Publications

2025

• Equality of tropical rank and dimension for tropical linear series
  
  • Amini Omid
  • Gaubert Stéphane
  • Gierczak Lucas
  , 2024. The tropical rank of a semimodule of rational functions on a metric graph mirrors the concept of rank in linear algebra. Defined in terms of the maximal number of tropically independent elements within the semimodule, this quantity has remained elusive due to the challenges of computing it in practice. In this note, we establish that the tropical rank is, in fact, precisely equal to the topological dimension of the semimodule, one more than the dimension of the associated linear system of divisors. Moreover, we show that the equality of divisorial and tropical ranks in the definition of tropical linear series is equivalent to the pure dimensionality of the corresponding linear system. We conclude with several complementary results and questions on combinatorial, topological, and computability properties of the tropical rank.
• Burnett's conjecture in generalized wave coordinates
  
  • Huneau Cécile
  • Luk Jonathan
  , 2024. We prove Burnett's conjecture in general relativity when the metrics satisfy a generalized wave coordinate condition, i.e., suppose $\{g_n\}_{n=1}^\infty$ is a sequence of Lorentzian metrics (in arbitrary dimensions $d \geq 3$) satisfying a generalized wave coordinate condition and such that $g_n\to g$ in a suitably weak and "high-frequency" manner, then the limit metric $g$ satisfies the Einstein--massless Vlasov system. Moreover, we show that the Vlasov field for the limiting metric can be taken to be a suitable microlocal defect measure corresponding to the convergence. The proof uses a compensation phenomenon based on the linear and nonlinear structure of the Einstein equations. (10.48550/arXiv.2403.03470)
  DOI : 10.48550/arXiv.2403.03470
• The Deligne-Riemann-Roch isomorphism
  
  • Eriksson Dennis
  • Freixas Montplet Gerard
  , 2025. This paper is the second in a series devoted to Deligne's conjectural program on refined versions of the Grothendieck-Riemann-Roch theorem via the determinant of the cohomology. We prove a general form of the Deligne-Riemann-Roch isomorphism, lifting the degree-one part of the Grothendieck-Riemann-Roch formula to a canonical isomorphism of line bundles. This extends previous constructions and is formulated and proven in a flexible reinterpretation of Elkik's theory of intersection bundles introduced in the first paper of the series. This resolves the geometric aspect of Deligne's program. Among the applications, we derive a natural isomorphism relating the BCOV bundle and the Hodge bundle of a family of Calabi-Yau varieties, which is part of the mathematical formulation of the genus one mirror symmetry conjecture proposed in a previous work with Mourougane.
• Small eigenvalues of Toeplitz operators, Lebesgue envelopes and Mabuchi geometry
  
  • Finski Siarhei
  , 2025. <div><p>We study small eigenvalues of Toeplitz operators on polarized complex projective manifolds. For Toeplitz operators whose symbols are supported on proper subsets, we prove the existence of eigenvalues that decay exponentially with respect to the semiclassical parameter. We moreover, establish a connection between the logarithmic distribution of these eigenvalues and the Mabuchi geodesic between the fixed polarization and the Lebesgue envelope associated with the polarization and the non-zero set of the symbol. As an application of our approach, we also obtain analogous results for Toeplitz matrices. Table of contents 1 Introduction 1 2 Logarithm of a Toeplitz operator is asymptotically Toeplitz 6 3 Determining measures and non-negligible psh envelopes 9 4 Lebesgue dense points and psh envelopes 13 5 Bernstein-Markov type properties for the Lebesgue measures 15 6 From Toeplitz to Transfer operators and back 19 7 Asymptotic class of the weighted L 2 -norms 21 8 Mabuchi geometry and the contact set 25 9 Growth of balls of holomorphic sections and Lebesgue envelopes 27 10 Generalized Toeplitz operators and Toeplitz matrices 28</p></div>
• On the Localization of the Bergman Kernel and applications to Toeplitz theory
  
