90 problems

Let $n$ and $k$ satisfy $k>n+1$, and let $f\in S_{2k-n}(\Gamma_1)$ be a normalized elliptic Hecke eigenform. Let $L(s,F;st)$ denote the standard $L$-function of a Hecke eigenform…

Let $F\in S_k(\operatorname{Sp}_4(\mathbb{Z}))$ be orthogonal to forms of classical Saito–Kurokawa type, and let $a(T)$ denote its Fourier coefficients for positive-definite half-i…

Let $c\in\mathbf R$, let $k_0\in\mathbf N$ be a scalar weight, and let $F\in M_k^n(\Gamma)$, where $\Gamma\subset\operatorname{Sp}_n(\mathbf Z)$ is a congruence subgroup. For all…

Bergman-kernel growth conjecture. With this notation,

Let $D$) be the quaternion algebra over $\mathbb{Q}$ that is non-split over $\mathbb{R}$ and $\mathbb{Q}_p$ for a prime $p$ and split at all other places. Let $G=GU_2(D)$ be the as…

Let $k$ be such that the indicated spaces are defined, and let $f \in S_{2k-4}(\operatorname{SL}_2(\mathbb{Z}))$ and $g \in S_k(\operatorname{SL}_2(\mathbb{Z}))$ be normalized Heck…

Siegel modularity conjecture. For every $g\in M_{24}$, the function $\Phi_g(\Omega)$ is a Siegel modular form, possibly with a nontrivial multiplier, for a subgroup of…

Let $H_1,H_2,\ldots,H_n$ be the algebraically independent generators of the algebra of invariant differential operators on Siegel upper half-space, and let $q_j$ denote the corresp…

Let $m\geq 1$, and let $f$ and $g$ be Siegel modular forms of genus $2m$, of weights $k>2m$ and $l=k-2m$, with Satake parameters $(\alpha_0,\alpha_1,\ldots,\alpha_{2m})$ and…

Let $f$ and $g$ be Siegel modular forms of genus $2$, of weights $k>4$ and $l=k-2$, with Satake parameters $(\alpha_0,\alpha_1,\alpha_2)$ and $(\beta_0,\beta_1,\beta_2)$, respectiv…

Vanishing-coefficient conjecture. The coefficients of $E(X)$ at $X$ and at $X^{2^n-3}$ are always equal to zero.

Let $(l,m)$ be regular. The endoscopic contribution $e_{\rm endo}({\mathcal A}_2,\mathbb V_{l,m})$ is defined using the cohomology of the moduli space ${\mathcal A}_2$ of principal…

Let $k$ and $n$ be natural numbers with $k-n$ even. Let $f\in S_{2k}(\Gamma_1)$ be a normalized Hecke eigenform, and let $F_{2n}\in S_{k+n}(\Gamma_{2n})$ be the Ikeda lift of $f$.…

Let $\pi$ be unramified at $p$, let $\iota_p$ be the chosen $p$-adic embedding, and let $\xi_J$ be the algebraic integers indexed by subsets $J\subset\{1,\ldots,g\}$, with $a_i$ th…

Let $g=2$, let $\pi$ be an automorphic representation that is neither CAP nor endoscopic, and let $\rho_\pi$ be its associated $p$-adic Galois representation. Write…

Kohnen–Skoruppa conjecture. It is conjectured that

Positive-density refinement. A positive answer to the Fourier-coefficient growth conjecture holds when the growth condition is imposed on a set $S$ with $\delta_+(S)>0$. The paper…

Let $\rho$ be a representation and let $\widetilde{M}_n(k,\rho,\widetilde{\Gamma})$ denote the corresponding space of vector-valued Siegel modular forms. Vector-valued generalizati…

Let $1\leq r\leq n-1$, let $F\in M_k^n(\Gamma)$, and let $\mathcal M$ range over indices in $\operatorname{GL}_r(\mathbf Z)\backslash\Lambda_r$. For a fixed $r$, suppose that the F…

Let $\Gamma_n=\operatorname{Sp}_n(\mathbf Z)$, let $T$ be a positive-definite index for the Siegel Poincaré series $P_{T,\Gamma_n}$, and assume the usual parity condition for scala…

Level-aspect sup-norm conjecture. With $F$ and $\Gamma$ as above, as $N\to\infty$,

Weight-aspect sup-norm conjecture. If $F$ is an eigenfunction of all Hecke operators, then as $k\to\infty$,

Let $n\geq 1$, let $\Gamma\subset\operatorname{Sp}_n(\mathbf Z)$ be a congruence subgroup of level $N$, and let $S^{(n)}_k(\Gamma)$ and $\mathbb B_{k,\Gamma}(Z)$ be as above. Level…

Weight-aspect Bergman-kernel size conjecture. As $k\to\infty$,

Let $k$ and $\nu$ be even integers with $k>4$. Let … be a normalized Hecke eigenform, meaning that $a(1,f)=1$. Let $L(k-1,f,\mathrm{St})$ denote the relevant standard $L$-function,…