602 problems

Let $\phi$ be a suitable test function, and let Theorem denote the density theorem established in the paper for the family of elliptic curves. Unrestricted-support conjecture. Theor…

Let $G_r$ be an unramified base-change form of the quasi-split group $G$, with Hecke algebras $\mathcal{H}(G_r)$ and $\mathcal{H}(G)$, and let $\mathfrak{Z}^{\textnormal{st}}(G_r)$…

Let $G$ be a reductive group satisfying the stated restriction that, for every root $\check{\alpha}\in\Lambda^{\vee}$, the map $\Lambda\to\mathbb{Z}$ given by…

Let $L/\mathbb Q$ be an imaginary quadratic extension with discriminant $-D$, let $\Lambda$ be a character of $L^\times\backslash\mathbb A_L^\times$ trivial on $\mathbb A^\times$,…

Let $G=\mathrm{GSp}_4$, let $\pi=\otimes_v\pi_v$ be an irreducible, unitary, cuspidal, generic automorphic representation of $G(\mathbb A)$ with trivial central character, and let…

Let $\phi$ be a Hecke–Maass form for the full modular group, let $K$ be a compact subset of the modular surface $\mathbb{X}$, let $2<p\le\infty$, and let $\epsilon>0$. Then Iwaniec…

Let $G$ be the reductive group considered in the paper, let $\Gamma\subset G(\mathbb{Q})$ be a congruence subgroup, and let $\Pi(K_\infty)$ denote the set of $K_\infty$-types. For…

Let $m,n\in \mathbb Z$ with $1\le m\le n-1$ and let $\delta\in \mathbb Z_{\ge0}$. For $x\in\mathbb C$ and $q\in\mathbb Z_{\ge0}$, write … Let $\what{n}{\alpha}{s}$ be the Mellin tr…

Let $\Delta=(T_i,d_i,\lambda_i,\eta_i)_{1\leq i\leq k}\neq\Sigma_{\lambda,\eta}$ satisfy the setup and conditions of the cited theorems, and let $R_0(\Delta)$ be the refined expone…

Refined exponent conjecture. Under all the same conditions, the bound in Theorem may be tightened to

Let $\Delta=(\Delta_i)_i=(T_i,d_i,\eta_i,\lambda_i)_i$ be a shape of rank $N$, and let $\varphi_v$ be a test function on the twisted group $\widetilde{G}(N)$. Stable Transfer conje…

Let $\Delta=((T_i,d_i,\eta_i,\lambda_i))_i$ be a shape and let $\varphi_v$ be a trace-positive test function on $G_v$. For every $\psi=\bigoplus_i\psi_i[d_i]$ in $\Delta$, with…

Michel–Venkatesh adelic mixing conjecture. If $|D_i|\to\infty$ and $q_i\to\infty$ as $i\to\infty$, then

Let $d\to\fty$ along ${\mathbb D}^\flat$, let ${\sf R}_d^\Delta([\mathfrak s])$ be the shifted diagonal Heegner packet, let $\mu$ be the quotient measure on ${\sf S}^2$, and let…

Let $n\geq 1$. Let $F^+$ be a totally real field, and let $\pi$ be a cuspidal automorphic representation of $\mathrm{GL}_2({\mathbf A}_{F^+})$ without CM, with $\pi_\infty$ essenti…

Let $n\geq 1$. Let $F^+$ be a totally real field, and let $(\pi,\chi)$ be a RAESDC automorphic representation of $\mathrm{GL}_2({\mathbf A}_{F^+})$ without CM. Symmetric power func…

Let $r\geq 1$. Let $F$ be a CM field, and let $(\pi_1,\psi_1)$ and $(\pi_2,\psi_2)$ be RAECSDC automorphic representations of $\mathrm{GL}_2({\mathbf A}_F)$ and…

Let $M$ be the Rapoport–Zink space of strict $\pi$-divisible $O_F$-modules with quasi-isogeny to $\mathbb{X}$, and let $Z_1,Z_2\to M$ be the closed immersions associated with the m…

Multiplicity conjecture. The unique distinguished element belongs to $\Pi_\phi(G)$ if and only if

Let $w\in{}^MW$, $\kappa\in X^\star(T^c)^{M_\mu,+}$, and let $\chi:T(\mathbb{Z}_p)\to {\overline{F}}^\times$ be a finite-order character. For a character $\lambda$ of $T^+$ or…

Let $k$ be the number field and $G$ a $k$-split reductive group, with complex dual group $G^\vee(\mathbb{C})$. Let $\rho\colon G^\vee(\mathbb{C})\to\operatorname{GL}_n(\mathbb{C})$…

Let $X$ be an affine spherical variety over $k$ with trivial arithmetic multiplicity. Let $\mathcal S(X(\mathbb A_k))$ be a Schwartz space in the sense described in the source, let…

Corrected Ruelle zeta-function conjecture. For any $x_1,\dots,x_k\in\mathbb F_q\setminus\{0,1,t\}$ with $k\ge 1$, the series

Equivariant bundle conjecture. There exists an $F$-equivariant vector bundle $(\mathcal E,g)$ of rank $N$, defined over $\mathbb Q$, such that for $z\in X_n$ the eigenvalue of…

Crystallinity conjecture. If $pi$ is unramified at $p$, then