  • Finski Siarhei
  , 2025. <div><p>For a compact complex manifold endowed with a big line bundle and a Radon measure, we study the localization phenomena of the associated Bergman (or Christoffel-Darboux) kernel. For Bernstein-Markov measures, this results in the determination of the limiting off-diagonal Bergman measure, thereby confirming a conjecture of Zelditch. We then turn to applications in the theory of Toeplitz operators, showing in particular that they form an algebra under composition. Building on this, we then show that for Bernstein-Markov measures, the spectrum of Toeplitz operators equidistributes.</p></div>
• Hommage à Peter Lax (1926-2025)
  
  • Allaire Grégoire
  • Berestycki Henri
  • Golse François
  • Rauch Jeffrey
  Matapli, Société de Mathématiques Appliquées et Industrielles (SMAI), 2025, 138, pp.101-124. Peter David Lax est décédé le 16 mai 2025 à l'âge de 99 ans. C'est une figure historique des mathématiques et du calcul scientifique qui nous a quittés après une vie extrêmement riche et féconde au cours de laquelle il a posé les bases mathématiques de la mécanique des fluides compressibles, du calcul numérique des ondes de choc, de la théorie du scattering, des systèmes intégrables, des solitons et de l'analyse numérique des équations aux dérivées partielles. Son influence a été et est toujours immense, aussi bien en mathématiques que dans d'autres disciplines comme, par exemple, la mécanique des fluides numérique. Après une première section qui donnera quelques repères biographiques et tracera son portrait, les trois sections suivantes essaieront de montrer sur quelques exemples la profondeur de ses idées et leur impact durable qui façonnent les mathématiques appliquées aujourd'hui encore.
• On the rigidity of the stable norm and Mather's β-function for geodesic flows
  
  • Florio Anna
  • Leguil Martin
  • Sorrentino Alfonso
  , 2025. We investigate rigidity phenomena associated to the stable norm and Mather's β-function for Riemannian geodesic flows on closed manifolds. Given two metrics g1 and g2, we compare these objects pointwise at individual homology classes. Our main result establishes that if Mather's β-function (or the stable norm) of g2 at a non-zero homology class h equals that of g1 at h multiplied by a suitable factor determined by the metrics, then the two metrics are homothetic on the Mather set of homology h associated to g1. In the case of conformally equivalent metrics, this yields a pointwise criterion for homothety on the projected Mather set. Some consequences are discussed, including a pointwise rigidity result on the 2-torus implying that if a metric has the same Mather's β-function at some non-zero homology class as a normalized flat metric in the same conformal class, then the metric must be flat. This result can be considered a pointwise version of a similar global result by Bangert. Finally, an extension of these results to Mañé's perturbations of general Tonelli Lagrangians is discussed.
• On the injective norm of random fermionic states and skew-symmetric tensors
  
  • Dartois Stephane
  • Radpay Parham
  , 2025. We study the injective norm of random skew-symmetric tensors and the associated fermionic quantum states, a natural measure of multipartite entanglement for systems of indistinguishable particles. Extending recent advances on random quantum states, we analyze both real and complex skew-symmetric Gaussian ensembles in two asymptotic regimes: fixed particle number with increasing one-particle Hilbert space dimension, and joint scaling with fixed filling fraction. Using the Kac-Rice formula on the Grassmann manifold, we derive high-probability upper bounds on the injective norm and establish sharp asymptotics in both regimes. Interestingly, a duality relation under particle-hole transformation is uncovered, revealing a symmetry of the injective norm under the action of the Hodge star operator. We complement our analytical results with numerical simulations for low fermion numbers, which matches the predicted bounds.
• High frequency limit for Klein-Gordon-Maxwell equations
  
