Generalizing the recent results of Bellaïche and Khare for the level-$1$ case, we study the structure of the local components of the shallow Hecke algebras (i.e., Hecke algebras without $U_p$ and $U_{\ell }$ for all primes $\ell$ dividing the level $N$) acting on the space of modular forms modulo $p$ for ${\Gamma }_0\left(N\right)$ and ${\Gamma }_1\left(N\right)$. We relate them to pseudodeformation rings and prove that in many cases, the local components are regular complete local algebras of dimension $2$.

For prime powers $q$, let $split\left(q\right)$ denote the probability that a randomly chosen principally polarized abelian surface over the finite field ${\mathbb{F}}_q$ is not simple. We show that there are positive constants $c_1$ and $c_2$ such that, for all $q$,

$$c_1{\left(logq\right)}^{-3}{\left(loglogq\right)}^{-4}<split\left(q\right)\sqrt{q}<c_2{\left(logq\right)}^4{\left(loglogq\right)}^2,$$

and we obtain better estimates under the assumption of the generalized Riemann hypothesis. If $A$ is a principally polarized abelian surface over a number field $K$, let ${\pi }_{split}\left(A∕K,z\right)$ denote the number of prime ideals $\mathfrak{p}$ of $K$ of norm at most $z$ such that $A$ has good reduction at $\mathfrak{p}$ and $A_{\mathfrak{p}}$ is not simple. We conjecture that, for sufficiently general $A$, the counting function ${\pi }_{split}\left(A∕K,z\right)$ grows like $\sqrt{z}∕logz$. We indicate why our theorem on the rate of growth of $split\left(q\right)$ gives us reason to hope that our conjecture is true.

Chambert-Loir and Ducros have recently introduced a theory of real valued differential forms and currents on Berkovich spaces. In analogy to the theory of forms with logarithmic singularities, we enlarge the space of differential forms by so called $\delta$-forms on the nonarchimedean analytification of an algebraic variety. This extension is based on an intersection theory for tropical cycles with smooth weights. We prove a generalization of the Poincaré–Lelong formula which allows us to represent the first Chern current of a formally metrized line bundle by a $\delta$-form. We introduce the associated Monge–Ampère measure $\mu$ as a wedge-power of this first Chern $\delta$-form and we show that $\mu$ is equal to the corresponding Chambert-Loir measure. The $\ast$-product of Green currents is a crucial ingredient in the construction of the arithmetic intersection product. Using the formalism of $\delta$-forms, we obtain a nonarchimedean analogue at least in the case of divisors. We use it to compute nonarchimedean local heights of proper varieties.

Deligne conjectured that a single $\ell$-adic lisse sheaf on a normal variety over a finite field can be embedded into a compatible system of ${\ell }^{\prime }$-adic lisse sheaves with various ${\ell }^{\prime }$. Drinfeld used Lafforgue’s result as an input and proved this conjecture when the variety is smooth. We consider an analogous existence problem for a regular flat scheme over $\mathbb{Z}$ and prove some cases using Lafforgue’s result and the work of Barnet-Lamb, Gee, Geraghty, and Taylor.

Let $R$ be a strictly local ring complete for a discrete valuation, with fraction field $K$ and residue field of characteristic $p>0$. Let $X$ be a smooth, proper variety over $K$. Nicaise conjectured that the rational volume of $X$ is equal to the trace of the tame monodromy operator on $\ell$-adic cohomology if $X$ is cohomologically tame. He proved this equality if $X$ is a curve. We study his conjecture from the point of view of logarithmic geometry, and prove it for a class of varieties in any dimension: those having logarithmic good reduction.

In algebraic statistics, Jukes–Cantor and Kimura models are of great importance. Sturmfels and Sullivant generalized these models by associating to any finite abelian group $G$ a family of toric varieties $X\left(G,K_{1,n}\right)$. We investigate the generators of their ideals. We show that for any finite abelian group $G$ there exists a constant $\varphi$, depending only on $G$, such that the ideals of $X\left(G,K_{1,n}\right)$ are generated in degree at most $\varphi$.