We study functions f on $\mathbb{Q}$ which satisfy a ‘quantum modularity’ relation of the shape $ f(x+1)=f(x)$ and $f(x) - {| {x} |}^{-k} f(-1/x) = h(x)$, where $h:\mathbb{R}_{\neq 0} \to \mathbb{C}$ is a function satisfying various regularity conditions. We study the case $\operatorname{Re}(k)\neq 0$. We prove the existence of a limiting function, denoted by $f^\triangleleft$ or $f^\triangleright$, depending on the sign of $\operatorname{Re}(k)$, which extends continuously f to $\mathbb{R}$ in some sense. This means, in particular, that in the $\operatorname{Re}(k)\neq0$ case, the quantum modular form itself has to have at least a certain level of regularity. We deduce that the values $\{f(a/q), 1\leqslant a<q, (a, q)=1\}$, appropriately normalized, tend to equidistribute along the graph of $f^\triangleleft$ or $f^\triangleright$, and we prove that, under natural hypotheses, the limiting measure is diffuse. We apply these results to obtain limiting distributions of values and continuity results for several arithmetic functions known to satisfy the above quantum modularity: higher weight modular symbols associated to holomorphic cusp forms; Eichler integral associated to Maaß forms; a function of Kontsevich and Zagier related to the Dedekind $\eta$-function; and generalized cotangent sums.

We correct an error in the statement of [LVW24, Theorem 3.2] by adding the omitted hypothesis that the covers are arithmetically Gorenstein.

We show that each local field $\mathbb{F}_q(\!(t)\!)$ of characteristic $p > 0$ is characterised up to isomorphism within the class of all fields of imperfect exponent at most 1 by (certain small quotients of) its absolute Galois group together with natural axioms concerning the p-torsion of its Brauer group. This complements previous work by Efrat and Fesenko, who analysed fields whose absolute Galois group is isomorphic to that of a local field of characteristic p.

We show that the fundamental group of every enumeratively rationally connected closed symplectic manifold is finite. In other words, if a closed symplectic manifold has a non-zero Gromov–Witten invariant with two point insertions, then it has finite fundamental group. We also show that if the spherical homology class associated with such a non-zero Gromov–Witten invariant is holomorphically indecomposable, then the rational second homology of the symplectic manifold has rank one.

The linear arithmetic fundamental lemma (AFL) is a conjectural identity of intersection numbers on Lubin–Tate deformation spaces and derivatives of orbital integrals. It was introduced for elliptic orbits by Li, and Howard and Li. For elliptic orbits, the relevant intersection problem is formulated for the basic isogeny class. In the present article, we extend the conjecture to all orbits and all isogeny classes. Our main result is a reduction of the non-basic cases of the AFL to the basic ones, which relies on an analysis of the connected-étale sequence. Our results will be relevant in the global setting, where also locally non-elliptic orbits may contribute in a non-trivial way.

We prove a conjecture of Emerton, Gee and Hellmann concerning the overconvergence of étale $(\varphi,\Gamma)$-modules in families parametrized by topologically finite-type $\mathbf{Z}_{p}$-algebras. As a consequence, we deduce the existence of a natural map from the rigid fiber of the Emerton–Gee stack to the rigid analytic stack of $(\varphi,\Gamma)$-modules.

This corrigendum fixes an error in the statement of Lemma 8.3 in the published version of the paper.