13 problems

Shapiro's conjecture. $(A,\sigma)$ admits a decomposition as a tensor product of $r$ quaternion algebras with involution in which each quaternion algebra is split.

Main conjecture. An irreducible smooth representation $\theta$ of $\operatorname{G}_n$ with $n>2$ is $\operatorname{H}_{1,n-1}$-distinguished if and only if $\theta$ is either…

Let $D$ and $\bar D$ be the quaternion algebras introduced earlier, let $\Sigma$ be the specified finite set of places, let $\mathbf{F}$ be the space of mod-$p$ algebraic functions…

Let $\dim=3$, $4$, or $5$. Let $\mathcal{A}$ be either an imaginary quadratic field when $\dim=3$, a rational definite quaternion algebra equipped with an orthogonal involution…

Chinburg–Reid–Stover's conjecture. In the notation above,

Let $D_1,D_2$ be coprime fundamental discriminants, let $\tau_1,\tau_2$ be the corresponding real quadratic points, and let $p\leq 13$ be prime. Let $J_p(\tau_1,\tau_2)$ be the Dar…

Let $F$ be a totally real number field of narrow class number $1$, let $\mathfrak p$ be a prime of $F$, and let $K_i/F$ be quadratic extensions in which $\mathfrak p$ is inert, wit…

Let $p$ be a prime and let $B_p$ be the quaternion algebra over $\mathbb{Q}$ ramified precisely at $p$ and $\infty$. Let $\mathcal{O}$ and $\mathcal{O}'$ be maximal orders of $B_p$…

Let $B$ be a definite quaternion algebra ramified at exactly one finite prime $p$, and let $N=pM$ with $p mid M$ be square-free. Write … For $D\in\mathbb{N}$, let $-D$ be a fundame…

Let $p$ be the prime defining the quaternion algebra $B_p$, and let $\mathcal{O},\mathcal{O}'\subset B_p$ be maximal orders of different types. For an integer $b>0$, say that…

Let $F$ be a totally real number field, let $\boldsymbol{c}$ be an odd square-free level, let $g\in S_{\mathbf{2}}(\boldsymbol{c})$ be a newform, and let $v_g$ be a Hecke eigenvect…

Let $R$ and $F$ be as in the Krull-dimension-three non-finite-generation conjecture, so $R$ is a finitely generated integral domain of Krull dimension at least $3$ and $F$ is its f…

Let $\Sigma=\{q,\infty\}$, where $q$ is a prime as in the preceding construction, and let $B^{(1)}$ be the norm-one subgroup of the quaternion algebra $B$. Write $C(\Sigma)$ for th…