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In a famous paper, Asai indicated how to develop a theory of Eisenstein series for arbitrary number fields, using hyperbolic 3-space to take care of the complex places. Unfortunately he limited himself to class number 1. The present paper gives a detailed exposition of the general case, to be used for many applications. First, it is shown that the Eisenstein series satisfy the authors’ definition of regularized products satisfying the generalized Lerch formula, and the basic axioms which allow the systematic development of the authors’ theory, including the Cramér theorem. It is indicated how previous results of Efrat and Zograf for the strict Hilbert modular case extend to arbitrary number fields, for instance a spectral decomposition of the heat kernel periodized with respect to SL2 of the integers of the number field. This gives rise to a theta inversion formula, to which the authors’ Gauss transform can be applied. In addition, the Eisenstein series can be twisted with the heat kernel, thus encoding an infinite amount of spectral information in one item coming from heat Eisenstein series. The main expected spectral formula is stated, but a complete exposition would require a substantial amount of space, and is currently under consideration.

Consider an elliptic curve defined over an imaginary quadratic field $K$ with good reduction at the primes above $p\geq 5$ and has complex multiplication by the full ring of integers $\mathcal{O}_K$ of $K$. In this paper, we construct $p$-adic analogues of the Eisenstein-Kronecker series for such elliptic curve as Coleman functions on the elliptic curve. We then prove $p$-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function., v2. The current version is the synthesis of {\S}1-{\S}3 of the first version of this article with the content of arXiv:0807.4008 "The Kronecker limit formulas via the distribution relation." {\S}4,{\S}5 of the first version of this paper will be treated in a subsequent article

In this paper we study the Weil-Petersson geometry of $\overline{\mathcal{M}_{g,n}}$, the compactified moduli space of Riemann surfaces with genus g and n marked points. The main goal of this paper is to understand the growth of the diameter of $\overline{\mathcal{M}_{g,n}}$ as a function of $g$ and $n$. We show that this diameter grows as $\sqrt{n}$ in $n$, and is bounded above by $C \sqrt{g}\log g$ in $g$ for some constant $C$. We also give a lower bound on the growth in $g$ of the diameter of $\overline{\mathcal{M}_{g,n}}$ in terms of an auxiliary function that measures the extent to which the thick part of moduli space admits radial coordinates., 26 pages, 7 figures

This paper establishes new estimates for linear Schroedinger equations in R^3 with time-dependent potentials. Some of the results are new even in the time-independent case and all are shown to hold for potentials in scaling-critical, translation-invariant spaces. The proof of the time-independent results uses a novel method based on an abstract version of Wiener's Theorem., 49 pages; this is an expanded and improved version of the older paper

In this paper we show how to compute the �� norm , �� 0, using the dyadic grid. This result is a consequence of the description of the Hardy spaces H p (R N ) in terms of dyadic and special atoms. Recently, several novel methods for computing the BMO norm of a function f in two dimensions were discussed in (9). Given its importance, it is also of interest to explore the possibility of computing the norm of a BMO function, or more generally a function in the Lipschitz class �α, using the dyadic grid in R N. It turns out that the BMO question is closely related to that of approximating functions in the Hardy space H 1 (R N ) by the Haar system. The approximation in H 1 (R N ) by affine systems was proved in (2), but this result does not apply to the Haar system. Now, if H A (R) denotes the closure of the Haar system in H 1 (R), it is not hard to see that the distance d(f, H A ) of f ∈ H 1 (R) to H A is ∼ � R ∞ 0 f(x) dx �, see (1). Thus, neither dyadic atoms suffice to describe the Hardy spaces, nor the evaluation of thenorm in BMO can be reduced to a straightforward computation using the dyadic intervals. In this paper we address both of these issues. First, we give a characterization of the Hardy spaces H p (R N ) in terms of dyadic and special atoms, and then, by a duality argument, we show how to compute the norm in �α(R N ), α ≥ 0, using the dyadic grid. We begin by introducing some notations. Let J denote a family of cubes Q in R N , and Pd the collection of polynomials in R N of degree less than or equal to d. Given α ≥ 0, Q ∈ J, and a locally integrable function g, let pQ(g) denote the unique polynomial in P(α) such that (g − pQ(g)) χQ has vanishing moments up to order (α). For a locally square-integrable function g, we consider the maximal function M ♯,2 α,J g(x) given by

