$p$-adic values of rational points on elliptic curves

Question

The following question came up naturally whilst studying diophantine equations: given an elliptic curve $E$ of the form $Y^2 + aY = X^3 +bX^2 + cX + d$ defined over $\mathbb{Q}$, consider the subset $C \subseteq \mathbb{Q}$ of numbers which appear as either the first or the second coordinate of a rational point on $E$.

If we assume that $E$ has infinitely many points then $C$ is infinite. I would like to understand how 'large' $C$ can get, in particular: can we choose $E$ such that $C$ has unbounded $p$-adic value for all prime numbers $p$? Maybe we can at least choose $E$ such that $C$ has unbounded $p$-adic value for all primes in a given finite set of prime numbers? I know almost nothing about the topic, so any pointers you might have to articles or books studying the set $C$ would be very helpful.

Answer 1

I'm not sure when you say unbounded $p$-adic value (absolute value?) you mean above or below?

Here is something that might interest you at least. Whenever $E(\mathbf Q)$ is infinite the set $C$ will always contain rationals with arbitrarily large $p$-adic absolute value, so denominator highly divisible by $p$. In fact these all come from the $x$ coordinate alone.

The trick here is the $p$-adic filtration on $E$, we may define for any $p$

$$ E_n = \{P \in E(\mathbf Q_p) : v_p(x(P)) \le -2n \} $$

then this is a descending sequence of subgroups of $E(\mathbf Q_p)$ which we think of as the subgroups of $p$-adic points which are $p$-adically close to the point at infinity. For this sequence the magic is that we have

$$E(\mathbf Q_p)/E_1 \cong E(\mathbf F_p)$$

and

$$E_n/E_{n+1} \cong \mathbf F_p$$

as groups. For detail you can look at Husemoller's book chapter 14. The point is if $P \in E(\mathbf Q)$ is of infinite order then $(\#E(\mathbf F_p)) \cdot P \in E_1$ and moreover $p^n \cdot (\#E(\mathbf F_p)) \cdot P \in E_n$ so we have a point with large negative $p$-adic valuation.

And here is some Sage code demonstrating this because I like Sage code:

sage: E = EllipticCurve("37a1")
sage: E.rank()
1
sage: P = E.gens()[0]
sage: P
(0 : -1 : 1)
sage: P.order()
+Infinity
sage: E.ap(5)
-2
sage: 6 - E.ap(5) # this is the number of points over F_5
8
sage: 8*P
(21/25 : -56/125 : 1)
sage: 5*8*P
(263817293110494867593838666854208001/292736325329248127651484680640160000 : -34188880637325550305106055730237610829874076311530751/158385319626308443937475969221994173751192384064000000 : 1)
sage: (5*8*P)[0]
263817293110494867593838666854208001/292736325329248127651484680640160000
sage: ((5*8*P)[0]).valuation(5)
-4
sage: ((5^2*8*P)[0]).valuation(5)
-6
sage: ((5^3*8*P)[0]).valuation(5)
-8
sage: ((5^4*8*P)[0]).valuation(5)
-10

You should probably be a little careful with the definition of the filtration if $E$ has bad reduction at $p$ but it should still work (I can show an example of this if you like).

Answer 2

I believe most of your questions can be answered using the material from Silverman's 'The Arithmetic of Elliptic curves'. The relevant section for your questions on the $p$-adic valuation of the rational solutions would be the section on elliptic curves over local fields.

Here are some highlights of the theory: if $E$ is an elliptic curve over $\mathbb{Q}_p$ (or any finite extension of such field, but if you're only interested in $\mathbb{Q}$ these suffice) defined by a Weierstrass equation, there is a filtration of subgroups of $E(\mathbb{Q}_p)$: $$E(\mathbb{Q}_p)\supset E_0(\mathbb{Q}_p) \supset E_1(\mathbb{Q}_p) \supset E_2(\mathbb{Q}_p) \supset \cdots$$ with the following properties:

• each of the successive quotients is finite: more precisely $E_r(\mathbb{Q}_p)/E_{r+1}(\mathbb{Q}_p) \simeq (\mathbb{F}_p,+)$ if $r\geq 1$, $E_0(\mathbb{Q}_p)/E_1(\mathbb{Q}_p) \simeq \tilde{E}_{ns}(\mathbb{F}_p)$ where $\tilde{E}_{ns}$ are the nonsingular points on the curve $E$ 'reduced modulo p' (i.e. by looking at the Weierstrass equation modulo $p$) and $E(\mathbb{Q}_p)/E_0(\mathbb{Q}_p)$ is trivial when $E$ has good reduction (i.e. when $E$ modulo $p$ has no singular points) but can be a nontrivial (finite) group when $E$ has bad reduction.
• if $p\geq 3$ then there's an isomorphism $E_1(\mathbb{Q}_p) \simeq (\mathbb{Z}_p,+)$ of topological groups. If $p = 2$ then likewise $E_2(\mathbb{Q}_2) \simeq (\mathbb{Z}_2,+)$.
• For $r\geq 1$ we can explicitly describe $E_r(\mathbb{Q}_p)$ as $$E_r(\mathbb{Q}_p) = \left\{(x,y) \in E(\mathbb{Q}_p) \mid v(x) \leq -2r , v(y) \leq -3r \right\} \cup \{O_E\} $$ (where $O_E$ is the identity of $E$ i.e. the unique point at infinity) and $$E(\mathbb{Q}_p) \setminus E_1(\mathbb{Q}_p) = \{(x,y) \in E\mid v(x),v(y)\geq 0 \}$$

Now for your questions: if $P \in E(\mathbb{Q}_p)$ is of infinite order and $r\geq 1$ then $n.[E(\mathbb{Q}_p):E_r(\mathbb{Q}_p)].P$ is in $E_r(\mathbb{Q}_p)$ for $n\geq 1$ and it doesn't equal $O_E$. It follows that for every prime $p$ we have that in your notation $S _p =\{v_p(x) \mid x\in C\}$ is never bounded below if the rank of $E$ is positive. It seems that the set $S_p$ should be bounded above but I don't see an immediate argument why this is the case. Hope this helps.