Annales Mathematiques du Quebec最新文献

In this paper, we develop an integral refinement of the Perrin-Riou theory of exponential maps. We also formulate the Perrin-Riou theory for anticyclotomic deformation of modular forms in terms of the theory of the Serre–Tate local moduli and interpolate generalized Heegner cycles p-adically.

We classify the (({mathfrak {g}},K))-modules generated by nearly holomorphic Hilbert–Siegel modular forms by the global method. As an application, we study the image of projection operators on the space of nearly holomorphic Hilbert–Siegel modular forms with respect to infinitesimal characters in terms of (({mathfrak {g}},K))-modules.

Let (pge 5) be a prime. We construct modular Galois representations for which the (mathbb {Z}_p)-corank of the p-primary Selmer group (i.e., its (lambda )-invariant) over the cyclotomic (mathbb {Z}_p)-extension is large. More precisely, for any natural number n, one constructs a modular Galois representation such that the associated (lambda )-invariant is (ge n). The method is based on the study of congruences between modular forms, and leverages results of Greenberg and Vatsal. Given a modular form (f_1) satisfying suitable conditions, one constructs a congruent modular form (f_2) for which the (lambda )-invariant of the Selmer group is large. A key ingredient in acheiving this is the Galois theoretic lifting result of Fakhruddin–Khare–Patrikis, which extends previous work of Ramakrishna. The results are illustrated by explicit examples.

Let K be an imaginary quadratic field and (K_infty ) be the ({textbf{Z}}_p^2)-extension of K. Answering a question of Ahmed and Lim, we show that the Pontryagin dual of the Selmer group over (K_infty ) associated to a supersingular polarized abelian variety admits an algebraic functional equation. The proof uses the theory of (Gamma )-system developed by Lai, Longhi, Tan and Trihan. We also show the algebraic functional equation holds for Sprung’s chromatic Selmer groups of supersingular elliptic curves along (K_infty ).

We study the adjoint Bloch–Kato Selmer groups attached to a classical point in the cuspidal eigenvariety associated with (textrm{GSp}_{2g}). Our strategy is based on the study of families of Galois representations on the eigenvariety, which is inspired by the book of J. Bellaiche and G. Chenevier.

Recently, D. Bucur and M. Nahon used boundary homogenisation to show the remarkable flexibility of Steklov eigenvalues of planar domains. In the present paper we extend their result to higher dimensions and to arbitrary manifolds with boundary, even though in those cases the boundary does not generally exhibit any periodic structure. Our arguments use a framework of variational eigenvalues and provide a different proof of the original results. Furthermore, we present an application of this flexibility to the optimisation of Steklov eigenvalues under perimeter constraint. It is proved that the best upper bound for normalised Steklov eigenvalues of surfaces of genus zero and any fixed number of boundary components can always be saturated by planar domains. This is the case even though any actual maximisers (except for simply connected surfaces) are always far from being planar themselves. In particular, it yields sharp upper bound for the first Steklov eigenvalue of doubly connected planar domains.

In this article, we study the Iwasawa theory for Hilbert modular forms over the anticyclotomic extension of a CM field. We prove an one-sided divisibility result toward the Iwasawa main conjecture in this setting. The proof relies on the first and second reciprocity laws relating theta elements to Heegner point Euler systems on Shimura curves. As a by-product we also prove a result towards the rank 0 case of certain Bloch–Kato conjecture and a parity conjecture.

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