In this paper we study biholomorphic maps between Teichmüller spaces and the induced linear isometries between the corresponding tangent spaces. The first main result in this paper is the following classification theorem. If $M$ and $N$ are two Riemann surfaces that are not of exceptional type, and if there exists a biholomorphic map between the corresponding Teichmüller spaces Teich($M$) and Teich($N$), then $M$ and $N$ are quasiconformally related. Also, every such biholomorphic map is geometric. In particular, we have that every automorphism of the Teichmüller space Teich($M$) must be geometric. This result generalizes the previously known results (see [2], [5], [7]) and enables us to prove the well-known conjecture that states that the group of automorphisms of Teich($M$) is isomorphic to the mapping class group of $M$ whenever the surface $M$ is not of exceptional type. In order to prove the above results, we develop a method for studying linear isometries between $L^1$-type spaces. Our focus is on studying linear isometries between Banach spaces of integrable holomorphic quadratic differentials, which are supported on Riemann surfaces. Our main result in this direction (Theorem 1.1) states that if $M$ and $N$ are Riemann surfaces of nonexceptional type, then every linear isometry between $A^1(M)$ and $A^1(N)$ is geometric. That is, every such isometry is induced by a conformal map between $M$ and $N$.

1. Introduction and main results. In this paper we investigate the relations between the rate of decay of solutions of Schrodinger equations, continuity properties of spectral measures of the corresponding operators, and dynamical properties of the corresponding quantum systems. The first main result of this paper shows that, in great generality, certain upper bounds on the rate of growth of L 2 norms of generalized eigenfunctions over expanding balls imply certain minimal singularity of the spectral measures. Consider an operator H � V defined by the differential expression H � V =− � + V( x) �

This paper is the second part of our work on differential Galois theory as we promised in [U3]. Differential Galois theory has a long history since Lie tried to apply the idea of Abel and Galois to differential equations in the 19th century (cf. [U3], Introduction). When we consider Galois theory of differential equation, we have to separate the finite dimensional theory from the infinite dimensional theory. As Kolchin theory shows, the first is constructed on a rigorous foundation. The latter, however, seems inachieved despite of several important contributions of Drach, Vessiot,…. We propose in this paper a differential Galois theory of infinite dimension in a rigorous and transparent framework. We explain the idea of the classical authors by one of the simplest examples and point out the problems.

According to the Langlands functoriality conjecture, broadened to the setting of spherical varieties (of which reductive groups are special cases), a map between $L$ -groups of spherical varieties should give rise to a functorial transfer of their local and automorphic spectra. The “beyond endoscopy” proposal predicts that this transfer will be realized as a comparison between limiting forms of the (relative) trace formulas of these spaces. In this paper, we establish the local transfer for the identity map between $L$ -groups, for spherical affine homogeneous spaces $X=H\backslash G$ whose dual group is $\operatorname{SL}_{2}$ or $\operatorname{PGL}_{2}$ (with $G$ and $H$ split). More precisely, we construct a transfer operator between orbital integrals for the $(X\times X)/G$ -relative trace formula, and orbital integrals for the Kuznetsov formula of $\operatorname{PGL}_{2}$ or $\operatorname{SL}_{2}$ . Besides the $L$ -group, another invariant attached to $X$ is a certain $L$ -value, and the space of test measures for the Kuznetsov formula is enlarged to accommodate the given $L$ -value. The transfer operator is given explicitly in terms of Fourier convolutions, making it suitable for a global comparison of trace formulas by the Poisson summation formula, hence for a uniform proof, in rank $1$ , of the relations between periods of automorphic forms and special values of $L$ -functions.

