We show some fundamental differences between the complex case. This is an analogy to riesz theorem on the selfduality for hilbert spaces. In general, p adic uniformization replaces complex symmetric spaces xwith p adic analytic spaces xone in each dimension. In order to prove the growth conditions of the appropriate complexvalued distributions we represent them as rankin type. 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. Invariant subspaces of p adic hilbert spaces request pdf. In general, padic hilbert spaces do not posses many of the fundamental characteristics of hilbert spaces. An introduction to hilbert spaces university of newcastle. Approach with p adic valued wave functions has been introduced by vladimirov and volovich, 1 and developed by khrennikov 14, 15.
The development of hilbert space, and its subsequent popularity, were a result of both mathematical and physical necessity. Introduction to p adic numbers an overview of ultrametric spaces and p adic numbers. In the p adic theory we study congruences between hilbert modular forms. They are thus functions on babb where 0 hilbert spaces 1. Notice that the isomorphism 1 can be extendedtothe space of c. Nevertheless, in a recent preprint titled on torsion in the cohomology of locally symmetric varieties, scholze showed that agp1is a wellbehaved object. On hilbert schmidt and schatten p class operators in p. Rigged hilbert space of the free coherent states and padic numbers article pdf available in theoretical and mathematical physics 52 may.
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. This was an extension of hidas construction of ordinary families. 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. P adic banach spaces and families of modular forms robert f. On the padic lfunction of hilbert modular forms at. Representation of a quantum field hamiltonian in p adic. Plusminus padic lfunctions for hilbert modular forms. It seems that hensels main motivation was the analogy between the ring of integers z, together with its.
It can be told as a part of the history of mathematics or as an integral stage in the development of modern physics. 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. Matti wurthen essen the proetale site of a rigid analytic space b101 52 wed. A subset cof a vector space xis said to be convex if for all. Mathematically it is natural to work over the p adic. 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.
Considering the padic lfunction of hilbert modular form as a function with the cyclotomic variable, i study the in. Hilbert modular forms and padic hodge theory cambridge core. Hensel though they are foreshadowed in the work of his predecessor e. The historical events and individuals responsible for the topics we will cover make up an interesting story. A hilbert space his a pre hilbert space which is complete with respect to the norm induced by the inner product. This space is rather large and pathological from the viewpoint of classical algebraic geometry. Halmos, a hilbert space problem book, springerverlag, new york. In particular, we have a natural decomposition of the adic weight space w f w rig f w f. The polynomial pd has integer coefficients, and is the minimal polynomial of the modular jvalue.
The goal of this paper is to construct plusminus padic lfunctions for hilbert modular forms for certain nonordinary primes p theorem 2. For the basic theory of toeplitz operators on the hardy space, the reader. As examples we know that cnwith the usual inner product 3. On padic properties of twisted traces of singular moduli 3 remark 1. There is a beautiful theory of representations over such and so there is no serious problem. 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. Pdf rigged hilbert space of the free coherent states and. Hilbert space theory and applications in basic quantum.
A linear space is a set xwith an associated scalar eld f in this course either. Mathematically it is natural to work over the padic. Rigged hilbert space of free coherent states and p adic. 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. The hamiltonians with singular potentials are realized as bounded symmetric operators in l2 space with respect to a p adic gaussian measure. In p adic hilbert space, representations of quantum hamiltonians for systems with an infinite number of degrees of freedom are constructed. Another way to put it is that a hilbert space is a banach space where the norm arises from some inner product. This applies to the study of congruences between special values of zeta functions of totally real elds. 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. A padic model of eternal in ation, a padic quantum. 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. Singular moduli are the values of the jfunction at the roots q2h of integral binary quadratic forms q. 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.
Smooth padic analytic spaces are locally contractible. Characterizing hilbert spaces using fourier transform over. This extends the result of panchishkin lecture notes in math. That space time geometry could be based on a padic or even a. Overconvergent cohomology of hilbert modular varieties and p. They are thus functions on babb where 0 p adic uniformization. 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. We investigate the rigged hilbert space of free coherent states. 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. On hilbert schmidt and schatten p class operators in padic. 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. Convergence of the cesarolike means on padic hilbert spaces.
Hilbert space theory and applications in basic quantum mechanics. A padic algorithm to compute the hilbert class polynomial. Introduction let pbe a prime, cp the completion of an algebraic closure of the p adicnumbers qp and ka. Hilbert schimdt and schatten p class operators on p adic hilbertspaces. Enrico beltrametti and his collaborators cassinelli and blasi in the late 1960s and early 1970s. Coleman this paper is dedicated to bernard dwork who has been a friend and an inspiration for many years. Pdf rigged hilbert space of the free coherent states and p. A classical approach of constructing elliptic curves that can be used for cryptographic purposes relies on the theory of complex multiplication.
The application of the theorytoelliptic curvesisdiscussed. Banachspace operators acting on semicircular elements. In this talk, we introduce the well known hilbert schimdt and schatten p class operators on p adic hilbertspaces. Hilbert spaces provide a userfriendly framework for the study of a wide range of subjects, from fourier analysis to quantum mechanics. Goren this paper is concerned with developing the theory of hilbert modular forms along the lines of the theory of elliptic modular forms. From the beginnning it was clear that the straightforward generalization of schr odinger equation is not very interesting physically and gradually. To approach the topic let us recall the riemann hilbert correspondence over the complex numbers. Tgd framework padic quantum theory utilizes padic hilbert space. In the padic theory we study congruences between hilbert modular forms. These values are algebraic integers, and their minimal polynomials are hilbert class polynomials. For v pwe take them to be elements of the schwartzbruhat space, namely, functions which are locally constant and compactly supported. We also show that the trace class operators in p adic hilbert spaces contains the class of completelycontinuousoperators,whichcontains the schatten class operators. Precisely, banach space x is isomorphic to a hilbert one if and only if fourier transform f.