A conjecture of Woods from 1972 is disproved: for $d\ge 30$, there are well-rounded unimodular lattices in ${\mathbb{R}}^d$ with covering radius greater than that of ${\mathbb{Z}}^d$.

We study special values of regularized theta lifts at complex multiplication (CM) points. In particular, we show that CM values of Borcherds products can be expressed in terms of finitely many Fourier coefficients of certain harmonic weak Maaß forms of weight $1$. As it turns out, these coefficients are logarithms of algebraic integers whose prime ideal factorization is determined by special cycles on an arithmetic curve. Our results imply a conjecture of Duke and Li and give a new proof of the modularity of a certain arithmetic generating series of weight $1$ studied by Kudla, Rapoport, and Yang.

We study analogues of $\mathtt{FI}$-modules where the role of the symmetric group is played by the general linear groups and the symplectic groups over finite rings, and we prove basic structural properties such as local Noetherianity. Applications include a proof of the Lannes–Schwartz Artinian conjecture in the generic representation theory of finite fields, very general homological stability theorems with twisted coefficients for the general linear and symplectic groups over finite rings, and representation-theoretic versions of homological stability for congruence subgroups of the general linear group, the automorphism group of a free group, the symplectic group, and the mapping class group.

In this article we prove a general bounded height result for specializations in finitely generated subgroups varying in families which complements and sharpens the toric Mordell–Lang theorem by replacing finiteness with emptiness, for the intersection of varieties and subgroups, all moving in a pencil, except for bounded height values of the parameters (and excluding identical relations). More precisely, an instance of the result is as follows. Consider the torus scheme ${{\mathbb{G}}_m^r}_{/\mathcal{C}}$ over a curve $\mathcal{C}$ defined over $\overline{\mathbb{Q}}$, and let $\Gamma$ be a subgroup scheme generated by finitely many sections (satisfying some necessary conditions). Further, let $V$ be any subscheme. Then there is a bound for the height of the points $P\in \mathcal{C}\left(\overline{\mathbb{Q}}\right)$ such that, for some $\gamma \in \Gamma$ which does not generically lie in $V$, $\gamma \left(P\right)$ lies in the fiber $V_P$. We further offer some direct Diophantine applications, to illustrate once again that the results implicitly contain information absent from the previous bounds in this context.