37 problems

Theta-lift nonvanishing conjecture. If $L(f,1)=0$, then

Let $G$) and $G^{\prime}$ be mutual centralizers forming a reductive dual pair inside a symplectic group over a local field $F$. Let $\omega$ be an oscillator representation of the…

Let $R_s(MG_1,MG_2)$ denote the relevant stable-range representation class, let $\theta_s(MG_1,MG_2)(\pi)$ be the stable theta lift, and let $\omega^c$ be the contragredient oscill…

Let $G=G(W_n)$ and let $\psi,\psi'\in\Psi^+(G_n)$ be local Arthur parameters with $\psi\geq_C\psi'$. Suppose that $\pi\in\Pi_\psi\cap\Pi_{\psi'}$. The closure-monotonicity conjectu…

The refined Adams conjecture for ABV-packets. If $\pi\in\Pi_\phi^{\mathrm{ABV}}$, then: (1) for $\alpha\gg0$, $\theta_{-\alpha}^\pm(\pi)\in\Pi_{\phi_\alpha}^{\mathrm{ABV}}$; (2) if…

Let $(G,H)$ be a dual pair, let $\pi$ be a representation of $G$ of Arthur type, and let $\psi^{\max}(\pi)$ be the maximal Arthur parameter attached to $\pi$. Let $\alpha_0$ be the…

Muić's conjecture. The big theta lift $\Theta(\pi)$ to $H(V_m)$ is irreducible, if non-zero.

Stable-range theta lift low-rank conjecture. The theta lift $\theta(\tau)$ is a low rank representation of $G$, meaning that its $U$-support contains no open $L$-orbit.

Let $n$ be such that there is a cuspidal Hecke eigenform of weight $2n+12$ for $\operatorname{SL}_{2}(\mathbb{Z})$, and let $\pi$ be the associated level-one algebraic automorphic…

Let $F$ be an archimedean local field, let $G$ be the group of type $\mathsf{G}_2$, and let $K/F$ be the quadratic algebra used to define $G'_J$. Let $\chi:K^\times\to S^1$ be a un…

Let $\pi$ be a representation of $G$ and let $\psi^{\max}(\pi)$ denote its maximal Arthur parameter. Let $d^\pm(\pi,\psi^{\max}(\pi))$ be the corresponding going-up or going-down i…

Kudla's arithmetic modularity conjecture. The formal generating function $Z(g,\varphi)_K$ converges absolutely and defines an element in

Let $F$ be the global field implicit in the automorphic setting, let $W$ be a symplectic space, let $V$ be a quadratic space, and let $\sigma\subset \mathcal{A}_{cusp}({\rm Sp}(W))$…

Let $E/k$ be the relevant quadratic extension, let ${\rm U}(V)\times {\rm U}(W)$ be a dual pair in the stable range, and let $\pi\subset\mathcal{A}_{\rm cusp}({\rm U}(V))$ have A-p…

Adams's conjecture. The representation $\theta_{-\alpha}(pi)$ is contained in the A-packet parametrized by

Let $\tau$ be as in the surrounding discussion, and let $\theta_{\Delta(\tau,\ell)}\in\Theta_{\Delta(\tau,\ell)}$ and…

Let $\theta_{\Delta(\tau,\ell)}$ and $\theta_{\Delta(\tau,2n+\ell)}$ be theta functions, and let $\Phi(\theta_{\Delta(\tau,\ell)},\theta_{\Delta(\tau,2n+\ell)},s)$ be the proposed…

Let $1 \leq ell<n$. Let $\theta{\Delta(\tau,ell)}$ and $\theta{\Delta(\tau,2n+ell)}$ be given theta functions, and let $f'{\Delta(\tau,2n),s}$ be a section of the relevant induced represent…

Theta-packet Arthur multiplicity conjecture. There is a decomposition

Theta-packet induction conjecture. The representation $I(\pi_0)$ is irreducible for every $\pi_0\in\Pi_{\psi_0}^\theta(G_0)$. Moreover, if $I(\pi_0)\simeq I(\pi'_0)$ for…

Let $F$ be a real or complex local field, let $G$ be an even orthogonal or unitary group over $F$, and let $\psi$ be a local $A$-parameter with bounded image on the Weil group. Let…

Let $({\rm G},{\rm G}')$ be an irreducible dual pair in ${\rm Sp}(W)$ with ${\rm rk}({\rm G})\leq {\rm rk}({\rm G}')$. Let ${\rm Chc}^{*}$ be the map transferring invariant distrib…

Folklore conjecture on theta correspondence. The inclusions

Let $\mathrm{GSp}^+(V_4)$ and $\mathrm{GSO}(U)$ be the similitude groups in the local Weil representation, and let $\widetilde{\Theta}(\pi^+)$ be the smooth theta lift associated w…

The refined character formula conjecture. One expects