  • Salvi Tony
  , 2025. The Klein-Gordon-Maxwell equations are used in several physical contexts, as a simple model for gauge theories,in quantum electrodynamics... As for other hyperbolic or elliptic equations, it is interesting to study the behaviorof high-frequency solutions. Especially at the limit, when the wavelength tends to 0.In the case of the vacuum Einstein equations, the Burnett conjecture states that a high-frequency limit of solutions (in a sense to be specified)is a solution to the massless Einstein-Vlasov equations. This phenomenon of backreaction is linked to the nonlinearities of the Einstein equations.For KGM, the nonlinearities are weaker and have a more flexible structure, which suggests the existence of more singular high-frequency solutions.Moreover, the presence of the Planck constant h (representative of the quantum effects in KGM) in the equationsallows the study of another type of high-frequency solutions, the ones appearing at the semi-classical limit,that is, when h tends to 0.The aim of this thesis will be to identify possible limits (semi-classical or not) for sequences of high-frequency solutionsto the Klein-Gordon-Maxwell equations and to understand which relativistic fluid or kinetic system are they solution to. To do so, several methods will be used: geometric optics,semi-classical analysis, modulated energy method, study of simplified equation (nonlinear Klein-Gordon)...
• Cohomologically tropical varieties
  
  • Aksnes Edvard
  • Amini Omid
  • Piquerez Matthieu
  • Shaw Kris
  Journal of the Institute of Mathematics of Jussieu, Cambridge University Press, 2025, 24 (6), pp.2543 - 2572. <div><p>Given the tropicalization of a complex subvariety of the torus, we define a morphism between the tropical cohomology and the rational cohomology of their respective tropical compactifications. We say that the subvariety of the torus is cohomologically tropical if this map is an isomorphism for all closed strata of the tropical compactification.</p><p>We prove that a schön subvariety of the torus is cohomologically tropical if and only if it is wunderschön and its tropicalization is a tropical homology manifold. The former property means that the open strata in the boundary of a tropical compactification are all connected and the mixed Hodge structures on their cohomology are pure of maximum possible weight; the latter property requires that, locally, the tropicalization verifies tropical Poincaré duality.</p><p>We study other properties of cohomologically tropical and wunderschön varieties, and show that in a semistable degeneration to an arrangement of cohomologically tropical varieties, the Hodge numbers of the smooth fibers are captured in the tropical cohomology of the tropicalization. This extends the results of Itenberg, Katzarkov, Mikhalkin and Zharkov.</p></div> (10.1017/s1474748025101114)
  DOI : 10.1017/s1474748025101114
• degeneration of complex rational maps
  
  • Gong Chen
  , 2025. We develop non-Archimedean techniques to analyze the degeneration of the dynamics of a sequence of holomorphic maps of the Riemann sphere.Given any sequence of rational maps f_n of degree d at least 2, we construct in the first chapter all its possible dynamical limits as rational maps over complete metrized (possibly non-Archimedean) fields. To do so, we avoid the use of pointed ultralimits of the hyperbolic 3-space as done by Y. Luo, and rely in a more systematic way on Berkovich theory. This allows us to prove a general statement on the convergence of the equilibrium measures and to control in a precise way the blow-up ofthe Lyapunov exponents of f_n. A key observation is that the scale at which f_n has to be renormalized to define a non-trivial limit of degree d is given by the Rumely’s minimal resultant function.In the second chapter, we focus more specifically on the link between f_n and the limit along a fixed ultrafilter. We establish a remarkable dichotomy on the blow-up of the multipliers of the periodic points of f_n: either almost all periodic points have uniformly bounded multipliers; or almost all multipliers diverge at a uniform rate given by the inverse of the minimal resultant.We then consider the set of all possible scales at which a multiplier of a periodic cycle of f_n explodes, and show that this set is finite of cardinality at most 2d-2. Finally we develop a complete theory of rescaling limits, expanding on the foundational work of J. Kiwi. By definition such an object is a (possibly non-Archimedean) rational map of degree at least 2 and of positive entropy obtained as a limit of a sequence of iterate of conjugates of f_n. Each of these limits comes with associated coordinate changes and a well-defined scales of zooming. We prove that the set of all rescaling limits can be naturally organized into a tree structure, and describe this tree for special families of rational maps, including polynomials, Bernoulli maps in the sense of Favre-Rivera-Letelier and quadratic maps. We conclude by comparing the set of scales associated with multipliers and with the set of scales arising from rescaling limits in the case of polynomial sequences.
• The spectral genus of an isolated hypersurface singularity and its relation to the Milnor number and analytic torsion
  
