17 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 $N$ be square-free and let $k$ be odd. For a newform $F_f\in \mathrm{SK}_{k+1}^{new}(N)$, let $F_f^\circ$ denote its pullback to a product of two modular curves. Its $L^2$-mass…

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…

For an odd integer $k>0$, let $g\in S_{2k}$ be a normalized Hecke eigenform for $\mathrm{SL}_2(\mathbf Z)$, let $F_g\in S^2_{k+1}$ be its Saito–Kurokawa lift, and define its pullba…

Conjectural formula for $C_\infty$.

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…

Let $F$ be a Siegel modular form of degree $2$, and let the Maass space consist of the forms whose Fourier coefficients satisfy … for all $n,m,r\in\mathbb{Z}$. A different Spezials…

A Siegel modular form has unimodular Satake parameters when all of its suitably normalized Satake parameters lie on the unit circle. Adjusted Ramanujan conjecture. If a Siegel modu…

Let $g\in S_{2k}$ be a holomorphic cuspidal Hecke eigenform with $k$ odd, let $F_g\in S_{k+1}(\operatorname{Sp}_4(\mathbb{Z}))$ be its Saito–Kurokawa lift, and let $N(F_g)$ denote…

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

Let $\mathcal{S}_{2k}^*$ be the relevant space of Siegel cusp forms, let $F\in\mathcal{S}_{2k}^*$ be not a Saito–Kurokawa lift, let $a(d,1;F)$ be its Fourier coefficient, let…

Let $\Gamma\subset \operatorname{GSp}_2(\mathbb{Q})$ be an arithmetic subgroup, and set … Assume that $a\ge b\ge0$ and that $V_\Gamma$ has Hodge numbers satisfying … Suppose that…

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…

Let $S^{\rm lift}_{j,k}(\Gamma_2[2])$ denote the subspace of lifted Siegel cusp forms, and put $k=2m+4$. Let $s[\lambda]$ denote the irreducible $\mathbb S_6$-representation indexe…