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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: 2021-10-08

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Abstract: This paper is a synopsis of the recent book [9]. The latter is dedicated to the stochastic Burgers equation as a model for 1d turbulence, and the paper discusses its content in relation to the Kolmogorov theory of turbulence. PubDate: 2021-10-05

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Abstract: In this computational paper we verify a truncated version of the Buzzard–Calegari conjecture on the Newton polygon of the Hecke operator \(T_2\) for all large enough weights. We first develop a formula for computing p-adic valuations of exponential sums, which we then implement to compute 2-adic valuations of traces of Hecke operators acting on spaces of cusp forms. Finally, we verify that if Newton polygon of the Buzzard–Calegari polynomial has a vertex at \(n\le 15\) , then it agrees with the Newton polygon of \(T_2\) up to n. PubDate: 2021-10-01

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Abstract: We give a criterion for the transversality of a curve embedded in a product of elliptic curves. We then apply our criterion to some explicit classes of curves. The transversality allows us to apply theorems that produce explicit and implementable bounds for the height of the rational points on the curves. PubDate: 2021-10-01

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Abstract: The hyperbolic center of mass of a finite measure on the unit ball with respect to a radially increasing weight is shown to exist, be unique, and depend continuously on the measure. Prior results of this type are extended by characterizing the center of mass as the minimum point of an energy functional that is strictly convex along hyperbolic geodesics. A special case is Hersch’s center of mass lemma on the sphere, which follows from convexity of a logarithmic kernel introduced by Douady and Earle. PubDate: 2021-10-01

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Abstract: In this article, we study the growth of (fine) Selmer groups of elliptic curves in certain infinite Galois extensions where the Galois group G is a uniform, pro-p, p-adic Lie group. By comparing the growth of (fine) Selmer groups with that of class groups, we show that it is possible for the \(\mu \) -invariant of the (fine) Selmer group to become arbitrarily large in a certain class of nilpotent, uniform, pro-p Lie extension. We also study the growth of fine Selmer groups in false Tate curve extensions. PubDate: 2021-10-01

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Abstract: Building on previous results [17, 35], we complete the classification of compact oriented Einstein 4-manifolds with \(\det (W^+) > 0\) . There are, up to diffeomorphism, exactly 15 manifolds that carry such metrics, and, on each of these manifolds, such metrics sweep out exactly one connected component of the corresponding Einstein moduli space. PubDate: 2021-10-01

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Abstract: We examine the spectrum of a family of Sturm–Liouville operators with regularly spaced delta function potentials parametrized by increasing strength. The limiting behavior of the eigenvalues under this spectral flow was described by Berkolaiko et al. (Lett Math Phys 109(7):1611–1623, 2019), where it was used to study the nodal deficiency of Laplacian eigenfunctions. Here we consider the eigenfunctions of these operators. In particular, we give explicit formulas for the limiting eigenfunctions, and also characterize the eigenfunctions and eigenvalues for all values for the spectral flow parameter (not just in the limit). We also develop spectrally accurate numerical tools for comparison and visualization. PubDate: 2021-10-01

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Abstract: Based on the new approach to modular forms presented in [6] that uses rational functions, we prove a dominated convergence theorem for certain modular forms in the Eisenstein space. It states that certain rearrangements of the Fourier series will converge very fast near the cusp \(\tau = 0\) . As an application, we consider L-functions associated to products of Eisenstein series and present natural generalized Dirichlet series representations that converge in an expanded half plane. PubDate: 2021-10-01

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Abstract: We study how the \(\ell \) -adic valuation of the number of spanning trees varies in regular abelian \(\ell \) -towers of multigraphs. We show that for an infinite family of regular abelian \(\ell \) -towers of bouquets, the \(\ell \) -adic valuation of the number of spanning trees behaves similarly to the \(\ell \) -adic valuation of the class numbers in \({\mathbb {Z}}_{\ell }\) -extensions of number fields. PubDate: 2021-10-01

