Determine the generic automorphism group in the remaining signature cases

Determine the generic automorphism group of the universal abelian variety over the basic locus of a split unitary Shimura variety when the signature $(m_0,n_0-m_0)$ does not satisfy the hypotheses of Theorem 1.1, and establish whether it is larger than $\{\pm 1\}$.

Background

The main theorem proves that, under specified restrictions on the prime $p$ and the signature $(m_0,n_0-m_0)$, the generic automorphism group of the universal abelian variety on the basic stratum is exactly $\{\pm1\}$. The paper does not resolve the complementary signatures excluded from that theorem.

The authors explicitly expect the generic automorphism group in those remaining cases to be larger than $\{\pm1\}$. The remark also identifies a potentially relevant exceptional situation: when $(g,p)=(2,2)$ and $m\in\{1,n-1\}$, the congruence subgroup $V_N$ may contain nontrivial $2$-torsion, whose finite subgroups are expected to contribute groups of the form $(\mathbb Z/2\mathbb Z)^k$.