16 problems

Let $m\geq 1$, let $N\geq 1$ be coprime to $m$, and let $k$ be an even integer with $k=12$ or $k\geq 16$. Write $T'_m(N,k)$ for the normalized Hecke operator. Rouse's generalized L…

Let $X$ be a compact manifold with a one-codimensional foliation ${\cal F}$ and an ${\cal F}$-compatible flow $\phi$. Assume that fixed points and periodic orbits are non-degenerat…

Let $G$ be a reductive group with maximal compact subgroup $K$, symmetric space $X=G/K$, and let $\Gamma\subset G$ be a discrete subgroup of finite covolume. Let $A$ be the split c…

Let $\phi_t$ be an Anosov flow with stable foliation $\mathcal{F}_s$. Write $\bar H^p(\mathcal{F}_s)$ for the reduced cohomology of the stable foliation, let…

Let $X$ be a compact manifold with a one-codimensional foliation ${\cal F}$ and an ${\cal F}$-compatible flow $\phi$, with the non-degeneracy assumptions on fixed points and closed…

Let $X$ be in the foliated dynamical setting above, let $\phi$ be the flow, and let $x$ be a fixed point, so that $Y_{\phi,x}=0$. For nonzero $t$, define the fixed-point contributi…

Let $h_k$ be an analytic function on $|\operatorname{Im}(r)|\leq 1+\delta$ satisfying $h_k(-r)=h_k(r)$ and $|h_k(r)|\leq M[1+|\operatorname{Re}(r)|]^{-4-\delta}$, and define … For…

Let $(M,\mathcal{F})$ be a closed smooth foliated manifold with $\operatorname{codim}\mathcal{F}=1$. Let $\phi=\{\phi^t\}$ be a foliated flow whose closed orbits are simple, whose…

Let $\Gamma$ be a Fuchsian group, let $\Sigma$ be a double coset of $\Gamma$, and let $X$ be a $\Gamma$-conjugacy class contained in $\Sigma$. For $M\in X$, define the discriminant…

Assume that $\mathsf{G}$ is a connected simply connected semisimple algebraic group over $\mathsf{F}$, with $\Xi=\\{1\\}$. Let $\pi_{w'}$ be a finite-length supercuspidal represent…

The local-global multiplicity conjecture. If $t\asymp X$ is admissible, then

Let $\Gamma(2)'$ be the commutator subgroup of $\Gamma(2)$, and let $\operatorname{Tr}(\Gamma(2)')$ denote its trace set. The commutator width of an element is the minimum number o…

Let $I(f)$ be the trace-formula distribution for a test function $f$, and let $I_{geom}(f)=m_{geom}(\theta_f)$ be its geometric expansion. Define $I_{spec}(f)$ by the cuspidal spec…

Let $(G,X)$ be a Shimura datum with reflex field $E$, let $K=K_pK^p\subset G(\mathbb A_f)$ be neat with $K_p$ hyperspecial at a prime $p$ coprime to $\lambda$, and let…

Let $n$ be the degree of a Siegel cusp form. In the trace formula for the dimension of spaces of Siegel cusp forms, the unipotent contributions corresponding to the partitions…

Let $G$ be the reductive group under consideration, let $\pi$ be a regular discrete series representation of $G(\mathbb{R})$, and let $e_{\pi}$ denote the stable cuspidal test func…