In this talk, we introduce the well known hilbert schimdt and schatten p class operators on p adic hilbertspaces. This space is rather large and pathological from the viewpoint of classical algebraic geometry. A subset cof a vector space xis said to be convex if for all. Request pdf invariant subspaces of p adic hilbert spaces in this paper we study the invariant subspaces of padic hilbert spaces. A linear space is a set xwith an associated scalar eld f in this course either. There is a beautiful theory of representations over such and so there is no serious problem.
Rigged hilbert space of free coherent states and p adic. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. This applies to the study of congruences between special values of zeta functions of totally real elds. The application of the theorytoelliptic curvesisdiscussed. We prove that this rigged hilbert space is isomorphic to the space of generalized functions over a p adic disk. Mathematically it is natural to work over the p adic. It can be told as a part of the history of mathematics or as an integral stage in the development of modern physics. Hilbert spaces provide a userfriendly framework for the study of a wide range of subjects, from fourier analysis to quantum mechanics. Nevertheless, in a recent preprint titled on torsion in the cohomology of locally symmetric varieties, scholze showed that agp1is a wellbehaved object. In general, padic hilbert spaces do not posses many of the fundamental characteristics of hilbert spaces.
Hensel though they are foreshadowed in the work of his predecessor e. Although we begin the oral lectures with a crash course on some basic highlights from rigidanalytic geometry in the sense of tate, some awareness of those ideas is taken as. P adic banach spaces and families of modular forms robert f. If the inner product space is complete in this norm or in other words, if it is complete in the metric arising from the norm, or if it is a banach space with this norm then we call it a hilbert space. The goal of this paper is to construct plusminus padic lfunctions for hilbert modular forms for certain nonordinary primes p theorem 2. Notice that the isomorphism 1 can be extendedtothe space of c. Full pdf abstract top we construct padic lfunctions in general case unbounded attached to motivic primitive hilbert cusp forms as a nonarchimedean mellin transform of the corresponding admissible measure.
May 07, 2002 rigged hilbert space of the free coherent states and padic numbers article pdf available in theoretical and mathematical physics 52 may 2002 with 30 reads how we measure reads. Introduction let pbe a prime, cp the completion of an algebraic closure of the p adicnumbers qp and ka. Hilbert space theory and applications in basic quantum. We also show that the trace class operators in p adic hilbert spaces contains the class of completelycontinuousoperators,whichcontains the schatten class operators.
Matti wurthen essen the proetale site of a rigid analytic space b101 52 wed. Representation of a quantum field hamiltonian in p adic. They are thus functions on babb where 0 hilbert spaces 1. This is an analogy to riesz theorem on the selfduality for hilbert spaces. Considering the padic lfunction of hilbert modular form as a function with the cyclotomic variable, i study the in.
In the p adic theory we study congruences between hilbert modular forms. Another way to put it is that a hilbert space is a banach space where the norm arises from some inner product. Coleman this paper is dedicated to bernard dwork who has been a friend and an inspiration for many years. But as there are no hilbert spaces we have to work in the category of banach spaces over the. Hilbert schimdt and schatten p class operators on p adic hilbertspaces. On the padic lfunction of hilbert modular forms at. That space time geometry could be based on a padic or even a. Convergence of the cesarolike means on padic hilbert spaces. Halmos, a hilbert space problem book, springerverlag, new york. Workshop padic riemann hilbert correspondence darmstadt, june 68, 2018 in this workshop we will study aspects of the padic riemann hilbert correspondence following ruochuan liu and xinwen zhu lz and some applications. In p adic hilbert space, representations of quantum hamiltonians for systems with an infinite number of degrees of freedom are constructed. A key ingredient in the algorithm is to compute the hilbert class polynomial pd for a suitable discriminant d. It seems that hensels main motivation was the analogy between the ring of integers z, together with its.
Banachspace operators acting on semicircular elements. Pdf rigged hilbert space of the free coherent states and. For v pwe take them to be elements of the schwartzbruhat space, namely, functions which are locally constant and compactly supported. Hilbert space theory and applications in basic quantum mechanics. In short, real numbers in decimal form can go on to the right forever and padic numbers in decimal form can go on to the left forever. A hilbert space his a pre hilbert space which is complete with respect to the norm induced by the inner product. Padic banach spaces and families of modular forms robert. Tgd framework padic quantum theory utilizes padic hilbert space.
