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Abstract: Abstract Let \(p\ge 5\) 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. PubDate: 2023-01-28

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Abstract: Abstract 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 \(K_\infty \) . PubDate: 2023-01-13

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Abstract: Abstract 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. PubDate: 2022-11-19

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Abstract: Abstract 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. PubDate: 2022-11-09 DOI: 10.1007/s40316-022-00207-8

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Abstract: Abstract 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. PubDate: 2022-11-09 DOI: 10.1007/s40316-022-00208-7

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Abstract: Abstract This article proves a case of the p-adic Birch and Swinnerton-Dyer conjecture for Garrett p-adic L-functions of [6], in the exceptional zero setting of extended analytic rank 2. PubDate: 2022-10-01 DOI: 10.1007/s40316-021-00166-6

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Abstract: Abstract Our objective in this series of two articles, of which the present article is the first, is to give a Perrin-Riou-style construction of p-adic L-functions (of Bellaïche and Stevens) over the eigencurve. As the first ingredient, we interpolate the Beilinson–Kato elements over the eigencurve (including the neighborhoods of \(\theta \) -critical points). Along the way, we prove étale variants of Bellaïche’s results describing the local properties of the eigencurve. We also develop the local framework to construct and establish the interpolative properties of these p-adic L-functions away from \(\theta \) -critical points. PubDate: 2022-10-01 DOI: 10.1007/s40316-021-00172-8

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Abstract: Abstract In this note we give a detailed construction of a \(\Lambda \) -adic \(\mathfrak d\) th Shintani lifting. We obtain a \(\Lambda \) -adic version of Kohnen’s formula relating Fourier coefficients of half-integral weight modular forms and special values of twisted L-series. As a by-product, we derive a mild generalization of such classical formulae, and also point out a relation between Fourier coefficients of \(\Lambda \) -adic \(\mathfrak d\) th Shintani liftings and Stark–Heegner points. PubDate: 2022-10-01 DOI: 10.1007/s40316-021-00182-6

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Abstract: R\'esum\'e 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. PubDate: 2022-07-27 DOI: 10.1007/s40316-022-00201-0

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Abstract: R\'esum\'e 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. PubDate: 2022-07-19 DOI: 10.1007/s40316-022-00205-w

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Abstract: Abstract 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 \(n^2+3n+9\) . PubDate: 2022-07-19 DOI: 10.1007/s40316-022-00204-x

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Abstract: Abstract 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. PubDate: 2022-07-08 DOI: 10.1007/s40316-022-00202-z

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Abstract: R\'esum\'e 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]. PubDate: 2022-07-08 DOI: 10.1007/s40316-022-00203-y

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Abstract: Abstract 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 \(\mathrm {GL}_2\times \mathrm {Res}_{K/\mathbb {Q}}\mathrm {GL}_1\) 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. PubDate: 2022-06-06 DOI: 10.1007/s40316-022-00197-7

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Abstract: Abstract 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. PubDate: 2022-05-26 DOI: 10.1007/s40316-022-00198-6

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Abstract: Abstract 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. PubDate: 2022-05-25 DOI: 10.1007/s40316-022-00200-1

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Abstract: Abstract 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. PubDate: 2022-05-21 DOI: 10.1007/s40316-022-00199-5

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Abstract: Abstract 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. PubDate: 2022-04-27 DOI: 10.1007/s40316-022-00195-9

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Abstract: Abstract 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. PubDate: 2022-04-27 DOI: 10.1007/s40316-022-00196-8