Annales scientifiques de l'Ecole normale supérieure

The Cauchy problem for the periodic Kadomtsev–Petviashvili–II equation below L^2

2026-04-01T15:08:25+02:00

We extend Bourgain's $L^2$-wellposedness result for the KP-II equation on $\T^2$ to initial data with negative Sobolev regularity. The key ingredient is a new linear $L^4$-Strichartz estimate which is effective on frequency-dependent time scales. The $L^4$-Strichartz estimates follow from combining an $\ell^2$-decoupling inequality recently proved by Guth--Maldague--Oh with semiclassical Strichartz estimates.

Moreover, we rely on a variant of Bourgain's bilinear Strichartz estimate on frequency-dependent times, which is proved via the Córdoba--Fefferman square function estimate.

On the density of strongly minimal algebraic vector fields

2026-04-01T15:08:25+02:00

Two theorems witnessing the abundance of geometrically trivial strongly minimal autonomous differential equations of arbitrary order are shown. The first one states that a generic algebraic vector field of degree $d\geq 2$ on the affine space of dimension $n \geq 2$ is strongly minimal and geometrically trivial. The second one states that if $X_0$ is the complement of a smooth hyperplane section $H_X$ of a smooth projective variety $X$ of dimension $n \geq 2$ then for $d \gg 0$, the system of differential equations associated with a generic vector field on $X_0$ with poles of order at most $d$ along $H_X$ is also strongly minimal and geometrically trivial.

Diagonals of algebraic power series and algebraicity modulo p

2026-04-01T15:08:25+02:00

In 1984, Deligne proved that for any prime number $p$, the reduction modulo $p$ of the diagonal of a multivariate algebraic power series
with integer coefficients is algebraic over the field of rational functions with coefficients in $\mathbb F_p$. Moreover, he conjectured that the
algebraic degrees $d_p$ of these functions should grow at most polynomially in $p$, {\it i.e.}, that $d_p=O(p^N)$ for some $N$ independent of $p$.
In this article, we provide a new elementary proof of Deligne's theorem, yielding the first general polynomial bound with a reasonable value of $N$,
thereby significantly improving all previously known results in this direction.

Liouville polarizations and the rigidity of their Lagrangian skeleta in dimension 4

2026-04-01T15:08:26+02:00

In 2000, Biran introduced polarizations of closed symplectic manifolds
and showed that their Lagrangian skeleta exhibit remarkable rigidity properties.
He in particular found that their complements contain only small balls.
In this paper, we introduce so-called Liouville polarizations of certain open 4-dimensional symplectic manifolds.
This leads to several symplectic embedding results,
that in turn lead to new Lagrangian non-removable intersections
and a novel phenomenon of Legendrian barriers.

We show for instance that given any connected symplectic 4-manifold $(M,\omega)$
and a 4-ball of smaller volume, their exists an explicit finite union of Lagrangian
discs in the 4-ball such that their complement symplectically embeds into $(M,\omega)$,
extending a result by Sackel-Song-Varolgunes-Zhu and Brendel.
Other applications are new Lagrangian intersection results and relative versions of the Arnold chord conjecture.

Automorphic density estimates and optimal approximation exponents

2026-04-01T15:08:26+02:00

The present paper is devoted to establishing an optimal approximation exponent for the action of an irreducible uniform lattice subgroup of a product group on its proper factors.

Previously optimal approximation exponents for lattice actions on homogeneous spaces were established under the assumption that the restriction of the automorphic representation to the stability group is suitably

tempered. However, for irreducible lattices in semisimple algebraic groups, either this property does not hold or it amounts to an instance of the

Ramanujan-Petersson-Selberg conjecture.

Sarnak's Density Hypothesis and its variants bounding the multiplicities of irreducible representations occurring in the decomposition of the automorphic representation can be viewed as a weakening of the

temperedness property. A refined form of this hypothesis has recently been established for uniform irreducible arithmetic congruence lattices arising from quaternion algebras. We employ this result in order to

establish - unconditionally - an optimal approximation exponent for the actions of these lattices on the associated symmetric spaces. We also give a general spectral criterion for the optimality of the approximation

exponent for irreducible uniform lattices in a product of arbitrary Gelfand pairs. Our methods involve utilizing the multiplicity bounds in the pre-trace formula, establishing refined estimates of the spherical

transforms, and carrying out an elaborate spectral analysis that bounds the Hilbert--Schmidt norms of carefully balanced geometric convolution operators.

Inductive systems of the symmetric group, polynomial functors and tensor categories

2026-04-01T15:08:26+02:00

We initiate the systematic study of modular representations of symmetric groups that arise via the braiding in (symmetric) tensor categories over fields of positive characteristic. We determine what representations appear for certain examples of tensor categories, develop general principles and demonstrate how this question connects with the ongoing study of the structure theory of tensor categories. We also formalise a theory of polynomial functors as functors which act coherently on all tensor categories. We conclude that the classification of such functors is a different way of posing the above question of which representation of symmetric groups appear. Finally, we extend the classical notion of strict polynomial functors from the category of (super) vector spaces to arbitrary tensor categories, and show that this idea is also a different packaging of the same information.

Exterior stability of Minkowski spacetime with borderline decay

2026-04-01T15:08:26+02:00

In 1993, the global stability of Minkowski spacetime has been proven in the celebrated work of Christodoulou and Klainerman. In 2003, Klainerman and Nicolo revisited Minkowski stability in the exterior of an outgoing null cone. In 2023, the author extended the results of Christodoulou and Klainerman to minimal decay assumptions. In this paper, we prove that the exterior stability of Minkowski holds with decay which is borderline compared to the minimal decay considered in by the author in 2023.

Strong disorder and very strong disorder are equivalent for directed polymers

2026-04-01T15:08:27+02:00

We show that if the normalized partition function $W^{\beta}_n$ of the directed polymer model on $\mathbb{Z}^d$ converges to zero, then it does so exponentially fast. This implies that there exists a critical temperature $\beta_c$ such that the renormalized partition function has a non-degenerate limit for all $\beta\in [0,\beta_c]$ -- weak disorder holds -- while for $\beta\in (\beta_c,\infty)$ it converges exponentially fast to zero -- very strong disorder holds. This solves a twenty-years-old conjecture formulated by Comets, Yoshida, Carmona and Hu. Our proof requires a technical assumption on the environment, namely, that it is bounded from above.