2 problems

Rationality conjecture. The value $L(\operatorname{ad}^0(M),1)$, after normalization by some suitable period, should lie in $\mathbf{Q}^{\times}$.

Let $E$ be an irreducible algebraic representation of $G$, let $\pi=\pi_\infty\otimes\pi_f$ be an irreducible cuspidal automorphic representation stable at infinity, and let…