  • Eriksson Dennis
  • Freixas I Montplet Gerard
  Documenta Mathematica, Universität Bielefeld, 2025. In this paper, we introduce the notion of spectral genus $\widetilde{p}_g$ of a germ of an isolated hypersurface singularity $(\mathbb{C}^{n+1} , 0) \to (\mathbb{C}, 0)$, defined as a sum of small exponents of monodromy eigenvalues. The number of these is equal to the geometric genus $p_{g}$, and hence $\widetilde{p}_{g}$ can be considered as a secondary invariant to it. We then explore a secondary version of the Durfee conjecture on $p_{g}$ , and we predict an inequality between $p_{g}$ and the Milnor number $\mu$, to the effect that $\widetilde{p}_{g}\leq\frac{\mu-1}{(n+2)!}$. We provide evidence by confirming our conjecture in several cases, including homogeneous singularities and singularities with large Newton polyhedra, and quasi-homogeneous or irreducible curve singularities. We also show that a weaker inequality follows from Durfee's conjecture, and hence holds for quasi-homogeneous singularities and curve singularities. Our conjecture is shown to relate closely to the asymptotic behavior of the holomorphic analytic torsion of the sheaf of holomorphic functions on a degeneration of projective varieties, potentially indicating deeper geometric and analytic connections. (10.4171/DM/1013)
  DOI : 10.4171/DM/1013
• Wave turbulence for a semilinear Klein-Gordon system
  
  • de Suzzoni Anne-Sophie
  • Stingo Annalaura
  • Touati Arthur
  , 2025. <div><p>In this article we consider a system of two Klein-Gordon equations, set on the d-dimensional box of size L, coupled through quadratic semilinear terms of strength ε and evolving from well-prepared random initial data. We rigorously derive the effective dynamics for the correlations associated to the solution, in the limit where L → ∞ and ε → 0 according to some power law. The main novelty of our work is that, due to the absence of invariances, trivial resonances always take precedence over quasi-resonances. The derivation of the nonlinear effective dynamics is justified up time to δT , where T = ε -2 is the appropriate timescale and δ is independent of L and ε. We use Feynmann interaction diagrams, here adapted to a normal form reduction and to the coupled nature of our real-valued system. We also introduce a frequency decomposition at the level of the diagrammatic and develop a new combinatorial tool which allows us to work with the Klein-Gordon dispersion relation.</p><p>After the normal form procedure, the correct timescale T to observe the effective dynamics can be proved to be ε -2 . One of the strength of our result is that we justify the approximation of the correlations by the effective dynamics up to times of order δT , where δ &gt; 0 is a small quantity independent of L and ε. In other words, our derivation extends the validity of the effective dynamics up to a nontrivial timescale. In the context of wave turbulence, such a rigorous derivation has only been obtained for the Schrödinger equation in the pioneer work [DH23b] (and subsequently improved in [DH23a]), where the authors reach the so-called kinetic timescale.</p><p>An overview of our proof will be given in Section 1.4, but we can already say that our result follows from a precise diagrammatic representation of the solution to the Klein-Gordon system, as in the aforementioned articles on other equations. However, the new features of the system we consider force us to develop a new diagrammatic adapted to the normal form procedure and an analysis adapted to the Klein-Gordon dispersion relation (which, as opposed to the Schrödinger one, forbids the application of number theory results). For this last aspect, we rely on a low/high-frequency analysis and we develop new combinatorial tools that take this decomposition into account.</p><p>Remark 1.4. Note that a low/high-frequency decomposition is also utilized in [LS11] and in the work on inhomogeneous settings [ACG21], but in both cases it does not show up in the diagrammatic. See also Remark 1.9 below.</p></div> <div>The Klein-Gordon system<p>In this section we present the system of two coupled Klein-Gordon equations we study in this article and introduce all the notations that are required to state Theorem 1.1.</p></div>
• THE GLOBAL STABILITY OF THE KALUZA-KLEIN SPACETIME
  
