Explicit unobstructed primes for modular deformation problems of squarefree level

Let f be a newform of weight k?3 with Fourier coefficients in a number field K. We give explicit bounds on the set of primes λ of K for which the deformation problem associated to the mod λ Galois representation of f is obstructed. We include some explicit examples.     

Newform theory for Hilbert Eisenstein series

In his thesis, Weisinger (Thesis, 1977) developed a newform theory for elliptic modular Eisenstein series. This newform theory for Eisenstein series was later extended to the Hilbert modular setting by Wiles (Ann. Math. 123(3):407–456, 1986). In this paper, we extend the theory of newforms for Hilbert modular Eisenstein series. In particular, we provide a strong multiplicity-one theorem in which we prove that Hilbert Eisenstein newforms are uniquely determined by their Hecke eigenvalues for any set of primes having Dirichlet density greater than $\frac{1}{2}$ . Additionally, we provide a number of applications of this newform theory. Let denote the space of Hilbert modular Eisenstein series of parallel weight k≥3, level $\mathcal{N}$ and Hecke character Ψ over a totally real field K. For any prime $\mathfrak{q}$ dividing $\mathcal{N}$ , we define an operator $C_{\mathfrak{q}}$ generalizing the Hecke operator $T_{\mathfrak{q}}$ and prove a multiplicity-one theorem for with respect to the algebra generated by the Hecke operators $T_{\mathfrak{p}}$ ( $\mathfrak{p}\nmid\mathcal{N}$ ) and the operators $C_{\mathfrak{q}}$ ( $\mathfrak{q}\mid\mathcal{N}$ ). We conclude by examining the behavior of Hilbert Eisenstein newforms under twists by Hecke characters, proving a number of results having a flavor similar to those of Atkin and Li (Invent. Math. 48(3):221–243, 1978).     

On the convergence of the rotated one-sided ergodic Hilbert transform

Sufficient conditions have been given for the convergence in norm and a.e. of the ergodic Hilbert transform (Gaposhkin in Theory Probab Appl 41:247–264, 1996; Cohen and Lin in Characteristic functions, scattering functions and transfer functions, pp 77–98, Birkhäuser, Basel, 2009; Cuny in Ergod Theory Dyn Syst 29:1781–1788, 2009). Here we apply these conditions to the rotated ergodic Hilbert transform \({\sum_{n=1}^\infty \frac{\lambda^n}{n} T^nf}\) , where λ is a complex number of modulus 1. When T is a contraction in a Hilbert space, we show that the logarithmic Hausdorff dimension of the set of λ’s for which this series does not converge is at most 2 and give examples where this bound is attained.     

S-adic L-Functions Attached to the Symmetric Square of a Newform

In order to apply the ideas of Iwasawa theory to the symmetricsquare of a newform, we need to be able to define non-archimedeananalogues of its complex L-series. The interpolated p-adic L-functionis closely connected via a "Main Conjecture" with certain Selmergroups over the cyclotomic Z_p-extension of Q. In the p-ordinarycase these functions are well understood. In this article we extend the interpolation to an arbitraryset S of good primes (not necessarily satisfying ordinarityconditions). The corresponding S-adic functions can be characterisedin terms of certain admissibility criteria. We also allow interpolationat particular primes dividing the level of the newform. One interesting application is to the symmetric square of amodular elliptic curve E defined over Q. Our constructions yieldp-adic L-functions at all primes of stable or semi-stable reduction.If p is ordinary or multiplicative the corresponding analyticfunction is bounded; if p is supersingular our function behaveslike log²(1 + T). 1991 Mathematics Subject Classification: 11F67,11F66, 11F33, 11F30     

Galois representations modulo p and cohomology of Hilbert modular varieties

The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let us mention:

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control of the image of Galois representations modulo p,

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Hida's congruence criterion outside an explicit set of primes,

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freeness of the integral cohomology of a Hilbert modular variety over certain local components of the Hecke algebra and Gorenstein property of these local algebras.

We study the arithmetic properties of Hilbert modular forms by studying their modulo p Galois representations and our main tool is the action of inertia groups at primes above p. In order to determine this action, we compute the Hodge-Tate (resp. Fontaine-Laffaille) weights of the p-adic (resp. modulo p) étale cohomology of the Hilbert modular variety. The cohomological part of our paper is inspired by the work of Mokrane, Polo and Tilouine on the cohomology of Siegel modular varieties and builds upon geometric constructions of Tilouine and the author.     

