Rigid Connections on the Projective Line with Elliptic Toral Singularities

We generalize two studies of rigid $G$-connections on $\pp$ which have an irregular singularity at origin and a regular singularity at infinity with unipotent monodromy: one is the work of Kamgarpour-Sage which classifies rigid homogeneous Coxeter $G$-connections with slope $\frac{r}{h}$, where $h$ is the Coxeter number of $G$, and the other is the work of Chen, which proves the existence of rigid homogeneous elliptic regular $G$-connections with slope $\frac{1}{m}$, where $m$ is an elliptic number for $G$. In our work, similar to Chen, we look for rigid homogeneous elliptic regular $G$-connections, but we allow the slope to have a numerator greater than $1$. However, for the present purpose, we essentially restrict to the case where $G$ is either $\Sp_{2n}$ or $\SO_{2n+1}$. For $\Sp_{2n}$, we show that Kamgarpour-Sage connections and Chen connections exhaust all the rigid homogeneous elliptic regular connections; furthermore, we write down explicit matrices for all rigid homogeneous elliptic regular $\Sp_{2n}$-connections. For $\SO_{2n+1}$-connections, having introduced the notion of generalized Chen connections, we classify all rigid connections of this type. We conjecture that any rigid homogeneous elliptic regular $\SO_{2n+1}$-connection is in this form. We also construct an explicit example of a rigid non-Coxeter $\SO_9$-connection with slope $\frac{3}{4}$, as well as an explicit construction of Chen $\SO_{2n+1}$-connections with slope $\frac{1}{2}$