11 problems

Nonsplit Bianchi quantum unique ergodicity conjecture. If $H\in C_b(\Gamma\backslash\mathbb{H})$ is fixed, then

Let $k$ be an odd positive integer, and let $W_{k,k}^{1}$ and $W_{k,k}$ denote the subspaces defined in the paper for the parabolic cocycle construction associated with…

Let $E$ be one of the CM elliptic curves $E=\mathbb{C}/\mathbb{Z}[i]$ or $E=\mathbb{C}/\mathbb{Z}[\rho]$, let $\mathfrak{p}$ be the relevant prime ideal, let $T_k$ be the coefficie…

Let $d$ be a positive square-free integer, let $\boldsymbol{\Gamma}_d$ denote the relevant Bianchi group, and let $\mathbb{H}^3$ be hyperbolic three-space. Define the finite set…

Let $K$ be an imaginary quadratic field, let $p$ be a rational prime, and let $G_K$ be the absolute Galois group of $K$. A mod $p$ eigenvalue system means a system of Hecke eigenva…

Let $\Gamma_0({\mathfrak p})$ be a congruence subgroup inside a fixed Bianchi group, where ${\mathfrak p}$ is a prime ideal of residue degree one. Write $N{\mathfrak p}$ for its no…

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…

Nonvanishing conjecture. This restricted differential is nonzero.

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

Cohomological elliptic-curve conjecture. Non-vanishing of $H^1(\Gamma_0(\mathfrak p),\mathbb C)$ is conjectured to be related to the existence of certain elliptic curves, or more g…