  • Huneau Cécile
  • Stingo Annalaura
  • Wyatt Zoe
  Journal of the European Mathematical Society, European Mathematical Society, 2025. In this paper we show the classical global stability of the flat Kaluza-Klein spacetime, which corresponds to Minkowski spacetime in R 1+4 with one direction compactified on a circle. We consider small perturbations which are allowed to vary in all directions including the compact direction. These perturbations lead to the creation of massless modes and Klein-Gordon modes. On the analytic side, this leads to a PDE system coupling wave equations to an infinite sequence of Klein-Gordon equations with different masses. The techniques we use are based purely in physical space using the vectorfield method. (10.4171/JEMS/1663)
  DOI : 10.4171/JEMS/1663
• Phénomènes de diffusion pour des équations dispersives
  
  • Maleze Cyril
  , 2025. Cette thèse est consacrée à l'étude de solutions d’équations dispersives, avec coloration aléatoire. Dans les chapitres 1,2 et 3, on étudie le caractère en temps long de solutions d’équations de Hartree avec point de vue probabiliste. Plus précisément, dans lechapitre 1, on travaille sur la stabilité d’un équilibre thermodynamique, `a travers un résultat de diffusion autour de cet équilibre, pour l’équation de Hartree quintique. Ce travail est une extension d’un résultat de diffusion pour l’équation de Hartree cubique, prouvépar Collot–de Suzzoni. Le résultat du chapitre 2 est un résultat de diffusion autour d’un équilibre thermodynamique pour l’équation de Hartree–Fock, obtenue en ajoutant un terme dit d’échange à l’équation de Hartree. Ce terme rajoute de nombreuses difficultés à l’étude de la stabilité des équilibres. Ce terme d’échange fait cependant apparaître certaines annulations qui permet de montrer en dimension 1 un résultat de diffusion autour de zéro : c’est le résultat principal du chapitre 3. Dans le chapitre 4, il est question de la construction de mesures de Gibbs pour l’équation de Dirac zonale. En particulier, on montre un résultat d’existence de solutions en construisant une mesure de Gibbs, après une renormalisation de l’équation.
• Reduction by stages for W-algebras and applications
  
  • Juillard Thibault
  , 2025. Affine W-algebra form a family of vertex algebras defined as quantum Hamiltonian reductions of affine Kac-Moody algebras. They are noncommutative and nonassociative algebras of infinite type, in one-to-one correspondence with nilpotent orbits in simple Lie algebras. Any affine W-algebra has an associative and finitely generated analogue, a finite W-algebra. The algebraic properties of affine or finite W-algebra are related to the geometric properties of some affine Poisson variety, a Slodowy slice. Given a pair of nilpotent orbits, one can associate of a pair of (finite oraffine) W-algebras. In this thesis, under some compatibility conditions on these orbits, we prove that one of these two W-algebras can be reconstructed as the quantum Hamiltonian reduction of the other one. This property is called reduction by stages. To prove reduction by stages for W-algebras,we first prove reduction by stages for the pair of Slodowy slices associated with the chosen pair of nilpotent orbits. Reduction by stages for finite W-algebras is proved by introducing filtrations on both W-algebras such that the associated graded algebras coincide with the Poisson algebras of polynomial functions on the Slodowy slices. As an application to to reduction by stages, we establish an analogue of the Skryabin equivalence of categories for the modules over these W-algebras.For affine W-algebras, the quantum Hamiltonian reduction is a achieved by the mean of BRST cohomology, implying new technical difficulties. We need to prove that each W-algebra can be defined by using several equivalent BRST cohomology constructions. Then, choosing the right BRST construction allows us to connect the two affine W-algebras in a natural way and deduce reduction by stages. We provide several examples of compatible pairs of nilpotent orbits for classical and exceptional simple Lie algebras. In type A, a fundamental example is when the chosen nilpotent orbits correspond to hook-type partitions. As an application, we give a new interpretation of the Kraft-Procesi isomorphisms between nilpotent Slodowy slices by using reduction by stages. We explain our future project of applying reduction by stages to prove isomorphisms of simple quotients of affine W-algebras that are analogous to these Kraft-Procesi isomorphisms.
• C. Favre - Dynamical degree 3
  