The aim of this paper is to provide an analogue of the Ball-Rivoal theorem for $p$-adic $L$-values of Dirichlet characters. More precisely, we prove for a Dirichlet character $\chi$ and a number field $K$ the formula $\dim_{K}(K+\sum_{i=2}^{s+1} L_p(i,\chi\omega^{1-i}) K )\geq \frac{(1-\epsilon)\log (s)}{2[K:\mathbb{Q}](1+\log 2)}$. As a byproduct, we establish an asymptotic linear independence result for the values of the $p$-adic Hurwitz zeta function., Comment: 26 pages, final version; Duke Math. J. (accepted)

Recently, Du, Guth and Li showed that the Schrodinger operator $e^{it\Delta }$ satisfies $\lim_{t\rightarrow 0}e^{it\Delta }f=f$ almost everywhere for all $f\in H^{s}(\mathbb{R}^{2})$, provided that $s>1/3$. In this paper, we discuss the rate of convergence on $e^{it\Delta }(f)$ by assuming more regularity on $f$. At $n=2$, our result can be viewed as an application of the Du–Guth–Li theorem. We also address the same issue on the cases $n=1$ and $n>2$.

It is known that the $3$ -manifold $\operatorname{SL}(2,\mathbb{Z})\backslash\operatorname{SL}(2,\mathbb{R})$ is diffeomorphic to the complement of the trefoil knot in $S^{3}$ . E. Ghys showed that the linking number of this trefoil knot with a modular knot is given by the Rademacher symbol, which is a homogenization of the classical Dedekind symbol. The Dedekind symbol arose historically in the transformation formula of the logarithm of Dedekind’s eta function under $\operatorname{SL}(2,\mathbb{Z})$ . In this paper we give a generalization of the Dedekind symbol associated to a fixed modular knot. This symbol also arises in the transformation formula of a certain modular function. It can be computed in terms of a special value of a certain Dirichlet series and satisfies a reciprocity law. The homogenization of this symbol, which generalizes the Rademacher symbol, gives the linking number between two distinct symmetric links formed from modular knots.

Our interest in this paper is a generalization of the additive Deligne-Simpson problem which is originally defined for Fuchsian differential equations on the Riemann sphere. We shall extend this problem to differential equations having an arbitrary number of unramified irregular singular points and determine the existence of solutions of the generalized additive Deligne-Simpson problems. Moreover we apply this result to the geometry of the moduli spaces of stable meromorphic connections of trivial bundles on the Riemann sphere. Namely, open embedding of the moduli spaces into quiver varieties is given and the non-emptiness condition of the moduli spaces is determined. Furthermore the connectedness of the moduli spaces is shown., Comment: 62 pages, some errors and typos are corrected

Let $\mu=\mu_{n_{1}}\times\mu_{n_{2}}$, where $\mu_{n_{1}}$ and $\mu_{n_{2}}$ are upper doubling measures on $\mathbb{R}^{n_{1}}$ and $\mathbb{R}^{n_{2}}$, respectively. Let the pseudo-accretive function $b=b_{1}\otimes b_{2}$ satisfy a bi-parameter Carleson condition. In this paper, we established the $L^{2}(\mu)$ boundedness of non-homogeneous Littlewood–Paley $g_{\lambda}^{*}$-function with non-convolution type kernels on product spaces. This was mainly done by means of dyadic analysis and non-homogenous methods. The result is new even in the setting of Lebesgue measures.

In this paper, we construct growing modes of the linearized Navier-Stokes equations about generic stationary shear flows of the boundary layer type in a regime of sufficiently large Reynolds number: $R \to \infty$. Notably, the shear profiles are allowed to be linearly stable at the infinite Reynolds number limit, and so the instability presented is purely due to the presence of viscosity. The formal construction of approximate modes is well-documented in physics literature, going back to the work of Heisenberg, C.C. Lin, Tollmien, Drazin and Reid, but a rigorous construction requires delicate mathematical details, involving for instance a treatment of primitive Airy functions and singular solutions. Our analysis gives exact unstable eigenvalues and eigenfunctions, showing that the solution could grow slowly at the rate of $e^{t/\sqrt {R}}$. A new, operator-based approach is introduced, avoiding to deal with matching inner and outer asymptotic expansions, but instead involving a careful study of singularity in the critical layers by deriving pointwise bounds on the Green function of the corresponding Rayleigh and Airy operators., 55 pages. arXiv admin note: substantial text overlap with arXiv:1402.1395