Hilbert modular forms and padic hodge theory cambridge core. Invariant subspaces of p adic hilbert spaces request pdf. Approach with p adic valued wave functions has been introduced by vladimirov and volovich, 1 and developed by khrennikov 14, 15. For the p adic galois representation associated to a hilbert modular form, carayol has shown that, under a certain assumption, its restriction to the local galois group at a finite place not dividing p is compatible with the local langlands correspondence. The development of hilbert space, and its subsequent popularity, were a result of both mathematical and physical necessity. The hamiltonians with singular potentials are realized as bounded symmetric operators in l2 space with respect to a p adic gaussian measure. The adic hilbert eigenvariety fabrizio andreatta adrian iovita vincent pilloni july 24, 2015 1 introduction robert coleman constructed nite slope families of overconvergent modular forms parametrized by rigid analytic spaces over q p. A classical approach of constructing elliptic curves that can be used for cryptographic purposes relies on the theory of complex multiplication. As examples we know that cnwith the usual inner product 3. This extends the result of panchishkin lecture notes in math.
This was an extension of hidas construction of ordinary families. On hilbert schmidt and schatten p class operators in p. We also show that the trace class operators in p adic hilbert spaces contains the class of. In the padic theory we study congruences between hilbert modular forms. The p adic general linear group acts on x, and, as with complex symmetric spaces, the quotients of x by discrete groups sometimes turn out to be algebraic varieties defined over number fields. We discuss the relation of the described isomorphism of rigged hilbert spaces and noncommutative geometry and show that the considered example realizes the isomorphism between the noncommutative line and the p adic disk. On padic properties of twisted traces of singular moduli 3 remark 1. Singular moduli are the values of the jfunction at the roots q2h of integral binary quadratic forms q. Introduction to p adic numbers an overview of ultrametric spaces and p adic numbers. Plusminus padic lfunctions for hilbert modular forms. Under the same assumption, we show that the same is true for the places dividing p, in the sense of padic hodge theory, as is shown for an. Goren this paper is concerned with developing the theory of hilbert modular forms along the lines of the theory of elliptic modular forms. Rigged hilbert space of the free coherent states and padic numbers article pdf available in theoretical and mathematical physics 52 may. L 2qp,x l 2qp,x in space of functions, which are squareintegrable in bochner sense and take value in x, is a bounded operator.
Mathematically it is natural to work over the padic. Characterizing hilbert spaces using fourier transform over. To approach the topic let us recall the riemann hilbert correspondence over the complex numbers. Rigged hilbert space of the free coherent states and p adic numbers article pdf available in theoretical and mathematical physics 52 may 2002 with 30 reads how we measure reads. We investigate the rigged hilbert space of free coherent states. A padic model of eternal in ation, a padic quantum. The polynomial pd has integer coefficients, and is the minimal polynomial of the modular jvalue. Smooth padic analytic spaces are locally contractible. For the basic theory of toeplitz operators on the hardy space, the reader.
A hilbert space is an inner product space h,h,i such that the induced hilbertian norm is complete. Background from padic functional analysis throughout this course kis a nite extension of the eld q p of padic numbers, jjdenotes the absolute value on k, o kthe ring of integers. They are thus functions on babb where 0 p adic uniformization. In order to prove the growth conditions of the appropriate complexvalued distributions we represent them as rankin type. The historical events and individuals responsible for the topics we will cover make up an interesting story. These values are algebraic integers, and their minimal polynomials are hilbert class polynomials. In particular, we have a natural decomposition of the adic weight space w f w rig f w f. From the beginnning it was clear that the straightforward generalization of schr odinger equation is not very interesting physically and gradually.
We show some fundamental differences between the complex case. A padic algorithm to compute the hilbert class polynomial. For the padic galois representation associated to a hilbert modular form, carayol has shown that, under a certain assumption, its restriction to the local galois group at a finite place not dividing p is compatible with the local langlands correspondence. An important paper of zagier zag02 shows that the coe cients an2d.