Is there an isotrivial elliptic surface of positive rank having a section of order $3$?

Question

Let $k$ be a field of characteristic $p > 3$. I cannot find any example of ordinary isotrivial elliptic $k$-surface $E$ (i.e., elliptic $k(t)$-curve, where $t$ is a variable) whose Mordell-Weil group $E(k(t))$ contains the subgroup $\mathbb{Z} \times \mathbb{Z}/3$. In addition, I assume that the $j$-invariant of $E$ is different from $0, 1728$. If I required only $E(k(t)) \supset \mathbb{Z}/3$, we could just take the constant surface $E$ associated with an elliptic $k$-curve $E_0$ having an order $3$ point.

Is it possible to construct such an elliptic surface in principle? Curiously, there is a desired non-isotrivial elliptic surface according to the classical article. Also, for any $j$-invariant, we have an example of the isotrivial $E$ such that $E(k(t)) \supset \mathbb{Z} \times \mathbb{Z}/2$. Indeed, given an elliptic $k$-curve $E_0\!: y^2 = f(x)$, it is enough to consider its twist $E\!: f(t)y^2 = f(x)$. As is known (see, e.g., Proposition 3.1 of Shioda's article), the Mordell-Weil group of this $E$ is naturally isomorphic to $\mathrm{End}(E_0) \times E_0(k)[2]$.

If one can construct the required surface $E$ over a finite field $k = \mathbb{F}_{\!q}$ satisfying the additional condition $3 \mid (q-1)$, then the smooth fibers $E_t$ with $t \in \mathbb{F}_{\!q}$ are pairwise $\mathbb{F}_{\!q}$-isomorphic (not just $\overline{\mathbb{F}_{\!q}}$-isomorphic). This is a nice feature. Indeed, there are only two elliptic $\mathbb{F}_{\!q}$-curves of a given $j$-invariant $\neq 0, 1728$. Since, the order $\#E_t(\mathbb{F}_{\!q}) = q+1 - a$ is divisible by $3$ (with $a$ as the Frobenius trace), $\#E_t^T(\mathbb{F}_{\!q}) = q+1 + a$ is not, where $E_t^T$ is the unique non-trivial (quadratic) $\mathbb{F}_{\!q}$-twist of $E_t$. Otherwise, $3 \mid 2(q+1)$, that is, $3 \mid (q+1)$, which is a contradiction to our assumption.

More generally, do you know how to obtain an ordinary elliptic $\mathbb{F}_{\!q}$-surface $E$ such that its smooth $\mathbb{F}_{\!q}$-fibers $E_t$ (except for a small number) are $\mathbb{F}_{\!q}$-isomorphic? Is it theoretically possible or not?

Answer 1

Let $k$ be a field and let $n\geq 3$ be an integer which is invertible on $k$.

The following lemma will allow you to conclude that an elliptic curve over $k(t)$ which is isotrivial with $j$-invariant not zero and a point of order $3$ is constant, assuming $k$ is not of characteristic three.

Lemma. An isotrivial elliptic curve over $k(t)$ with $j$-invariant not $0$ and a point of order $n$ is constant (i.e., can be defined over $k$).

Proof. Let $Y_1(n)$ be the fine moduli space of elliptic curves together with a point of order $n$; here we use that $n$ is invertible on $k$ and is at least three. Let $E$ be an elliptic curve over $k(t)$ with a point of order $n$. Let $\mathcal{E}\to B$ be an elliptic curve over a dense open $B$ of $\mathbb{A}^1_k$ extending $E$. The moduli map $B\to Y_1(n)$ is isotrivial (by the isotriviality assumption). Thus, the map to the coarse space is constant. However, the stack $Y_1(n)$ and its coarse space coincide outside the $j=0$. Therefore, the moduli map $B\to Y_1(n)$ factors over a point. This point of $Y_1(n)$ is a $k$-point corresponding to an elliptic curve $E_0$ over $k$ and gives an isomorphism of $\mathcal{E}$ with $E_0\times_k B$ (by the universal property of $Y_1(n)$). QED

Isotriviality is a "stacky" issue, i.e., it appears because the objects you study have automorphisms. Once you enrich a moduli problem in such a way that automorphisms disappear (as is the case of pairs $(E,p)$ with $E$ an elliptic curve and $p$ of high enough order), there is no difference between "isotrivial" and "constant".

So, for example, a principally polarized abelian variety over $k(t)$ with full level $n$ structure for some $n\geq 3$ is constant.

Something else: There is yet another reason for which your elliptic curve can not exist. This has nothing to do with the existence of a point of order three (nor with the assumption that $E$ is ordinary). If $E$ is an isotrivial elliptic curve over $k(t)$ with $j$-invariant not 0 or 1728, then the monodromy around a singular fibre (of the minimal regular model $\mathcal{E}\to \mathbb{P}^1_k$ of your elliptic curve) is of order at most two. Now, if you look at the Kodaira classification, you see that the types II, III, IV, II*, III*, IV* do not have such a monodromy action (they are of order 3, 4, or 6). Now, this of course leaves the possibility of I_n-fibres. However, in this case, the $j$-invariant is infinite at the singular fibres, contradicting the fact that the $j$-invariant is generically constant. Thus, the only possibility is that the singular fibres are multiple fibres, but this has been excluded by the fact that $\mathcal{E}\to \mathbb{P}^1$ has a section.)