Examples of curves with fixed genus, number of rational points, and large Jacobian rank

Question

I am asking whether there are known constructions for curves with the following criteria: curves $C$ defined over $\mathbb{Q}$ with genus $g \geq 2$, $|C(\mathbb{Q})| \leq 1$, $C$ is locally soluble, and $r = \text{rank}(J(\mathbb{Q})) \geq g/2$?

There are known results that bound $|C(\mathbb{Q})|$ in terms of $\text{rank}(J(\mathbb{Q}))$ alone, first using the Chabauty-Coleman method (this depends on a relation between $g$ and $r = \text{rank}(J(\mathbb{Q}))$, usually $g \leq r-\kappa$ for some $\kappa \geq 0$) or much more recently, the success in proving Mazur's Conjecture B due to Dimitrov-Gao-Habegger and Khune (independently) which gives a bound of the form $B^{r+1}$ where $B$ is some universal constant. Thus one can surmise that $|C(\mathbb{Q})|$ will be small if $r$ is small. However, there is no known bound of $r$ in terms of $|C(\mathbb{Q})|$, nor is there expected to be one (indeed, it is possible for $r > 0$ while $C(\mathbb{Q}) = \emptyset$).

Answer 1

Yes, at least if you assume the Birch and Swinnerton-Dyer conjecture. Choose a large prime number $p$ that is inert in every imaginary quadratic field of class number $1$ and take $C = X_{{\rm sp}}^{+}(p)$, the modular curve parametrizing elliptic curves whose mod $p$ image of Galois is contained in the normalizer of the split Cartan subgroup. Here are properties of this curve.

$\bullet$ We have $|C(\mathbb{Q})| = 1$. Bilu, Parent, and Rebolledo proved in "Rational points on $X_{0}^{+}(p^{r})$" that this modular curve, which is isomorphic to $X_{0}^{+}(p^{2})$, has no non-cuspidal, non-CM rational points provided that $p = 11$ or $p > 13$. The assumption that $p$ is inert in every imaginary quadratic field of class number $1$ forces every CM elliptic curve $E/\mathbb{Q}$ to have mod $p$ image contained in the normalizer of a non-split Cartan subgroup mod $p$ and hence there are no rational CM points on $C$. Further, $X_{{\rm sp}}^{+}(p)$ has exactly one rational cusp, and so this is the unique rational point on $C$ (and in particular, $C$ is locally soluble).

$\bullet$ The genus of $C$ is $\sim \frac{p^{2}}{24}$ and $J(C)$ is isogenous to $J_{0}(p) \times J_{0}^{{\rm new}, +}(p^{2})$. The first factor has dimension $\sim \frac{p}{12}$. Moreover, every simple factor of $J_{0}^{{\rm new},+}(p^{2})$ is isogenous to $A_{f}$, where $f$ is a newform of level $p^{2}$ with sign of functional equation equal to $-1$. In particular, the analytic rank of $J_{0}^{{\rm new},+}(p^{2})$ is at least its dimension, and so assuming the Birch and Swinnerton-Dyer conjecture, the rank of $J(C)$ is quite close to its genus. It is reasonable to ask if this can be made unconditional (which would require bounds on the number of level $p^{2}$ newforms $f$ with sign $-1$ and for which $L'(f,1/2) \ne 0$). In Kowalski, Michel, and VanderKam's 2000 Crelle paper, they prove that (for prime level) the proportion of newforms with sign $-1$ for which $L'(f,1/2) \ne 0$ is asymptotically at least $7/8$ and this gives a lower bound on the rank of $J(C)$ of about $7g/8$. However, generalizing Kowalski, Michel and VanderKam's results to non-prime level is not so easy. (Some generalizations of Kowalski, Michel and VanderKam's work has been done. There is a 2018 paper of Balkanova and Frolenkov in Monatschefte für Mathematik that handles prime power level, but only for the question of nonvanishing of $L(f,1/2)$, not $L'(f,1/2)$.)

For what it's worth, $C = X_{0}(p)$ almost works. For $p > 163$, it has exactly two rational points and rank close to $g/2$.