  • Favre Charles
  • Bastien Fanny
  , 2025. Dynamical degrees were introduced by Russakovskii and Shiffman in 1997. They measure the growth of preimages of subvarieties of a fixed dominant rational map of a projective variety, and control many dynamical features of this map. In this first set of lectures, we shall define dynamical degrees using positivity properties of divisor classes, and explain how to compute them in low dimension and in some significant examples.
• C. Favre - Dynamical degree 4
  
  • Favre Charles
  • Bastien Fanny
  , 2025. Dynamical degrees were introduced by Russakovskii and Shiffman in 1997. They measure the growth of preimages of subvarieties of a fixed dominant rational map of a projective variety, and control many dynamical features of this map. In this first set of lectures, we shall define dynamical degrees using positivity properties of divisor classes, and explain how to compute them in low dimension and in some significant examples.
• C. Favre - Dynamical degree 2
  
  • Favre Charles
  • Bastien Fanny
  , 2025. Dynamical degrees were introduced by Russakovskii and Shiffman in 1997. They measure the growth of preimages of subvarieties of a fixed dominant rational map of a projective variety, and control many dynamical features of this map. In this first set of lectures, we shall define dynamical degrees using positivity properties of divisor classes, and explain how to compute them in low dimension and in some significant examples.
• C. Favre - Dynamical degree 1
  
  • Favre Charles
  • Bastien Fanny
  , 2025. Dynamical degrees were introduced by Russakovskii and Shiffman in 1997. They measure the growth of preimages of subvarieties of a fixed dominant rational map of a projective variety, and control many dynamical features of this map. In this first set of lectures, we shall define dynamical degrees using positivity properties of divisor classes, and explain how to compute them in low dimension and in some significant examples.
• Three results on holonomic D-modules
  
  • Sabbah Claude
  , 2025. In this text, we illustrate the use of local methods in the theory of (irregular) holonomic D-modules. I. (The Euler characteristic of the de~Rham complex) We show the invariance of the global or local Euler characteristic of the de~Rham complex after localization and dual localization of a holonomic D-module along a hypersurface, as well as after tensoring with a rank one meromorphic connection with regular singularities. II. (Local generic vanishing theorems for holonomic D-modules) We prove that the natural morphism from the proper pushforward to the total pushforward of an algebraic holonomic D-module by an open inclusion is an isomorphism if we first twist the D-module structure by suitable closed algebraic differential forms. III. (Laplace transform of a Stokes-filtered constructible sheaf of exponential type) Motivated by the construction in [YZ24], we~propose a slightly different construction of the Laplace transform of a Stokes-perverse sheaf on the projective line and show directly that it corresponds to the Laplace transform of the corresponding holonomic D-module via the Riemann-Hilbert-Birkhoff-Deligne-Malgrange correspondence. This completes the presentation given in [Sab13, Chap. 7]}, where only the other direction of the Laplace transformation is analyzed. We~also compare our approach with the construction made previously in [YZ24].
• On the approximation of the von Neumann equation in the semi-classical limit. Part I : numerical algorithm
  
  • Filbet Francis
  • Golse François
  Journal of Computational Physics, Elsevier, 2025, 527, pp.113810. We propose a new approach to discretize the von Neumann equation, which is efficient in the semi-classical limit. This method is first based on the so called Weyl’s variables to address the stiffness associated with the equation. Then, by applying a truncated Hermite expansion of the density operator, we successfully handle this stiffness. Additionally, we develop a finite volume approximation for practical implementation and conduct numerical simulations to illustrate the efficiency of our approach. This asymptotic preserving numerical approximation, combined with the use of Hermite polynomials, provides an efficient tool for solving the von Neumann equation in all regimes, near classical or not. (10.1016/j.jcp.2025.113810)
  DOI : 10.1016/j.jcp.2025.113810
• Global well-posedness of a 2D fluid-structure interaction problem with free surface
  
