A (final) remark (9/29). Now that the question is open for bounty anyway, and hence cannot be deleted even though everything boils down to the line added on 9/28, I may as well record -- for anyone who might stumble upon this post -- a more general statement in algebraic dynamics to which I was led by some heuristic considerations. This, and not Moret-Bailly's result quoted below, was my true motivation. I had considered the following statement to be simply outrageous, leading me to ask the question in the contrapositive sense, to see if there was any point in pursuing this.

Statement. Fix a prime $p$ and an integer $q > 1$. Consider varying number fields $K,L$ and a varying rational iteration (self-map) $f : \mathbb{P}_K^1 \to \mathbb{P}_K^1$ of degree $q$, defined over $K$, which has a weak Neron model (in the sense considered by Silverman and Hsia) at every prime of $K$ dividing $p$. (This hypothesis includes, in particular, all rational iterations having good reduction at the primes dividing $p$, as well as the Lattes maps associated to an arbitrary elliptic curve. So it is a more general setup than the one considered in my original question.)

Then the number of $L$-rational eventually periodic points for $f$ should be bounded by a uniform quantity only depending on the prime $p$, the degree $q$ of $f$, the global degree $[K:\mathbb{Q}]$ of the field of definition of $f$, and the set of local $p$-adic degrees of the field $L$.

In view of examples with chaotic $p$-adic behavior such as $z \mapsto (z^p-z)/p$, the hypothesis on the existence of a "weak Neron model" is essential. This hypothesis, however, does include the setup of arbitrary elliptic curves.

A similar remark applies to the dynamical height (Call-Silverman): under the assumption of a "weak Neron model," the height of an $L$-rational point of infinite order should be bounded below by a positive number only depending on the specified parameters. This is now open for both the elliptic curve and good reduction cases. In the example of $z \mapsto (z^p-z)/p$, note that there are infinitely many totally $p$-adic eventually periodic points, as well as totally $p$-adic algebraic points of arbitrarily small and positive dynamical height.

(9/28). OK, I was notified that the negative answer to 2 is actually a trivial consequence of the expected uniform open image conjecture ... Just note that the Galois action on $E[N]$ is transitive almost all of the time, while $\mathbb{Q}(E[N]) \supset \mathbb{Q}(\mu_N)$! My apology for having overlooked this prior to asking the question.

Original post. (Of the original title: Constructing elliptic curves over $\mathbb{Q}$ having a totally $2$-adic point of an arbitrarily high finite order.)

A theorem of L. Moret-Bailly (Groupes de Picard et problemes de Skolem I, II, Ann. Sci. ENS Ser. 4, 1989) states, in a particular case, that a variety over $\mathbb{Q}$ has totally real (resp. totally $p$-adic) algebraic points as soon as it has a point in $\mathbb{R}$ (resp. in $\mathbb{Q}_p$). This also follows by the work of R. Rumely on Fekete-Szego theorems with splitting conditions.

Let us consider the implication of this for the (affine) modular curve $Y_1(N)$ of level $N$, classifying pairs $(E,P)$ of a non-degenerate elliptic curve and a point of order $N$. While Mazur's theorem implies $Y_1(N)(\mathbb{Q}) = \emptyset$ for $N > 11$, the theory of the Tate curve supplies points on $Y_1(N)$ with values in every completion of $\mathbb{Q}$. Given a prime $p$ of $\mathbb{Q}$, possibly $p = \infty$, it follows from the quoted result that for any $N$ there is a totally $p$-adic finite extension $K/\mathbb{Q}$ (meaning that $p$ is unramified and splits completely in $K$) and an elliptic curve $E/K$ having a $K$-rational point of exact order $N$.

This however does not guarantee that we may take $E$ to be itself defined over $\mathbb{Q}$. It is well-known on the other hand that each given $E$ has finite totally $p$-adic torsion, and we are led to the question of the title (where I take $p = 2$ for concreteness, though I will be just as interested in an answer concerning totally real points):

  1. For every $N$, are there elliptic curves over $\mathbb{Q}$ which have a totally $2$-adic point of exact order $N$? [Answer: No! ]

  2. Can we construct, at least, elliptic curves over $\mathbb{Q}$ with an arbitrarily high number of totally $2$-adic torsion points? [Answer: No! ]

  3. Or, on the other extreme, could it actually be that $[\mathbb{Q}(j(E)):\mathbb{Q}] \to \infty$ as $N \to \infty$ and $E$ runs through the elliptic curves over $\bar{\mathbb{Q}}$ having a totally $2$-adic point of exact order $N$? More generally, considering number fields $K,L$ and elliptic curves $E/K$, should $|E(L)_{\mathrm{tors}}|$ be bounded only in terms of the global degree $[K:\mathbb{Q}]$ of $K$ and the set of local degrees of $L$ at the prime $2$?


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