26 problems

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…

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 $G$ be a reductive group over the local field $F$, let $H$ be an endoscopic group, and let $\gamma\in H(F)$ be strongly $G$-regular semisimple. Write $\Lambda_{G,H}(\gamma)$ fo…

The fundamental lemma. The orbital integrals satisfy

Endoscopic equivalence conjecture. There exists an endoscopic equivalence as in Theorem $$ without any hypothesis on $\ell$.

Let $F$ be a non-archimedean local field, let $G$ be a connected reductive $F$-group, and let $\Pi_{\varphi}(G)$ be a discrete series $L$-packet of irreducible smooth representatio…

Let $G=G(V)$ be the classical group associated to a quadratic space $(V,q_V)$, and let $\psi$ be a local Arthur parameter of $G$. For an endoscopic group $G'$ of $G$ and a local Ar…

The epsilon-character conjecture. The function $\omega_{\phi',\rho_X}$ is well defined, independent of the extended endoscopic triple and the lifting, descends to a function on…

Let $(G,H)$ be a strongly tempered spherical pair that is the Whittaker induction of $(G_0,H_0,\xi)$, with $G$ admitting a quasi-split pure inner form $G_{qs}$. Assume that…

Let $\phi:L_F\to{}^LG$ be a tempered Langlands parameter, let $\tilde s\in\tilde S_\phi^{[z]}$ be semisimple, and let $\mathfrak e$ be the associated endoscopic datum. Choose a $z$…

Let $G$ be a reductive group with cover $G(F)_{x_G}$, let $(\dot s,\mathcal H)$ be a refined endoscopic datum for $G$, and let $H(F)_{x_H}$ be the corresponding endoscopic cover. L…

Let $G$ be a reductive group, let $\mu$ be a cocharacter with reflex field $E_{\mu}$, and let $(H,\mathcal{H},s,\eta)$ be an endoscopic datum such that an $L$-parameter $\phi$ fact…

Weighted Langlands–Shelstad transfer conjecture. For each $G$ and $M$, there is a function $\hat{S}^{{\tilde G}'}_{{\tilde M}'}(\delta',f')$ on … such that

Let $(G,b,\mu)$ be the local Shimura data appearing in the paper, let $\mathcal{H}^{\mathfrak{e}}$ be an endoscopic datum, and let $\operatorname{Mant}_{G,b,\mu}$ and…

Smooth transfer conjecture. For any relative endoscopic datum $(\xi,\alpha,\beta)$ and any $f\in C_c^\infty(\mathfrak g_1)$, there exists…

Let $F$ be a non-archimedean local field, let $G$ be a reductive group that splits over a tame extension of $F$, and assume that the residue characteristic is sufficiently large fo…

Endoscopic classification conjecture. The set $\Pi_{irr,temp}({\mathrm{GU}}_{2n})$ is a disjoint union of finite local tempered Vogan $L$-packets:

Waldspurger's conjecture. (1) For every $f\in C_c^\infty(\mathfrak{g})$, there exists a transfer $f^H\in C_c^\infty(\mathfrak{h})$. (2) For compatible Fourier transforms…

Let $G$ be an arbitrary reductive group, let $G^*$ be its quasisplit inner form, and let $J\subset G$ be an arbitrary parahoric subgroup. Put $L=J/J^+$ and let $W_L$ be the Weyl gr…

Let $\mathcal{A}_2$ be the moduli space of principally polarized abelian surfaces, let $\mathbb{V}(\lambda)$ be the local system of weight $\lambda=(l,m)$, and let…

Structural conjecture for the strict endoscopic part. The strict endoscopic part contributes as a finite part of theta-lifts of cuspidal automorphic representations of…

Let $\tau$ be the self-dual cuspidal automorphic datum and let the global Arthur parameters $\psi_1$, $\psi_2$, and $\psi=\psi_1\boxplus\psi_2$ determine the relevant classical gro…

Stable Bernstein center character identity conjecture. For any $\eta$-endoscopic transfer $h^{\mathbf{H}}$ of $h$, the function $z_\tau^{\mathbf{H}}\ast h^{\mathbf{H}}$ is a twiste…

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…