Criteria for biholomorphic convex mappings on the unit ball in Hilbert spaces

In this paper, we give a necessary and sufficient condition that a locally biholomorphic mapping f on the unit ball B in a complex Hilbert space X is a biholomorphic convex mapping, which improves some results of Hamada and Kohr and solves the problem which is posed by Graham and Kohr. From this, we derive some sufficient conditions for biholomorphic convex mapping. We also introduce a linear operator in purpose to construct some concrete examples of biholomorphic convex mappings on B in Hilbert spaces. Moreover, we give some examples of biholomorphic convex mappings on B in Hilbert spaces.     

Iwasawa invariants of Λ-adic representation

It is a classical result that the number of primes ? for which τ(?) vanishes has Dirichlet density 0, where τ(?) is the Ramanujan τ function. We study an analogous question that arises in studying the Λ-adic representation whose image is full. In particular, we show that the set of primes ? for which the trace of the Frobenius at ? has positive μ-invariant has Dirichlet density 0. We also discuss the analogous Dirichlet densities related to λ-invariants.     

The Tamagawa number conjecture of adjoint motives of modular forms

Let f be a newform of weight k?2, level N with coefficients in a number field K, and A the adjoint motive of the motive M associated to f. We carefully discuss the construction of the realisations of M and A, as well as natural integral structures in these realisations. We then use the method of Taylor and Wiles to verify the λ-part of the Tamagawa number conjecture of Bloch and Kato for L(A,0) and L(A,1). Here λ is any prime of K not dividing Nk!, and so that the mod λ representation associated to f is absolutely irreducible when restricted to the Galois group over where λ|?. The method also establishes modularity of all lifts of the mod λ representation which are crystalline of Hodge-Tate type (0,k−1).     

Weyl's theorem in several variables

In this note we consider Weyl's theorem and Browder's theorem in several variables. The main result is as follows. Let T be a doubly commuting n-tuple of hyponormal operators acting on a complex Hilbert space. If T has the quasitriangular property, i.e., the dimension of the left cohomology for the Koszul complex Λ(T−λ) is greater than or equal to the dimension of the right cohomology for Λ(T−λ) for all λ∈Cn, then ‘Weyl's theorem’ holds for T, i.e., the complement in the Taylor spectrum of the Taylor Weyl spectrum coincides with the isolated joint eigenvalues of finite multiplicity.     

Finite Hilbert stability of (bi)canonical curves

We prove that a generic canonically or bicanonically embedded smooth curve has semistable mth Hilbert points for all m≥2. We also prove that a generic bicanonically embedded smooth curve has stable mth Hilbert points for all m≥3. In the canonical case, this is accomplished by proving finite Hilbert semistability of special singular curves with $\mathbb{G}_{m}$ -action, namely the canonically embedded balanced ribbon and the canonically embedded balanced double A _2k+1-curve. In the bicanonical case, we prove finite Hilbert stability of special hyperelliptic curves, namely Wiman curves. Finally, we give examples of canonically embedded smooth curves whose mth Hilbert points are non-semistable for low values of m, but become semistable past a definite threshold.     

Stratonovich-Weyl correspondence for compact semisimple Lie groups

Let G be a compact Lie group, M a G-homogeneous space and π a unitary representation of G realized on a Hilbert space of functions on M. We give a general presentation of the Stratonovich-Weyl correspondence associated with π. In the case when G is a compact semisimple Lie group and π _λ an irreducible representation of G with highest weight λ, we study the Stratonovich-Weyl symbol of the derived operator d π _λ (X) for X in the Lie algebra of G and its behavior as λ goes to infinity.     

Elementary localization theorems for nonlinear eigenproblems

The minimax formula for linear eigenvalues of a linear operator is used to estimate the parameter values (λ) for which the self-adjoint operator L(λ) on Hilbert space to itself fails to have a bounded inverse. Such λ compose the “nonlinear spectrum” of L. The parameter spaces include regions in real or complex n-space. The localization theorems are used to demonstrate certain necessary conditions for stability of linear integro-partial-differential delay equations.     

