14 problems

Let … be a Siegel embedding of a Hodge-type Shimura datum, and let $K_1\subset G_1(\mathbb{A}_f)$ be compact open. Let $\mathrm{Sh}=\mathrm{Sh}_{K_1}(G_1,X_1)_E$ be the canonical m…

Let $E$) be a number field, let $A/E$ be an abelian variety, and let $\mathrm{MT}(A)$ denote its Mumford–Tate group. A point of an algebraic group is unipotent if it is unipotent i…

A Fuchsian group with cusps has a boundary map acting piecewise on ${\mathbb S}$, whose partition points determine the rectangular structure of the natural extension on geodesics.…

Serre's conjecture. The variety $V$ has infinitely many primes of good ordinary reduction.

Let $B$ be a smooth variety over a field $k$, let $K=k(B)$, and let $A$ be an abelian $K$-variety. Assume that $A$ has semistable reduction at every point of $B^{(1)}$. For any pro…

Let $A$ be an abelian variety over a number field $F$. A reduction of $A$ at a non-archimedean place is ordinary when it has good ordinary reduction, and let $V(A,L)$ denote a set…

Let $\Omega$ be an arbitrary Hessenberg type, and let $NRS(\Omega)$ denote the set of NRS-matrices of type $\Omega$. Call a matrix $\varsigma$-reduced according to the paper's redu…

Let $NRS\big(\langle 1,2|1,1,3\rangle\big)$ be the family of Hessenberg matrices of type $\langle 1,2|1,1,3\rangle$, and call a matrix $\varsigma$-reduced according to the paper's…

Let $NRS\big(\langle 0,1|1,0,2\rangle\big)$ be the family of Hessenberg matrices of type $\langle 0,1|1,0,2\rangle$, and call a matrix $\varsigma$-reduced according to the paper's…

Let $K$ be a number field and let $\phi:{\mathbb P}_K^1\to{\mathbb P}_K^1$ be a rational map. Its minimal resultant is an integral ideal of ${\mathcal O}_K$ supported on the places…

Ordinary reduction conjecture. The ordinary reduction theorem holds for all abelian varieties of all dimensions defined over a number field.

Let $NRS\big(\langle 1,2|1,1,3\rangle\big)$ denote the NRS-matrices of Hessenberg type $\langle 1,2|1,1,3\rangle$. A Hessenberg matrix in this family is called reduced when it sati…

Let $H_{\langle 0,1|1,0,2 \rangle}^{(1,0,1)}(m,n)$ be the family of Hessenberg matrices of type $\langle 0,1|1,0,2 \rangle$, with integer parameters $m$ and $n$. A matrix in this f…

Let $R$ be a local Gorenstein ring with infinite residue field, let $I$ be an $R$-ideal with $g=\operatorname{ht} I>0$ and $\ell=\ell(I)$. Assume $I$ satisfies $G_{\ell}$ and … Let…