  • Alazard Thomas
  • Shao Chengyang
  • Yang Haocheng
  , 2025. This paper is devoted to the analysis of the incompressible Euler equation in a time-dependent fluid domain, whose interface evolution is governed by the law of linear elasticity. Our main result asserts that the Cauchy problem is globally well-posed in time for any irrotational initial data in the energy space, without any smallness assumption. We also prove continuity with respect to the initial data and the propagation of regularity. The main novelty is that no dissipative effect is assumed in the system. In the absence of parabolic regularization, the key observation is that the system can be transformed into a nonlinear Schrödinger-type equation, to which dispersive estimates are applied. This allows us to construct solutions that are very rough from the point of view of fluid dynamics-the initial fluid velocity has merely one-half derivative in L2. The main difficulty is that the problem is critical in the energy space with respect to several key inequalities from harmonic analysis. The proof incorporates new estimates for the Dirichlet-to-Neumann operator in the low-regularity regime, including refinements of paralinearization formulas and shape derivative formulas, which played a key role in the analysis of water waves.
• Equidistribution and counting of periodic tori in the space of Weyl chambers
  
  • Dang Nguyen-Thi
  • Li Jialun
  , 2025. Let G be a semisimple Lie group without compact factor and Γ &lt; G a torsion-free, cocompact, irreducible lattice. According to Selberg, periodic orbits of regular Weyl chamber flows live on tori. We prove that these periodic tori equidistribute exponentially fast towards the quotient of the Haar measure. From the equidistribution formula, we deduce a higher rank prime geodesic theorem. (10.4171/CMH/594)
  DOI : 10.4171/CMH/594
• Classification, structure, and associated local zeta functions for a class of p-adic symmetric spaces.
  
  • Harinck Pascale
  • Rubenthaler Hubert
  , 2025. {\bf Part I}:\,\, {\it Let $F$ be a p-adic field of characteristic $0$. Let $\widetilde{\mathfrak {g}}$ be a reductive Lie algebra over $F$ which is endowed with a short $\mathbb {Z}$-grading: $\widetilde{\mathfrak {g}}=\widetilde{\mathfrak g}_{-1}\oplus \widetilde{\mathfrak g}_{0}\oplus \widetilde{\mathfrak g}_{1}$. It is known that the representation $(\widetilde{\mathfrak g}_{0},\widetilde{\mathfrak g}_{1})$ is always prehomogeneous. Under some additional conditions we classify these objects using weighted Satake-Tits diagrams. Moreover we study the orbits of $\widetilde {G}_{0}$ in $\widetilde{\mathfrak g}_{1}$, where $\widetilde {G}_{0}$ is an algebraic group defined over $F$, whose Lie algebra is $\widetilde{\mathfrak g}_{0}$. It turns out that the open orbits are symmetric spaces. We then investigate the $P$-orbits in $\widetilde{\mathfrak g}_{1}$, where $P$ is a minimal $\sigma$-split parabolic subgroup of $\widetilde {G}_{0}$ ($\sigma$ being the involution defining the symmetric spaces).} {\bf Part II}:\,\, {\it In the second part we define and study the zeta functions associated to the minimal spherical principal series of representations (of $\widetilde {G}_{0}$) for the reductive $p$-adic symmetric spaces described in Part I. We prove that these zeta functions satisfy a functional equation which is given explicitly (see Theorem 8.3.9 and Theorem 8.4.5). Moreover, for a subclass of these spaces, we define $L$-functions and $\varepsilon$-factors associated to the representations.} This paper is essentially the synthesis of two preceding papers by the authors (arXiv: 2003.05764 and arXiv: 2407.06667) with some additions and some corrections.