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.
We compute Benois ({mathscr {L}})-invariants of weight 1 cuspforms and of their adjoint representations and show how this extends Gross’ p-adic regulator to Artin motives which are not critical in the sense of Deligne. Benois’ construction depends on the choice of a regular submodule which is well understood when the representation is p-regular, as it then amounts to the choice of a “motivic” p-refinement. The situation is dramatically different in the p-irregular case, where the regular submodules are parametrized by a flag variety and thus depend on continuous parameters. We are nevertheless able to show in some examples, how Hida theory and the geometry of the eigencurve can be used to detect a finite number of choices of arithmetic and “mixed-motivic” significance.
We obtain non-Euclidean versions of classical theorems due to Hardy and Littlewood concerning smoothness of the boundary function of an analytic mapping on the unit disk with an appropriate growth condition.
We give all normal integral bases for the simplest cubic field (L_n) generated by the roots of Shanks’ cubic polynomial when these bases exist, that is, (L_n/{mathbb {Q}}) is tamely ramified. Furthermore, as an application of the result, we give an explicit relation between the roots of Shanks’ cubic polynomial and the Gaussian periods of (L_n) in the case that (L_n/{mathbb {Q}}) is tamely ramified, which is a generalization of the work of Lehmer, Châtelet and Lazarus in the case that the conductor of (L_n) is equal to (n^2+3n+9).
Let (E/{mathbb {Q}}) be a CM elliptic curve and p a prime of good ordinary reduction for E. We show that if (text {Sel}_{p^infty }(E/{mathbb {Q}})) has ({mathbb {Z}}_p)-corank one, then (E({mathbb {Q}})) has a point of infinite order. The non-torsion point arises from a Heegner point, and thus ({{,mathrm{ord},}}_{s=1}L(E,s)=1), yielding a p-converse to a theorem of Gross–Zagier, Kolyvagin, and Rubin in the spirit of [49, 54]. For (p>3), this gives a new proof of the main result of [12], which our approach extends to all primes. The approach generalizes to CM elliptic curves over totally real fields [4].
The group (Ham(M,omega )) of all Hamiltonian diffeomorphisms of a symplectic manifold ((M,omega )) plays a central role in symplectic geometry. This group is endowed with the Hofer metric. In this paper we study two aspects of the geometry of (Ham(M,omega )), in the case where M is a closed surface of genus 2 or 3. First, we prove that there exist diffeomorphisms in (Ham(M,omega )) arbitrarily far from being a k-th power, with respect to the metric, for any (k ge 2). This part generalizes previous work by Polterovich and Shelukhin. Second, we show that the free group on two generators embeds into the asymptotic cone of (Ham(M,omega )). This part extends previous work by Alvarez-Gavela et al. Both extensions are based on two results from geometric group theory regarding incompressibility of surface embeddings.
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