For all Zarski dense Anosov subgroups of a semisimple real algebraic group, we prove that their limit sets are Ahlfors regular for intrinsic conformal premetrics. As a consequence, we obtain that a Patterson–Sullivan measure is Ahlfors regular (and hence equal to the Hausdorff measure) if and only if the associated linear form is symmetric. We also discuss several applications, including analyticity of (p,q)-Hausdorff dimensions on the Teichmüller spaces, new upper bounds on the growth indicator, and $L^2$-spectral properties of associated locally symmetric manifolds.

We prove an arithmetic refinement of the Yau–Zaslow formula by replacing the classical Euler characteristic in Beauville’s argument by a motivic Euler characteristic, related to the work of Levine. Our result implies similar formulas for other related invariants, including a generalisation of a formula of Kharlamov and Răsdeaconu on counting real rational curves on real K3 surfaces, and Saito’s determinant of cohomology.

Let $K/k$ be a finite Galois extension of global function fields. Let E be a Drinfeld module over k. We state and prove an equivariant refinement of Taelman’s analogue of the analytic class number formula for $(E,K/k)$, and derive explicit consequences for the Galois structure of the Taelman class group of E over K.

Colmez, Dospinescu et Nizioł démontrent que les seules représentations p-adiques de $\mathrm{Gal}(\bar{{\mathbb{Q}}}_p/{\mathbb{Q}}_p)$ qui apparaissent dans la cohomologie étale p-adique de la tour de revêtements du demi-plan de Drinfeld sont les représentations cuspidales (i.e. potentiellement semi-stables, dont la représentation de Weil-Deligne associée est irréductible) de dimension 2, à poids de Hodge-Tate 0 et 1 et que leur multiplicité est donnée par la correspondance de Langlands p-adique. Nous étendons ce résultat en poids quelconque, en considérant la cohomologie étale p-adique à coefficients dans les puissances symétriques du système local universel sur la tour de Drinfeld. Une différence notable est que toutes les représentations potentiellement semi-stables de dimension 2 non cristabélines, pas seulement les cuspidales, apparaissent avec les multiplicités attendues. Le point clé est que les systèmes locaux que l’on considère sont particulièrement simples : ce sont des ‘opers isotriviaux’ sur une courbe. Nous donnons une recette pour calculer la cohomologie proétale de tels systèmes locaux à partir de la cohomologie de Hyodo-Kato de la courbe et du complexe de de Rham du fibré plat filtré associé au système local.

Given an $E_1$-ring A and a class $a \in \pi_{mk}(A)$ satisfying a suitable hypothesis, we define a map of $E_1$-rings $A\to A(\!\sqrt[m]{a})$ realizing the adjunction of an mth root of a. We define a form of logarithmic topological Hochschild homology (THH) for $E_1$-rings, and show that root adjunction is log-THH-étale for tamely ramified root adjunctions, which provides a formula for $\operatorname{THH}(A(\!\sqrt[m]{a}))$ in terms of THH and log THH of A. If A is connective, we prove that the induced map $K(A) \to K(A(\!\sqrt[m]{a}))$ in algebraic K-theory is the inclusion of a wedge summand. Using this, we compute $V(1)_*K(ko_p)$ for $p\gt 3$ and also deduce that if K(A) exhibits chromatic redshift, so does $K(A(\!\sqrt[m]{a}))$. We exhibit several maps of ring spectra as root adjunctions, and use this to obtain a new proof of the fact that Lubin–Tate spectra satisfy the redshift conjecture.

This note corrects two errors in the paper Symplectomorphisms and spherical objects in the conifold smoothing [Compositio Math. 160 (2024), 2738–2773]. Neither affects the main results of the paper (in particular, neither affects any result stated in the Introduction of the paper). We apologise to our readers for these inaccuracies.