102 problems

Oliver's p-group conjecture.

Lapid–Mao's local conjecture. For every $\eta\in E_v^\times$ such that $\bar\eta=-\eta$, one has

Let $\chi=\chi^{\phi,\lambda}$ be a primary irreducible character of $\mathrm{GU}_n(q)$, where $\phi$ is an orbit other than $\{\mathbf{1}\}$ and $\mathbf{1}$ is the trivial charac…

Let $(r,s)$ be a signature and let $I\subset\{0,\dotsc,m\}$ be nonempty. Let $M_I^{\mathrm{naive}}$ be the naive local model, $M_I$ the closed subscheme cut out by the wedge and st…

Let $F$ be a $p$-adic field, let $E$ and $K$ be quadratic field extensions of $F$, and let $V$ be an $n$-dimensional skew-hermitian space relative to $E/F$. Set $V_K=V\otimes_F K$,…

Arithmetic transfer conjecture for odd type $(r,0)$. There exists $\varphi'\in C_c^\infty(G')$ transferring to $(c_r^2\\,\mathbf{1}_{K_n^{[r]}\times K_{n+1}},0)$ such that, for eve…

Let $\varphi'\in\mathcal{H}_{K'^{{{{{{{{\flat}}}}}}}}}\otimes\mathcal{H}_{K'}$, and let…

Amenability equivalence conjecture. The following conditions are equivalent:

Let $F_0$ be the $p$-adic field with ring of integers $O_{F_0}$ and residue-field cardinality $q$. Let…

Let $p$ be an odd prime, let $\mathscr{W}$ be the rigid analytic space parametrizing continuous characters of $\mathbb{Z}_p^\times$, and write $T(w)=w(\exp(p))-1$ for the $T$-coord…

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

Let $\pi$ be an automorphic representation with associated Galois representation $\rho_\pi$, let $\chi$ be a character, and let $\mathcal{K}$ be the relevant number field. Write…

Let $F$ be a CM field, let $\Sigma$ be a CM type, and let $F^+$ be the maximal totally real subfield of $F$. For each signature $I:\Sigma\to\{0,1,\ldots,n\}$, let $U_I$ be the corr…

Let $\phi_n \in \Phi_\mathrm{temp}(\mathrm{U}_{n}(\mathbb{R}))$ and $\phi_{n+1} \in \Phi_\mathrm{temp}(\mathrm{U}_{n+1}(\mathbb{R}))$ have decompositions … … where…

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 $\Delta=((T_i,d_i,\eta_i,\lambda_i))_i$ 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…

Fix an imaginary CM field $F$ in which $p$ is unramified, let $F^+$ be its maximal totally real subfield, and set … Let $({\mathbb Z}^2_+)^S_0$ consist of tuples $a$ satisfying…

Pappas's spin flatness conjecture. The scheme $M_I$ is flat over $\mathcal O_E$; equivalently,

Let $E$ be the quadratic imaginary field in the paper, let $G_E$ be its absolute Galois group, and let $\pi$ be an automorphic representation of $\operatorname{U}(m)$ with regular…

Let $F$ be a $p$-adic field, let $E/F$ be a degree-two extension, let ${\rm U}(n)$ be the corresponding unitary group, and let $\pi$ be a representation of ${\rm GL}_n(E)$. A repre…

Genericity transfer conjecture. If $\pi_1\otimes\pi_2$ is an irreducible subrepresentation of this function space and $\pi_1$ is generic, then $\pi_2$ is also generic.

Let $\Gamma=([n],w)$ be an arbitrary hypergraph with non-negative weights. Let $\mathcal{KMP}_k$ be the discrete KMP process with $k$ indistinguishable particles on $\Gamma$, and l…

Let $\Gamma=([n],w)$ be an arbitrary hypergraph with non-negative weights. For each irreducible representation $\rho$ of $\mathrm{U}(n)$, let $\lambda_{\min}(\Gamma,\rho)$ denote t…

Let $\mathrm{GL}^\varepsilon_n$ denote the general linear group when $\varepsilon=1$ and the general unitary group when $\varepsilon=-1$, and let $\mathrm{GL}^\varepsilon_1$ be its…