In this paper, we consider a deformation of Plancherel measure linked to Jack polynomials. Our main result is the description of the first and second-order asymptotics of the bulk of a random Young diagram under this distribution, which extends celebrated results of Vershik-Kerov and Logan-Shepp (for the first order asymptotics) and Kerov (for the second order asymptotics). This gives more evidence of the connection with Gaussian $\beta$-ensemble, already suggested by some work of Matsumoto. Our main tool is a polynomiality result for the structure constant of some quantities that we call Jack characters, recently introduced by Lassalle. We believe that this result is also interested in itself and we give several other applications of it., Comment: 71 pages. Minor modifications from version 1. An extended abstract of this work, with significantly fewer results and a different title, is available as arXiv:1201.1806

The present authors introduced the notion of \emph{weakly unobstructed} Lagrangian submanifolds and constructed their \emph{potential function} $\mathfrak{PO}$ purely in terms of $A$-model data in [FOOO2]. In this paper, we carry out explicit calculations involving $\mathfrak{PO}$ on toric manifolds and study the relationship between this class of Lagrangian submanifolds with the earlier work of Givental [Gi1] which advocates that quantum cohomology ring is isomorphic to the Jacobian ring of a certain function, called the Landau-Ginzburg superpotential. Combining this study with the results from [FOOO2], we also apply the study to various examples to illustrate its implications to symplectic topology of Lagrangian fibers of toric manifolds. In particular we relate it to Hamiltonian displacement property of Lagrangian fibers and to Entov-Polterovich's symplectic quasi-states., Comment: 84 pages, submitted version ; more examples and new results added, exposition polished, minor typos corrected; v3) to appear in Duke Math.J., Example 10.19 modified, citations from the book [FOOO2,3] updated accoding to the final version of [FOOO3] to be published

As a continuation of the previous paper [7], we shall consider in this paper the problem of prediction given bounded intervals and obtain integral representations of predictors and prediction errors. For that purpose we shall introduce innovation processes well matched with bounded intervals. We follow the notation and terminology in [6].

In his paper [2], [3], D. A. Hejhal investigated the variational theory of linear polynomic functions. In this paper we are concerned with the variational theory of higher-order differential equations. To be more precise, consider a compact Riemann surface having genus g > 1. As is well known, we can choose a projective coordinate covering U = (Ua, za). Fix this coordinate covering of X. We shall be concerned with linear ordinary differential operators of order n defined in each projective coordinate open set Ua

The main purpose of this paper is to give a method to evaluate the actual value of the filtering error which arises in the transmission of a signal process, using the optimal coding, over a Gaussian channel. In his earier papers ([4] and [7]), the author has shown a method to construct an optimal causal coding for which the filtering error is minimized and at the same time the mutual information is maximized.

has a solution g that is unique modulo m. The exponent d of g modulo m is the least common multiple of 4(pl), ..., 0(p,), where sb(n) is the Euler function. In this paper we consider the problem of determining the number N(g) of solutions in s, t of the congruence gS + 1 gt (mod m), where the values of s and t are each selected from the integers 0, 1, . ., d 1. We discuss for the most part the case in which m is the product of two distinct odd primes p and q. If e denotes the greatest common divisor of p 1 and q 1, then d = (p 1)(q 1)/e. Let g denote a common primitive root of p and q, and let N(g) = Ne(g) denote the number of solutions in s,t of the congruence gs + 1 gt (mod pq), where 0 < s, t < d 1. In [3] the function Ne(g) is required for the construction of certain types of residue difference sets. Indeed, it is this application of Ne(g) that provided the motivation for the present paper (see the corollary of Theorem 3 in ?5). The case e = 2 is treated in ?4. We find that

We [4], [5] examined some properties of balayaged measures in the theory of a Dirichlet space. In those papers, we showed that the singular measure of a Dirichlet space plays some important roles. In this paper, we shall precisely examine some properties of the singular measure of a Dirichlet space. Let X be a locally compact Hausdorff space in which there exists a positive Radon measure ξ which is everywhere dense in X.

The paper generalizes the theory given in a previous paper for the Branching Process with Random Environments (BPRE) to cover situations where a certain kind of Markov dependence holds between successive environmental variables. The new process is called a Branching Process with Markovian Environments (BPME).

