5 problems

Let $f_d$ be a quadratic twist and let $\alpha_{f_d}(p)$ and $\beta_{f_d}(p)$ be its Satake parameters at a prime $p$. Ramanujan–Petersson conjecture. The conjecture states that ……

Let $F_1,F_2\in S_k(\Gamma_2)$ be Siegel cusp forms with Fourier–Jacobi expansions … Here $\langle\phi_n,\psi_n\rangle$ denotes the Petersson scalar product of the Jacobi cusp form…

Let $\Gamma$ be a non-arithmetic lattice, and let $\psi_{j,m}^{(k)}(s)$ denote the Fourier coefficients of its Eisenstein series, with $m$ a Fourier-index parameter and $s$ in the…

Ramanujan--Petersson conjecture. The value

For each place $v$, let $G_v$ be the group of local points of a semisimple algebraic group $G$, and let $\mathfrak{q}_v(G)$ denote the spherical integrability exponent of the repre…