Weyl–Schanuel law for cuspidal automorphic representations of general linear groups

Let $F$ and $n$ be fixed, and let $4\mathfrak{F}(Q)4$ denote the universal family of cuspidal automorphic representations under consideration with analytic conductor at most $Q$. Let ${\rm vol}(\mathrm{GL}_n)$ be the volume of $\mathrm{GL}_n(F)\backslash\mathrm{GL}_n(\mathbb{A}_F)^1$, and let $\widehat{\tau}_\mathfrak{F}(\mathrm{GL}_n)$ be the Tamagawa volume of the universal family.

Weyl–Schanuel law. As $Q\rightarrow\infty$,

$$|\mathfrak{F}(Q)|\sim \mathscr{C}(\mathfrak{F}) Q^{n+1},$$

where

$$\mathscr{C}(\mathfrak{F})={\rm vol}(\mathrm{GL}_n)\cdot \frac{1}{n+1} \widehat{\tau}_\mathfrak{F}(\mathrm{GL}_n).$$

This conjecture predicts the asymptotic number of members of the universal family ordered by analytic conductor, giving an automorphic analogue of Weyl’s law and Schanuel’s theorem on rational points of bounded height. Its status is not specified in the supplied source.