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Abstract: For a CM-field K and an odd prime number p, let \(\widetilde{K}'\) be a certain multiple \(\mathbb {Z}_p\) -extension of K. In this paper, we study several basic properties of the unramified Iwasawa module \(X_{\widetilde{K}'}\) of \(\widetilde{K}'\) as a \(\mathbb {Z}_p\llbracket \mathrm{Gal}(\widetilde{K}'/K)\rrbracket \) -module. Our first main result is a description of the order of a Galois coinvariant of \(X_{\widetilde{K}'}\) in terms of the characteristic power series of the unramified Iwasawa module of the cyclotomic \(\mathbb {Z}_p\) -extension of K under a certain assumption on the splitting of primes above p. The second result is that if K is an imaginary quadratic field and if p does not split in K, then, under several assumptions on the Iwasawa \(\lambda \) -invariant and the ideal class group of K, we determine a necessary and sufficient condition such that \(X_{\widetilde{K}}\) is \(\mathbb {Z}_p\llbracket \mathrm{Gal}(\widetilde{K}/K)\rrbracket \) -cyclic. Here, \(\widetilde{K}\) is the \(\mathbb {Z}_p^2\) -extension of K. PubDate: 2021-10-01

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Abstract: Sommaire A complete classification of continuous \(\text {SL}(n)\) covariant vector-valued valuations on \(L^{p}({\mathbb {R}}^{n}, x dx)\) is obtained without any homogeneity assumptions. The moment vector is shown to be essentially the only such valuation. PubDate: 2021-10-01

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Abstract: Given a prime p, a number field \({K}\) and a finite set of places S of \({K}\) , let \({K}_S\) be the maximal pro-p extension of \({K}\) unramified outside S. Using the Golod–Shafarevich criterion one can often show that \({K}_S/{K}\) is infinite. In both the tame and wild cases we construct infinite subextensions with bounded ramification using the refined Golod–Shafarevich criterion. In the tame setting we are able to produce infinite asymptotically good extensions in which infinitely many primes split completely, and in which every prime has Frobenius of finite order, a phenomenon that had been expected by Ihara. We also achieve new records on Martinet constants (root discriminant bounds) in the totally real and totally complex cases. PubDate: 2021-10-01

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Abstract: Given a compactly supported area-preserving diffeomorphism of the disk, we prove an integral formula relating the asymptotic action to the asymptotic winding number. As a corollary, we obtain a new proof of Fathi’s integral formula for the Calabi homomorphism on the disk. PubDate: 2021-09-23

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Abstract: Let G be a semisimple compact connected Lie group. An N-fold reduced product of G is the symplectic quotient of the Hamiltonian system of the Cartesian product of N coadjoint orbits of G under diagonal coadjoint action of G. Under appropriate assumptions, it is a symplectic orbifold. Using the technique of nonabelian localization and the residue formula of Jeffrey and Kirwan, we investigate the symplectic volume of an N-fold reduced product of G. Suzuki and Takakura gave a volume formula for the N-fold reduced product of \( \mathbf {SU}(3) \) in [25] by using geometric quantization and the Riemann–Roch formula. We compare our volume formula with theirs and prove that our volume formula agrees with theirs in the case of triple reduced products of \( \mathbf {SU}(3) \) . PubDate: 2021-07-30

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Abstract: This is a translation of the paper “Статистические свойства собственных функций” which appeared in the Proceedings of the All-USSR School in Differential Equations with Infinite Number of Independent Variables and in Dynamical Systems with Infinitely Many Degrees of Freedom, Dilijan, Armenia, May 21–June 3, 1973; published by the Armenian Academy of Sciences, Yerevan, 1974. Translated from the Russian original by Semyon Dyatlov. PubDate: 2021-07-30

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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: 2021-07-01

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Abstract: We discuss Shnirelman’s Quantum Ergodicity Theorem, giving an outline of a proof and an overview of some of the recent developments in mathematical Quantum Chaos. PubDate: 2021-06-24

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Abstract: We study the algebraicity of the central critical values of twisted triple product L-functions associated to motivic Hilbert cusp forms over a totally real étale cubic algebra in the totally unbalanced case. The algebraicity is expressed in terms of the cohomological period constructed via the theory of coherent cohomology on quaternionic Shimura varieties developed by Harris. As an application, we generalize our previous result with Cheng on Deligne’s conjecture for certain automorphic L-functions for \({\text {GL}}_3 \times {\text {GL}}_2\) . PubDate: 2021-06-22

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Abstract: Let \(f(z)=q+\sum _{n\ge 2}a(n)q^n\) be a weight k normalized newform with integer coefficients and trivial residual mod 2 Galois representation. We extend the results of Amir and Hong in Amir and Hong (On L-functions of modular elliptic curves and certain K3 surfaces, Ramanujan J, 2021) for \(k=2\) by ruling out or locating all odd prime values \( \ell <100\) of their Fourier coefficients a(n) when n satisfies some congruences. We also study the case of odd weights \(k\ge 1\) newforms where the nebentypus is given by a quadratic Dirichlet character. PubDate: 2021-06-16