 0$ and $$ v_p\left(p^{-1}\frac{x}{y}\right) = v_p\left(p^{-1}\frac{me}{n}\right) = v_p(p^{-1}e) \geq 0 $$ So it's a $p$-adic integer.