15 problems

Improved error-term conjecture. In all terms above, the error term $o(X)$ can be replaced by $O(X^{1/2+\varepsilon})$ for any $\varepsilon>0$, where the implied constant only depen…

Meromorphic-power conjecture. Then there is an integer $N$ such that

Langlands–Shelstad transfer conjecture. For every $G_v'\in{{\mathscr E}}(G_v)$, this transfer function exists and the resulting map sends ${{\mathscr H}}(G_v,\zeta_v)$ continuously…

Trace formula. Assume that $X$ is proper and $K$ is admissible. Then

Let $\mathfrak a_{\mathbb R}^{*,+}$ be the positive dual cone, let $\overline{\mathfrak a_{\mathbb R}^{*,+}}$ be its closure, and let…

Let $q=p^r$ be the cardinality of the residue field of ${\mathfrak p}$, and let $T(m,f)$, ${\mathfrak{KP}}_{\operatorname{cla}}(q^m)$, $\alpha$, $c_1$, $c_2$, $O_\gamma$,…

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 $\widetilde{X}$ be the symmetric space associated with $G(\mathbb{R})^1$, let…

Let $Z$ be a spatially compact stationary globally hyperbolic spacetime, and let $D_Z$ have discrete spectrum in $\operatorname{Char}(\Box)$ with eigenvalue counting function … Ass…

Let $E \geq F' \geq F$ be a tower of number fields, let $S$ be a finite set of places of $F$ containing the infinite places, and let $S'$ and $S_0$ be the places of $F'$ and $E$ ab…

Let $F$ and $E$ be number fields as in the paper, let $n$ be a positive integer, and let $_{E/F}(\Sigma_{\phi}^{S_0}(X))$ denote the associated base-change test function. For a cus…

Let $\mathbf G$ be a connected reductive linear algebraic group over $\mathbb Q$, let $\eta:\mathbf G\to\operatorname{GL}(n)$ be a faithful rational representation, and let…

The weak orthogonality relation. For positive integers $m_1,\ldots,m_{N-1}$,

The orthogonality relation. For positive integers $m_i,n_i$,

Rapoport–Langlands conjecture. The special points of the Shimura variety satisfy