We use the expression Integral geometry in the sense given it by Blaschke in “Vorlesungen über Integrtüçeomelrie”. In a previous paper (“Integral formulas in Croflon style on the sphere and some inequalities referring to spherical curve” (1942), Duke Mathematical Journal, vol. 9, pp. 707) we generalized to the sphere many formulas of plane integral geometry and at the same time applied these to the demonstration of certain inequalities referring to spherical curves. The present paper considers analogous qüestions for surfaces of constant negative curvature and consequently for hyperbolic geometry https://doi.org/10.1215/S0012-7094-43-01064-6

In this paper we establish some theorems about the group of affine transformations on a Riemannian manifold. First we prove a decomposition theorem (Theorem 1) of the largest connected group of affine transformations on a simply connected complete Riemannian manifold, which corresponds to the decomposition theorem of de Rham [4] for the manifold. In the case of the largest group of isometries, a theorem of the same type is found in de Rham’s paper [4] in a weaker form. Using Theorem 1 we obtain a sufficient condition for an infinitesimal affine transformation to be a Killing vector field (Theorem 2). This result includes K. Yano’s theorem [13] which states that on a compact Riemannian manifold an infinitesimal affine transformation is always a Killing vector field. His proof of the theorem depends on an integral formula which is valid only for a compact manifold. Our method is quite different and is based on a result [11] of K. Nomizu.

In this paper we consider certain tensors associated with differentiable mappings of Riemannian manifolds and apply the results to a p-mapping, which is a special case of a subprojective one in affinely connected manifolds (cf. [1], [7]). The p-mapping in Riemannian manifolds is a generalization of a conformal mapping and a projective one. From a point of view of differential geometry an analogy between these mappings is well known. On the other hand it is interesting that a stereographic projection of a sphere onto a plane is conformal, while a central projection is projectve, namely geodesic-preserving. This situation was clarified partly in [6]. A p-mapping defined in this paper gives a precise explanation of this and also affords a certain mapping in the euclidean space which includes a similar mapping and an inversion as special cases.

In this paper we establish function field versions of two classical conjectures on prime numbers. The first says that the number of primes in intervals (x,x+x^epsilon] is about x^epsilon/log x and the second says that the number of primes p1 and m\geq 3 if q is even and deg f' \leq 1. We show that this estimation fails in the neglected cases. Let \pi_q(k) be the number of monic prime polynomials of degree k with coefficients in the finite field with q elements \FF_q. For relatively prime polynomials f,D\in \FF_q[t] we prove that the number N' of monic prime polynomials g that are congruent to f modulo D and of degree k satisfies |N'-\pi_q(k)/\phi(D)|\leq c(k)\pi_q(k)q^{-1/2}/\phi(D), as long as 1\leq \deg D\leq k-3 (or \leq k-4 if p=2 and (f/D)' is constant). We also generalize these results to other factorization types., Comment: Changes in the introduction, accepted to Duke

Let L be a divergence form elliptic operator with complex bounded measurable coefficients, let ω be a positive Musielak-Orlicz function on (0, ∞) of uniformly strictly critical lower-type pω ∈ (0, 1], and let ρ(x,t) = t−1/ω−1 (x,t−1) for x ∈ ℝn, t ∊ (0, ∞). In this paper, we study the Musielak-Orlicz Hardy space Hω,L(ℝn) and its dual space BMOρ,L* (ℝ n), where L* denotes the adjoint operator of L in L2 (ℝ n). The ρ-Carleson measure characterization and the John-Nirenberg inequality for the space BMOρ,L (ℝn) are also established. Finally, as applications, we show that the Riesz transform ∇L−1/2 and the Littlewood–Paley g-function gL map Hω,L(ℝn) continuously into L(ω).

A conjecture of Kadison and Kastler from 1972 asks whether sufficiently close operator algebras in a natural uniform sense must be small unitary perturbations of one another. For $n\geq 3$ and a free ergodic probability measure preserving action of $SL_n(\mathbb Z)$ on a standard nonatomic probability space $(X,\mu)$, write $M=((L^\infty(X,\mu)\rtimes SL_n(\mathbb Z))\,\overline{\otimes}\, R$, where $R$ is the hyperfinite II$_1$ factor. We show that whenever $M$ is represented as a von Neumann algebra on some Hilbert space $\mathcal H$ and $N\subseteq\mathcal B(\mathcal H)$ is sufficiently close to $M$, then there is a unitary $u$ on $\mathcal H$ close to the identity operator with $uMu^*=N$. This provides the first nonamenable class of von Neumann algebras satisfying Kadison and Kastler's conjecture. We also obtain stability results for crossed products $L^\infty(X,\mu)\rtimes\Gamma$ whenever the comparison map from the bounded to usual group cohomology vanishes in degree 2 for the module $L^2(X,\mu)$. In this case, any von Neumann algebra sufficiently close to such a crossed product is necessarily isomorphic to it. In particular, this result applies when $\Gamma$ is a free group., Comment: 33 pages. Paper restructured. Some of the material removed will appear in a future article. Duke Math. J., to appear

