11 problems

Murty–Patankar conjecture. The abelian variety $B$ has absolutely simple reduction for a density one set of primes up to a finite extension if and only if its endomorphism ring is…

Let $F$ be the local field and let $\lambda$ index a category $\operatorname{Rep}(\operatorname{GL}_n(F))^{\lambda}$. Let $E_{\lambda}$ be the endomorphism ring of the associated p…

Let $g$ be even and let $\mathsf{A}^{(g)}_a$ be the Jacobian in the family described above, with $a$ ranging over the parameter space. The geometric endomorphism ring is expected t…

Coleman's discriminant conjecture. The quantity $\operatorname{discr}(A)$ is uniformly bounded for abelian varieties $A$ of bounded dimension defined over number fields of bounded…

Let $C$ be a symmetrizable Cartan matrix, $D$ a symmetrizer, $\Omega$ an orientation, and $H=H(C,D,\Omega)$. Let $\mathbf{r}$ be an $H$-Schur root, so that…

Let $K$ be a number field and let $A/K$ be a principally-polarizable abelian surface whose absolute endomorphism ring satisfies … Let $\pi_{\operatorname{split}}(A/K,z)$ count prim…

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…

Let ${\bf R}$ be a Noetherian local ring, let $E$ be a local ${\bf R}$-module, and let ${\Lambda}=\operatorname{End}_{\bf R}(E)$. Suppose that $E$ is ${\Lambda}$-cyclic and has fin…

Let $X$ be an absolutely simple abelian variety over a number field $K$. Say that a prime of $K$ is a splitting prime for $X$ when the reduction of $X$ at that prime is not simple.…

Let $M$ be the dimension-$n$, rank-$2n$ T-motive defined by $Te_* = \theta e_*+A\tau e_*+\tau^2e_*$, where $A\in M_n(\mathbb C_\infty)$ is skew-symmetric, and let…