**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):

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

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

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 $2$?