8 problems

Let $F$ be a number field, let $G=GL_n$, and let $E$ be a coefficient field. Write $X^+_{00}(\operatorname{Res}_{F/\mathbb{Q}}(T_0)\times E)$ for the set of strongly-pure dominant…

Let $\mathscr{C}$ be one of the four types of categories considered in the paper, and let $\mathfrak{cu}(\mathscr{C})$ denote its cuspidal cohomology. The simple-root conjecture. I…

Let $\mathscr{C}$ be one of the four types of categories considered in the paper. Let $L$ be its grading lattice, let $\mathfrak{cu}(\mathscr{C})$ be the cuspidal cohomology, and l…

Fix a quiver $Q$, a Serre subcategory $\mathcal{S}$, and a dimension vector $\mathbf{d}$ for which a simple $\mathbf{d}$-dimensional $\Pi_Q$-module exists. Let…

Strong purity criterion. Cuspidal cohomology with these coefficients is nonzero for some finite level if and only if $\mu$ is strongly pure:

Let $K$ be an imaginary quadratic field, let ${\mathcal O}_K$ be its ring of integers, and let $\Gamma_0({\mathfrak p})$ denote the corresponding congruence subgroup for a prime id…

Long–Maclachlan–Reid conjectures. 1. There exist infinitely many pairs of prime ideals $\mathfrak{p}_1,\mathfrak{p}_2$ in $mathbb Z[i]$ such that

Nonvanishing conjecture. There are infinitely many prime ideals $\mathfrak{p}$ of residue degree one such that