Uniqueness conjecture for submaximal curves on principally polarized abelian surfaces

Let $X$ be the principally polarized abelian surface under consideration, and let $\operatorname{End}(X)$ denote its endomorphism ring. A curve $C$ is submaximal for an ample $\mathbb Q$-line bundle $L$ if its Seshadri quotient is strictly less than the maximal value $\sqrt{L^2}$. The relevant possible endomorphism rings are the orders $\mathbb Z[\sqrt{e}]$ and $\mathbb Z[{\textstyle\frac{1}{2}}+{\textstyle\frac{1}{2}}\sqrt{e}]$ in real quadratic fields.

Uniqueness conjecture. For every ample $\mathbb Q$-line bundle $L$ on $X$, there exists at most one irreducible curve $C$ that is submaximal for $L$ if and only if either

$$\operatorname{End}(X)=\mathbb Z[\sqrt{e}]$$

for a non-square integer $e>0$, or

$$\operatorname{End}(X)=\mathbb Z[{\textstyle\frac{1}{2}}+{\textstyle\frac{1}{2}}\sqrt{e}]$$

for a non-square integer $e>0$ with $e\equiv 1\pmod 4$ and with a prime factor $p$ satisfying $p\equiv 5$ or $7\pmod 8$.

The conjecture characterizes precisely when submaximal curves are unique. The preceding theorem and computer-assisted examples establish the motivation for the claim, while the equivalence in full generality remains to be proved.