Jtdylktuy

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.

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).

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 $rgeq 1$, $E_0(mathbb{Q}_p)/E_1(mathbb{Q}_p) simeq tilde{E}_{ns}(mathbb{F}_p)$ 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 $pgeq 3$ 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 $rgeq 1$ 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 $$E(mathbb{Q}_p) setminus E_1(mathbb{Q}_p) = {(x,y) in Emid v(x),v(y)geq 0 }$$
  

Now for your questions: if $P in E(mathbb{Q}_p)$ is of infinite order and $rgeq 1$ then $n.[E(mathbb{Q}_p):E_r(mathbb{Q}_p)].P$ is in $E_r(mathbb{Q}_p)$ for $ngeq 1$ and it doesn't equal $O_E$. It follows that for every prime $p$ we have that in your notation $S _p ={v_p(x) mid xin C}$ 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.