This paper describes the intrinsic geometry of a leaf $\mathcal{A}(L)$ of the absolute period foliation of the Hodge bundle $\Omega\overline{\mathcal{M}}_{g}$ : its singular Euclidean structure, its natural foliations, and its discretized Teichmüller dynamics. We establish metric completeness of $\mathcal{A}(L)$ for general $g$ and then turn to a study of the case $g=2$ . In this case the Euclidean structure comes from a canonical meromorphic quadratic differential on $\mathcal{A}(L)\cong{ \mathbb{H}}$ whose zeros, poles, and exotic trajectories are analyzed in detail.

The pentagram map was introduced by R. Schwartz in 1992 for convex planar polygons. Recently, V. Ovsienko, R. Schwartz, and S. Tabachnikov proved Liouville integrability of the pentagram map for generic monodromies by providing a Poisson structure and the sufficient number of integrals in involution on the space of twisted polygons. In this paper we prove algebraic-geometric integrability for any monodromy, i.e., for both twisted and closed polygons. For that purpose we show that the pentagram map can be written as a discrete zero-curvature equation with a spectral parameter, study the corresponding spectral curve, and the dynamics on its Jacobian. We also prove that on the symplectic leaves Poisson brackets discovered for twisted polygons coincide with the symplectic structure obtained from Krichever-Phong's universal formula., 33 pages, 1 figure; v3: substantially revised

Introduced in the late 1960's, the asymmetric exclusion process (ASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites with open boundaries. It has been cited as a model for traffic flow and protein synthesis. In the most general form of the ASEP with open boundaries, particles may enter and exit at the left with probabilities alpha and gamma, and they may exit and enter at the right with probabilities beta and delta. In the bulk, the probability of hopping left is q times the probability of hopping right. The first main result of this paper is a combinatorial formula for the stationary distribution of the ASEP with all parameters general, in terms of a new class of tableaux which we call staircase tableaux. This generalizes our previous work for the ASEP with parameters gamma=delta=0. Using our first result and also results of Uchiyama-Sasamoto-Wadati, we derive our second main result: a combinatorial formula for the moments of Askey-Wilson polynomials. Since the early 1980's there has been a great deal of work giving combinatorial formulas for moments of various other classical orthogonal polynomials (e.g. Hermite, Charlier, Laguerre, Meixner). However, this is the first such formula for the Askey-Wilson polynomials, which are at the top of the hierarchy of classical orthogonal polynomials.

We introduce loop spaces (in the sense of derived algebraic geometry) into the representation theory of reductive groups. In particular, we apply the theory developed in our previous paper arXiv:1002.3636 to flag varieties, and obtain new insights into fundamental categories in representation theory. First, we show that one can recover finite Hecke categories (realized by D-modules on flag varieties) from affine Hecke categories (realized by coherent sheaves on Steinberg varieties) via S^1-equivariant localization. Similarly, one can recover D-modules on the nilpotent cone from coherent sheaves on the commuting variety. We also show that the categorical Langlands parameters for real groups studied by Adams-Barbasch-Vogan and Soergel arise naturally from the study of loop spaces of flag varieties and their Jordan decomposition (or in an alternative formulation, from the study of local systems on a Moebius strip). This provides a unifying framework that overcomes a discomforting aspect of the traditional approach to the Langlands parameters, namely their evidently strange behavior with respect to changes in infinitesimal character., A strengthened version of the second half of arXiv:0706.0322, with significant new material. v2: minor revisions. v3: more minor revisions