Intersection bodies and generalized cosine transforms

The paper is focused on intimate connection between geometric properties of intersection bodies in convex geometry and generalized cosine transforms in harmonic analysis. A new concept of λ-intersection body, that unifies some known classes of geometric objects, is introduced. A parallel between trace theorems in function theory, restriction onto lower-dimensional subspaces of the spherical Radon transforms and the generalized cosine transforms, and sections of λ-intersection bodies is established. New integral formulas for different classes of cosine transforms are obtained and examples of λ-intersection bodies are given. We also revisit some known facts in this area and give them new simple proofs.     

Palindromic companion forms for matrix polynomials of odd degree

The standard way to solve polynomial eigenvalue problems P(λ)x=0 is to convert the matrix polynomial P(λ) into a matrix pencil that preserves its spectral information — a process known as linearization. When P(λ) is palindromic, the eigenvalues, elementary divisors, and minimal indices of P(λ) have certain symmetries that can be lost when using the classical first and second Frobenius companion linearizations for numerical computations, since these linearizations do not preserve the palindromic structure. Recently new families of pencils have been introduced with the goal of finding linearizations that retain whatever structure the original P(λ) might possess, with particular attention to the preservation of palindromic structure. However, no general construction of palindromic linearizations valid for all palindromic polynomials has as yet been achieved. In this paper we present a family of linearizations for odd degree polynomials P(λ) which are palindromic whenever P(λ) is, and which are valid for all palindromic polynomials of odd degree. We illustrate our construction with several examples. In addition, we establish a simple way to recover the minimal indices of the polynomial from those of the linearizations in the new family.     

Compact operators without extended eigenvalues

A complex number λ is called an extended eigenvalue of a bounded linear operator T on a Banach space B if there exists a non-zero bounded linear operator X acting on B such that XT=λTX. We show that there are compact quasinilpotent operators on a separable Hilbert space, for which the set of extended eigenvalues is the one-point set {1}.     

Local convergence in Fermat's problem

The general Fermat problem is to find the minimum of the weighted sum of distances fromm destination points in Euclideann-space. Kuhn recently proved that a classical iterative algorithm converges to the unique minimizing point , for any choice of the initial point except for a denumerable set. In this note, it is shown that although convergence is global, the rapidity of convergence depends strongly upon whether or not  is a destination. If  is not a destination, then locally convergence is always linear with upper and lower asymptotic convergence boundsλ andλ′ (λ ≥ 1/2, whenn=2). If  is a destination, then convergence can be either linear, quadratic or sublinear. Three numerical examples which illustrate the different possibilities are given and comparisons are made with the use of Steffensen's scheme to accelerate convergence.     

Spectrum of a network of Euler-Bernoulli beams

A network of N flexible beams connected by n vibrating point masses is considered. The spectrum of the spatial operator involved in this evolution problem is studied. If λ² is any real number outside a discrete set of values S and if λ is an eigenvalue, then it satisfies a characteristic equation which is given. The associated eigenvectors are also characterized. If λ² lies in S and if the N beams are identical (same mechanical properties), another characteristic equation is available. It is not the case for different beams: no general result can be stated. Some numerical examples and counterexamples are given to illustrate the impossibility of such a generalization. At last the asymptotic behaviour of the eigenvalues is investigated by proving the so-called Weyl's formula.     

Commutativity up to a factor for bounded and unbounded operators

In this paper, we further investigate the problem of commutativity up to a factor (or λ-commutativity) in the setting of bounded and unbounded linear operators in a complex Hilbert space. The results are based on a new approach to the problem. We finish the paper by a conjecture on the commutativity of self-adjoint operators.     

Elliptic boundary value problems with λ-dependent boundary conditions

In this paper second order elliptic boundary value problems on bounded domains Ω⊂Rn with boundary conditions on ∂Ω depending nonlinearly on the spectral parameter are investigated in an operator theoretic framework. For a general class of locally meromorphic functions in the boundary condition a solution operator of the boundary value problem is constructed with the help of a linearization procedure. In the special case of rational Nevanlinna or Riesz-Herglotz functions on the boundary the solution operator is obtained in an explicit form in the product Hilbert space L²(Ω)⊕(L²m(∂Ω)), which is a natural generalization of known results on λ-linear elliptic boundary value problems and λ-rational boundary value problems for ordinary second order differential equations.