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 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.
Let K be an imaginary quadratic field where the prime p splits. Our goal in this article is to prove results towards the Iwasawa main conjectures for p-nearly-ordinary families associated to (mathrm {GL}_2times mathrm {Res}_{K/mathbb {Q}}mathrm {GL}_1) with a minimal set of assumptions. The main technical input is an improvement on the locally restricted Euler system machinery that allows the treatment of residually reducible cases, which we apply with the Beilinson–Flach Euler system.
The rank one Gross conjecture for Deligne–Ribet p-adic L-functions was solved in works of Darmon-Dasgupta-Pollack and Ventullo by the Eisenstein congruence among Hilbert modular forms. The purpose of this paper is to prove an analogue of the Gross conjecture for the Katz p-adic L-functions attached to imaginary quadratic fields via the congruences between CM forms and non-CM forms. The new ingredient is to apply the p-adic Rankin–Selberg method to construct a non-CM Hida family which is congruent to a Hida family of CM forms at the (1+varepsilon ) specialization.
This paper begins a new approach to the r-trace formula, without removing the nontempered contribution to the spectral side. We first establish an invariant trace formula whose discrete spectral terms are weighted by automorphic L-functions. This involves extending the results of Finis, Lapid, and Müller on the continuity of the coarse expansion of Arthur’s noninvariant trace formula to the refined expansion, and then to the invariant trace formula, while incorporating the use of basic functions at unramified places.
We introduce multi-torsion, a spectral invariant generalizing Ray–Singer analytic torsion. We define multi-torsion for compact manifolds with a certain local geometric product structure that gives a bigrading on differential forms. We prove that multi-torsion is metric-independent in a suitable sense. Our definition of multi-torsion is inspired by an interpretation of each of analytic torsion and the eta invariant as a regularized integral of a closed differential form on a space of metrics on a vector bundle or on a space of elliptic operators. We generalize the Stokes’ theorem argument explaining the dependence of torsion and eta on the geometric data used to define them to the local product setting to prove our metric-independence theorem for multi-torsion.
We extend the definition of the bounded reduction property to endomorphisms of automatic group and find conditions for it to hold. We study endomorphisms with L-quasiconvex image and prove that those with finite kernel satisfy a synchronous version of the bounded reduction property. Finally, we use these techniques to prove L-quasiconvexity of the equalizer of two endomorphisms under certain (strict) conditions.
We ask several questions about substitution maps in the Robba ring. These questions are motivated by p-adic Hodge theory and the theory of p-adic dynamical systems. We provide answers to those questions in special cases, thereby generalizing results of Kedlaya, Colmez, and others.
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