This is the second in a pair of papers developing a framework to apply logarithmic methods in the study of stable maps and singular curves of genus $1$. This volume focuses on logarithmic Gromov–Witten theory and tropical geometry. We construct a logarithmically nonsingular and proper moduli space of genus $1$ curves mapping to any toric variety. The space is a birational modification of the principal component of the Abramovich–Chen–Gross–Siebert space of logarithmic stable maps and produces logarithmic analogues of Vakil and Zinger’s genus one reduced Gromov–Witten theory. We describe the nonarchimedean analytic skeleton of this moduli space and, as a consequence, obtain a full resolution to the tropical realizability problem in genus $1$.

Let $F$ be a totally real field in which $p$ is unramified. Let $\bar{r}:G_F\to {GL}_2\left({\overline{\mathbb{F}}}_p\right)$ be a modular Galois representation which satisfies the Taylor–Wiles hypotheses and is generic at a place $v$ above $p$. Let $\mathfrak{m}$ be the corresponding Hecke eigensystem. We show that the $\mathfrak{m}$-torsion in the $modp$ cohomology of Shimura curves with full congruence level at $v$ coincides with the ${GL}_2\left(k_v\right)$-representation $D_0\left(\bar{r}{|}_{G_{F_v}}\right)$ constructed by Breuil and Paškūnas. In particular, it depends only on the local representation $\bar{r}{|}_{G_{F_v}}$, and its Jordan–Hölder factors appear with multiplicity one. This builds on and extends work of the author with Morra and Schraen and, independently, Hu–Wang, which proved these results when $\bar{r}{|}_{G_{F_v}}$ was additionally assumed to be tamely ramified. The main new tool is a method for computing Taylor–Wiles patched modules of integral projective envelopes using multitype tamely potentially Barsotti–Tate deformation rings and their intersection theory.

We define a theta operator on $p$-adic vector-valued modular forms on unitary groups of arbitrary signature, over a quadratic imaginary field in which $p$ is inert. We study its effect on Fourier–Jacobi expansions and prove that it extends holomorphically beyond the $\mu$-ordinary locus, when applied to scalar-valued forms.

For the polynomial ring over an arbitrary field with twelve variables, there exists a prime ideal whose symbolic Rees algebra is not finitely generated.

We show that the $b$-constant (appearing in Manin’s conjecture) is constant on very general fibers of a family of algebraic varieties. If the fibers of the family are uniruled, then we show that the $b$-constant is constant on general fibers.

We extend Urban’s construction of eigenvarieties for reductive groups $G$ such that $G\left(ℝ\right)$ has discrete series to include characteristic $p$ points at the boundary of weight space. In order to perform this construction, we define a notion of “locally analytic” functions and distributions on a locally ${\mathbb{Q}}_p$-analytic manifold taking values in a complete Tate ${\mathbb{Z}}_p$-algebra in which $p$ is not necessarily invertible. Our definition agrees with the definition of locally analytic distributions on $p$-adic Lie groups given by Johansson and Newton.

We show that for a self-morphism of an abelian variety defined over an algebraically closed field of arbitrary characteristic, the second cohomological dynamical degree coincides with the first numerical dynamical degree.

We give an equivalence of categories between certain subcategories of modules of pro-$p$ Iwahori–Hecke algebras and modulo $p$ representations.