31 problems

Let $p$ be a prime, and let $S_2(\Gamma_0(p))$ denote the space of cusp forms on $\Gamma_0(p)$. For the definite quaternion division algebra of discriminant $p$, consider the theta…

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

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.

Coleman's finiteness conjecture. The set of premodular pairs $(D,m)$ over $\mathbb{Q}$ is finite.

Giampietro--Darmon conjecture. This value should still be algebraic, with its norm prescribed by Equation. The conjecture extends the result beyond the class-number-one setting, wh…

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 $\pi=\otimes_v\pi_v$ be an irreducible cuspidal automorphic representation generated by the fixed Hecke eigenform, and let $\{\mathcal{E}_v\}_{v\in S_\mathcal{O}}$ be local qua…

Let $D$ be a quaternion algebra over $\mathbb{Q}$, let $G$ be the associated algebraic group, and let $S_\mathcal{O}$ be the relevant finite set of places. Let $\pi$ be an irreduci…

Let $\mathcal{D}_2$ denote the set of discriminants under consideration, and consider quaternion algebras $B=\left(\frac{-1,-1}{F}\right)$ over number fields $F$ whose maximal orde…

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…

Let $dim=3$, $4$, or $5$. Let $R(\Q)$ be either an imaginary quadratic field (if $dim=3$), a rational, definite quaternion algebra equipped with an orthogonal involution $\ddagger$…

Let $d\ge1$ have prime-power factorization $d=p_1^{m_1}\cdots p_r^{m_r}$, and let $\mathcal{O}$ be the order described in the source. Let $C_d$ be the number of…

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 $F$ be a totally real number field of narrow class number $1$, let $\mathfrak p$ be a prime of $F$, and let $K_1,K_2$ be almost totally complex quadratic extensions of $F$ in w…

Let $F$ be a totally real number field of narrow class number $1$, let $\mathfrak p$ be a prime of $F$, and let $K/F$ be an almost totally complex quadratic extension in which…

Let $A$ be a quaternion algebra over the field $K$ considered in the paper, and let a maximal order in $A$ have a norm-one group, meaning the subgroup of elements of reduced norm o…

Let $E$ be a totally real field, let $J_E$ be its set of real embeddings, and let $\bot$ be a quaternion algebra over $E$. Let $$ be a cuspidal automorphic representation of…

Let $A/\mathbb Q$ be a modular abelian variety, let $K$ be an imaginary quadratic field, and let $\chi$ be an anticyclotomic character factoring through the ring class field $H_c$…

Chevyrev–Galbraith's linear-bound conjecture. In particular, they conjecture that

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 $p$ 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…