Annales scientifiques de l'Ecole normale supérieure

Actions of tensor categories on Kirchberg algebras

2025-12-10T09:29:26+01:00

We characterize the simplicity of Pimsner algebras for non-proper C*-correspondences. With the aid of this criterion, we give a systematic strategy to produce outer actions of unitary tensor categories on Kirchberg algebras. In particular, every countable unitary tensor category admits an outer action on the Cuntz algebra $\mathcal{O}_2$. We also study the realizability of modules over fusion rings as K-groups of Kirchberg algebras acted on by unitary tensor categories, which turns out to be generically true for every unitary fusion category. Several new examples are provided, among which actions on Cuntz algebras of 3-cocycle twists of cyclic groups are constructed for all possible 3-cohomological classes by answering a question asked by Izumi.

Topological entropy of a rational map over a complete metrized field

2025-12-10T09:29:28+01:00

We prove that the topological entropy of any dominant rational self-map of a projective variety defined over a complete non-Archimedean field is bounded from above by the maximum of its dynamical degrees, thereby extending a theorem of Gromov and Dinh-Sibony from the complex to the non-Archimedean setting. We proceed by proving that any regular self-map which admits a regular extension to a projective model defined over the valuation ring has necessarily zero entropy. To this end we introduce the ε- reduction of a Berkovich analytic space, a notion of independent interest.

On the geometric connected components of moduli spaces of p-adic shtukas and local Shimura varieties

2025-12-10T09:29:29+01:00

We study connected components of local Shimura varieties. Given local shtuka datum $(G, b, \mu)$, with $G$ unramified over $\mathbb{Q}_p and $(b, \mu)$ HN-irreducible, we determine $\pi_0(Sht_{(G,b,[\mu],\infty)}\times \mathbb{C}_p)$ with its $G(\mathbb{Q}_p)\times J_b (\mathbb{Q}_p)\times W_E$-action. This confirms new cases of a conjecture of Rapoport and Viehmann. We construct and study the specialization map for moduli spaces of p-adic shtukas at parahoric level whose target is an affine Deligne–Lusztig variety.

The étale local structure of algebraic stacks

2025-12-10T09:29:31+01:00

We prove that an algebraic stack, locally of finite presentation and quasi-separated over a quasi-separated algebraic space with affine stabilizers, is étale locally a quotient stack around any point with a linearly reductive stabilizer. This result generalizes the main result of [AHR20] to the relative setting and the main result of [AOV11] to the case of non-finite inertia. We also provide various coherent completeness and effectivity results for algebraic stacks as well as structure theorems for linearly reductive groups schemes. Finally, we provide several applications of these results including generalizations of Sumihiro's theorem on torus actions and Luna's étale slice theorem to the relative setting.

Lower bounds on the essential dimension of reductive groups

2025-12-10T09:29:32+01:00

We introduce a new technique for proving lower bounds on the essential dimension of split reductive groups. As an application, we strengthen the best previously known lower bounds for various split simple algebraic groups, most notably for the exceptional group $E_8$. In the case of the projective linear group $\operatorname{PGL}_n$, we recover A. Merkurjev's celebrated lower bound with a simplified proof. Our technique relies on decompositions of loop torsors over valued fields due to P. Gille and A. Pianzola.

Algebraization techniques and rigid-analytic Artin-Grothendieck vanishing

2025-12-10T09:29:32+01:00

First, we prove an algebraization result for rig-smooth algebras over a general noetherian ring; this positively answers the question raised in [Sta24, Tag 0GAX]. Then we prove a general partial algebraization result in non-archimedean geometry. The result says that we can always algebraize a geometrically reduced affinoid rigid-analytic space in "one direction" in an appropriate sense. As an application of this result, we show the remaining cases of the Artin-Grothendieck Vanishing for affinoid algebras, which were previously conjectured in [BM21, §7]. This allows us to deduce a stronger version of the rigid-analytic Artin-Grothendieck Vanishing Conjecture (see [Han20, Conj. 1.2]) over a field of characteristic 0. Using a completely different set of ideas, we also obtain a weaker version of this conjecture over a field of characteristic p>0.

The mixing conjecture under GRH

2025-12-10T09:29:33+01:00

We prove the Mixing Conjecture of Michel–Venkatesh for the class group action on Heegner points of large discriminant on compact arithmetic surfaces attached to maximal orders in rational quaternion algebras. The proof is conditional on the Generalized Riemann Hypothesis, and when the division algebra is indefinite we furthermore assume the Ramanujan conjecture. We establish the mixing conjecture for the discrete spectrum of the modular surface as well, under the same conditions. Our methods, which provide an effective rate, are based on the spectral theory of automorphic forms and their L-functions, as well as sieve methods and multiplicative functions.

The weak Beauville–Bogomolov decomposition in characteristic p>=0

2025-12-10T09:29:35+01:00

We prove a variant of the Beauville--Bogomolov decomposition for weakly ordinary, or generally globally $F$-split, varieties $X$ with $K_X \sim 0$, 
in characteristic $p>0$. We also show that the weakly ordinary assumption in our statement cannot be dropped. Additionally, if the assumption 
$K_X \sim 0$ is replaced by $-K_X$ being semi-ample, we show the weaker statement that all closed fibers of the Albanese morphism are isomorphic.

Finally, we apply our main theorem to draw consequences to the behavior of rational points and fundamental groups of weakly ordinary $K$-trivial 
varieties in positive characteristic.