Annales Mathematiques du Quebec最新文献

We study a p-adic Maass–Shimura operator in the context of Mumford curves defined by [15]. We prove that this operator arises from a splitting of the Hodge filtration, thus answering a question in [15]. We also study the relation of this operator with generalized Heegner cycles, in the spirit of [1, 4, 19, 28].

Let (ell ) be a rational prime. Previously, abelian (ell )-towers of multigraphs were introduced which are analogous to (mathbb {Z}_{ell })-extensions of number fields. It was shown that for towers of bouquets, the growth of the (ell )-part of the number of spanning trees behaves in a predictable manner (analogous to a well-known theorem of Iwasawa for (mathbb {Z}_{ell })-extensions of number fields). In this paper, we extend this result to abelian (ell )-towers over an arbitrary connected multigraph (not necessarily simple and not necessarily regular). In order to carry this out, we employ integer-valued polynomials to construct power series with coefficients in (mathbb {Z}_ell ) arising from cyclotomic number fields, different than the power series appearing in the prequel. This allows us to study the special value at (u=1) of the Artin–Ihara L-function, when the base multigraph is not necessarily a bouquet.

We show that extended graph 4-manifolds (as defined by Frigerio–Lafont–Sisto in [12]) do not support Einstein metrics.

We generalise Pollack’s construction of plus/minus L-functions to certain cuspidal automorphic representations of (mathrm {GL_{2n}}) using the p-adic L-functions constructed in work of Barrera Salazar et al. (On p-adic l-functions for (text {GL}_{2n}) in finite slope shalika families, 2021). We use these to prove that the complex L-functions of such representations vanish at at most finitely many twists by characters of p-power conductor.

We first recall various formulations and approximations for the motion of an incompressible fluid, in the well-known setting of the Euler equations. Then, we address incompressible motions in porous media, through the Muskat system, which is a friction dominated first order analog of the Euler equations for inhomogeneous incompressible fluids subject to an external potential.

We construct examples of p-adic L-functions over universal deformation spaces for ({{,mathrm{GL},}}_2). We formulate a conjecture predicting that the natural parameter spaces for p-adic L-functions and Euler systems are not the usual eigenvarieties (parametrising nearly-ordinary families of automorphic representations), but other, larger spaces depending on a choice of a parabolic subgroup, which we call ‘big parabolic eigenvarieties’.

Let (G=A *B) be a free product of freely indecomposable groups. We explicitly construct quasimorphisms on G which are invariant with respect to all automorphisms of G. We also prove that the space of such quasimorphisms is infinite-dimensional whenever G is not the infinite dihedral group. As an application we prove that an invariant analogue of stable commutator length recently introduced by Kawasaki and Kimura is non-trivial for these groups.

We consider the case of scattering by several obstacles in ({mathbb {R}}^d), (d ge 2) for the Laplace operator (Delta ) with Dirichlet boundary conditions imposed on the obstacles. In the case of two obstacles, we have the Laplace operators (Delta _1) and (Delta _2) obtained by imposing Dirichlet boundary conditions only on one of the objects. The relative operator (g(Delta ) - g(Delta _1) - g(Delta _2) + g(Delta _0)) was introduced in Hanisch, Waters and one of the authors in (A relative trace formula for obstacle scattering. arXiv:2002.07291, 2020) and shown to be trace-class for a large class of functions g, including certain functions of polynomial growth. When g is sufficiently regular at zero and fast decaying at infinity then, by the Birman–Krein formula, this trace can be computed from the relative spectral shift function (xi _mathrm {rel}(lambda ) = -frac{1}{pi } {text {Im}}(Xi (lambda ))), where (Xi (lambda )) is holomorphic in the upper half-plane and fast decaying. In this paper we study the wave-trace contributions to the singularities of the Fourier transform of (xi _mathrm {rel}). In particular we prove that ({hat{xi }}_mathrm {rel}) is real-analytic near zero and we relate the decay of (Xi (lambda )) along the imaginary axis to the first wave-trace invariant of the shortest bouncing ball orbit between the obstacles. The function (Xi (lambda )) is important in the physics of quantum fields as it determines the Casimir interactions between the objects.