Let $n\in\mathbb{N}$ be a positive integer, and let $[n] = \{1,\ldots,n\}$. We set $S_n$ for the set of all bijections $\varphi:[n]\to [n]$. Let $G= ([n], E)$ be a simple, undirected graph, and let $\varphi\in S_n$ We define the label distance number of $G$ in the following way: $$\lambda(G) = ...

Is there a way to characterize (e.g. by their local components) the cuspidal automorphic representations that arise from elliptic curves? There is a procedure to go from elliptic curves to cuspidal automorphic representations of $GL_2$ but can the image be characterized?

Let $k$ be a field of characteristic 0 (not necessarily algebraically closed), let $G$ be a connected split reductive group over $k$ and let $\mathfrak{g}$ be the Lie algebra of $G$. Let $X \in \mathfrak{g}$ be a nilpotent element. Does there exist a unipotent subgroup $U$ of $G$ such that $X$ ...