Annales Mathematiques du Quebec最新文献

We present a new and simpler proof of the fact that any Lagrangian ({mathbb {R}}P^2) in (T^*{mathbb {R}}P^2) is Hamiltonian isotopic to the zero section. Our proof mirrors the one given by Li and Wu for the Hamiltonian uniqueness of Lagrangians in (T^*S^2), using surgery to turn Lagrangian spheres into symplectic ones. The main novel contribution is a detailed proof of the folklore fact that the complement of a symplectic quadric in ({mathbb {C}}P^2) can be identified with the unit cotangent disc bundle of ({mathbb {R}}P^2).

The purpose of this note is to find explicit representatives in de Rham cohomology for the generators of the cohomology of the moduli space of parabolic bundles, analogous to the results of [5] for the moduli space of vector bundles. Further we use the explicit generators to compute the intersection pairing of its cohomology.

We investigate elliptic operators with a symmetry that forces their index to vanish. We study the secondary index, defined modulo 2. We examine Callias-type operators with this symmetry on non-compact manifolds and establish mod 2 versions of the Gromov–Lawson relative index theorem, the Callias index theorem, and the Boutet de Monvel’s index theorem for Toeplitz operators.

Let E be an elliptic curve over (mathbb {Q}). Greenberg has posed a question whether the structure of the fine Selmer group over the cyclotomic (mathbb {Z}_{p})-extension of (mathbb {Q}) can be described by cyclotomic polynomials in a certain precise manner. A recent work of Lei has made progress on this problem by proving that the fine Mordell–Weil group (in the sense of Wuthrich) does have this required property. The goal of this paper is to study analogous questions of Greenberg over various (mathbb {Z}_{p})-extensions of an imaginary quadratic field F. In particular, when the elliptic curve has complex multiplication by the ring of integers of the imaginary quadratic field, we obtain results that are analogous to those of Lei over the cyclotomic (mathbb {Z}_{p})-extension and anti-cyclotomic (mathbb {Z}_{p})-extension of F. In the event that the elliptic curve has good ordinary reduction at the prime p, we further obtain a result over the (mathbb {Z}_{p})-extension of F unramified outside precisely one of the prime of F above p. Finally, we study the situation of an elliptic curve over the anticyclotomic (mathbb {Z}_{p})-extension under the generalized Heegner hypothesis. Along the way, we establish an analogous result for the BDP-Selmer group. This latter result is then applied to obtain a relation between the BDP p-adic L-function and the Mordell–Weil rank growth in the anticyclotomic (mathbb {Z}_{p})-extension which may be of independent interest.

In this short note, we construct an explicit embedding of the rescaling of the p-sum (Koplus _p K^{circ }) of the centrally symmetric convex domain K and it’s polar (K^{circ }) to the product (K times K^{circ }). The rescaling constant is sharp in some cases. Additionally, we comment about the strong Viterbo conjecture for (Koplus _p K^{circ }).

We study the effects of a domain deformation to the nodal set of Laplacian eigenfunctions when the eigenvalue is degenerate. In particular, we study deformations of a rectangle that perturb one side and how they change the nodal sets corresponding to an eigenvalue of multiplicity 2. We establish geometric properties, such as number of nodal domains, presence of crossings, and boundary intersections, of nodal sets for a large class of boundary deformations and study how these properties change along each eigenvalue branch for small perturbations. We show that internal crossings of the nodal set break under generic deformations and obtain estimates on the location and regularity of the nodal sets on the perturbed rectangle.