In this paper, we construct (many) Kolyvagin systems out of Stickelberger elements utilizing ideas borrowed from our previous work on Kolyvagin systems of Rubin-Stark elements. The applications of our approach are twofold. First, assuming Brumer’s conjecture, we prove results on the odd parts of the ideal class groups of CM fields which are abelian over a totally real field, and we deduce Iwasawa’s main conjecture for totally real fields (for totally odd characters). Although this portion of our results has already been established by Wiles unconditionally (and refined by Kurihara using an Euler system argument, when Wiles’s work is assumed), the approach here fits well in the general framework the author has developed elsewhere to understand Euler/Kolyvagin system machinery when the core Selmer rank is r >1 (in the sense of Mazur and Rubin). As our second application, we establish a rather curious link between the Stickelberger elements and Rubin-Stark elements by using the main constructions of this article hand in hand with the “rigidity” of the collection of Kolyvagin systems proved by Mazur, Rubin, and the author., Scientific and Technological Research Council of Turkey (TÜBİTAK); Marie Curie grant

The real locus in the moduli space of complex algebraic curves of given genus consists of curves having real forms, that is, Riemann surfaces admitting a symmetry (anticonformal involution). The real locus is covered by subsets, each formed by curves having a given topological type determined by the number of connected components and the separability type of the real models of the curves. In this paper, we study the structure of the nerve corresponding to this covering, called the real nerve of complex algebraic curves, for even genera. We find its geometrical and homological dimension, and we obtain some results concerning its global geometrical properties. In the proofs, we use the equivalent language of Riemann surfaces and their symmetries.

We are concerned with the question how the capacity of the ideal boundary of a subsurface of a covering Riemann surface over a Riemann surface varies according to the variation of its branch points. In the present paper we treat the most primitive but fundamental situation that the covering surface is a two sheeted sphere with two branch points one of which is fixed and the other is moving and the subsurface is given as the complement of two disjoint continua each in different sheets of the covering surface whose projections are two disjoint continua in the base plane given in advance not touching the projections of branch points. We will derive a variational formula for the capacity and as one of its many useful consequences expected we will show that the capacity changes smoothly as one branch point moves in the subsurface.

In this paper we study the propagation of singularity for Schrodinger-type equations with variable coefficients. We introduce a new notion of wave propagation set, the homogeneous wave front set, which propagates along straight lines with finite speed away from x≠0. Then we show that it is related to the wave front set in a natural way. These results may be considered as a refinement of the microlocal smoothing property of Craig, Kappeler, and Strauss under more general assumptions.

For locally compact abelian groups it is known that if the product of the measures of the support of an $L^1$-function $f$ and its Fourier transform is less than $1$, then $f = 0$ almost everywhere. This is a weak version of the classical qualitative uncertainty principle. In this paper we focus on compact groups. We obtain conditions on the structure of a compact group under which there exists a lower bound for all products of the measures of the support of an integrable function and its Fourier transform, and conditions under which this bound equals $1$. For several types of compact groups, we determine the exact set of values which the product can attain.

In this paper the authors prove that a class of multilinear operators formed by the singular integral or fractional integral operators with homogeneous kernels are bounded operators from the product spaces Lp1 × Lp2 × · · · × LpK (ℝn) to the Hardy spaces Hq (ℝn) and the weak Hardy space Hq,∞(ℝn), where the kernel functions Ωij satisfy only the Ls-Dini conditions. As an application of this result, we obtain the (Lp, Lq) boundedness for a class of commutator of the fractional integral with homogeneous kernels and BMO function.

In this paper we use Lie group actions on noncompact Riemannian manifolds with calibrations to construct calibrated submanifolds. In particular, if we have an $(n-1)$-torus acting on a noncompact Calabi-Yau $n$-fold with a trivial first cohomology, then we have a special Lagrangian fibration on that $n$-fold. We produce several families of examples for this construction and give some applications to special Lagrangian geometry on compact almost Calabi-Yau manifolds. We also use group actions on noncompact $G_2$-manifolds to construct coassociative submanifolds, and we exhibit some new examples of coassociative submanifolds via this setup.

