6 problems

Let $h$ be a cuspidal Hecke eigenform on $\mathbf{SL}_{2}(\mathbb{Z})$ of weight $2\ell-n+2$, and let $F_h$ be the function on $\mathbf{G}(\mathbb{A})$ whose constant term along…

Let $G$ be the group supporting quaternionic modular forms, and let $\mathbb X$ be the representation of its Levi subgroup $M\cong\operatorname{GL}_2$ identified with…

For $N\in\mathrm{Sq}_{\mathrm{odd}}$, choose a maximal order $\mathcal O_N$ in the definite quaternion algebra $B_N$. For an eigenform $\varphi\in S(\mathcal O_N)$, its degree is t…

For $N\in\mathrm{Sq}_{\mathrm{odd}}$, choose a maximal quaternionic order $\mathcal O_N$ of level $N$ in $B_N$, and let a nontrivial zero be a zero of an eigenform in…

Let $\varphi\in S(\mathcal O_N)$ be an eigenform, and let $K_\varphi$ be the number field generated by its Hecke eigenvalues. Define the degree of $\varphi$ to be…

Let $r$ be odd. For each $N\in\mathrm{Sq}_r$, choose a maximal order $\mathcal O_N$ in the definite quaternion algebra $B_N$ of discriminant $N$. An eigenform in $S(\mathcal O_N)$…