8 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 $N$ tend to infinity through odd square-free integers. For an SK lift $F_f\in \mathrm{SK}_{k+1}^{new}(N)$, let $F_f^\circ$ be its pullback and let $N(F_f)$ be its normalized…

Let $\mathrm{SK}_k(N)$ denote the space of Saito-Kurokawa lifts of weight $k$ and level $N$, and let $F\in\mathrm{SK}_k(N)$ be an $L^2$-normalized newform. Write…

Let $\mathrm{SK}_k(N)$ denote the space of Saito-Kurokawa lifts of weight $k$ and level $N$, let $B_k^{SK}(N)$ be an orthonormal basis, and define … with…

Let $F\in\mathrm{SK}_k$ be a Hecke eigenform, let $F^\circ$ denote its pullback to the diagonal, and suppose that $F^\circ\ne0$ and $\lVert F^\circ\rVert_2=1$. The Saito–Kurokawa p…

Let $F$ be a totally real number field, let $n$ be even, and let $\pi\simeq\otimes'_v\pi_v$ be an irreducible cuspidal automorphic representation of $\mathrm{PGL}_2(\mathbb A)$ wit…

Asymptotic norm conjecture. As $k$ tends to infinity,

Let $\mathbb V_{m,m}$ be the equal-weight local system on $\mathcal A_2[2]$, set $k=2m+4$, and write $\tau_{N,k}=\dim S_k(\Gamma_0(N))^{\rm new}$ and…