In this paper, we prove that if a complete Riemannian manifold M has finitely many ends, each of which is a Harnack end, then the set of all energy finite bounded A-harmonic functions on M is one to one corresponding to Rl, where A is a nonlinear elliptic operator of type p on M and l is the number of p-nonparabolic ends of M. We also prove that if a complete Riemannian manifold M is roughly isometric to a complete Riemannian manifold with finitely many ends, each of which satisfies the volume doubling condition, the Poincaré inequality and the finite covering condition near infinity, then the set of all energy finite bounded A-harmonic functions on M is finite dimensional. This result generalizes those of Yau, of Donnelly, of Grigor’yan, of Li and Tam, of Holopainen, and of Kim and the present author, but with a barrier argument at infinity that the peculiarity of nonlinearity demands.

In this paper, firstly, we show the Bondi-mass type estimate of solutions of Calabi flow on closed 4-manifolds. Secondly, in our applications, we obtain the long time existence on closed 4-manifolds. In particular, we are able to show the asymptotic convergence of a subsequence of solutions of the Calabi flow on closed Einstein 4-manifolds.

On the half space Rn × R+, it has been known that harmonic Bergman space bp can contain a positive function only if . Thus, for , Poisson integrals can be bp-functions only by means of their boundary cancellation properties. In this paper, we describe what those cancellation properties explicitly are. Also, given such cancellation properties, we obtain weighted norm inequalities for Poisson integrals. As a consequence, under weighted integrability condition given by our weighted norm inequalities, we show that our cancellation properties are equivalent to the bp-containment of Poisson integrals for p under consideration. Our results are sharp in the sense that orders of our weights cannot be improved.

This paper considers a higher-dimensional generalization of the notion of Ramanujan graphs, defined by Lubotzky, Phillips, and Sarnak. Specifically the Ramanujan property is studied for cubical complexes which are uniformized by an ordered product of regular trees. We construct explicit arithmetic examples using quaternion algebras over totally real fields. Here we reduce the Ramanujan property to special cases of the Ramanujan-Petersson conjecture for holomorphic Hilbert modular forms, many of which are known.

The Schubert calculus for G/B can be completely determined by a certain matrix related to the Kostant polynomials introduced in section 5 of Bernstein, Gelfand, and Gelfand [Bernstein, I., Gelfand, I. & Gelfand, S. (1973) Russ. Math. Surv. 28, 1–26]. The polynomials are defined by vanishing properties on the orbit of a regular point under the action of the Weyl group. For each element w in the Weyl group the polynomials also have nonzero values on the orbit points corresponding to elements which are larger in the Bruhat order than w. The main theorem given here is an explicit formula for these values. The matrix of orbit values can be used to determine the cup product for the cohomology ring for G/B, using only linear algebra or as described by Lascoux and Schutzenberger [Lascoux, A. & Schutzenberger, M.-P. (1982) C. R. Seances Acad. Sci. Ser. A 294, 447–450]. Complete proofs of all the theorems will appear in a forthcoming paper.

Let n > 1, m ≥ 1, g ≥ 3 and γ be given integers. The purpose of this paper is to determine the relations of n, m, g and γ for the existence of the symmetric Riemann surfaces S of type (n, m) with genus g and species γ. If n is an odd prime, the relations are known in [3]. In the case that n is odd, we shall show the analogous result when E(S) is isomorphic to a cyclic group Z2n and when the quotient space S/E(S) is orientable.

We study in this paper a generalization of the notion of a discrete hypergroup with particular emphasis on the relation with systems of orthogonal polynomials. The concept of a locally compact hypergroup was introduced by Dunkl [8], Jewett [12] and Spector [25]. It generalizes convolution algebras of measures associated to groups as well as linearization formulae of classical families of orthogonal polynomials, and many results of harmonic analysis on locally compact abelian groups can be carried over to the case of commutative hypergroups; see Heyer [11], Litvinov [17], Ross [22], and references cited therein. Orthogonal polynomials have been studied in terms of hypergroups by Lasser [15] and Voit [31], see also the works of Connett and Schwartz [6] and Schwartz [23] where a similar spirit is observed.

The primary purpose of this paper is to give an affirmative answer to a problem posed by Ohtsuka [13] whether there exists a p-harmonic measure on the unit disc in the 2-dimensional Euclidean space R2 with an infinite p-Dirichlet integral for the exponent 1 < p < 2.