In p adic hilbert space, representations of quantum hamiltonians for systems with an infinite number of degrees of freedom are constructed. 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. Enrico beltrametti and his collaborators cassinelli and blasi in the late 1960s and early 1970s. Considering the padic lfunction of hilbert modular form as a function with the cyclotomic variable, i study the in. This was an extension of hidas construction of ordinary families.
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. 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. We show some fundamental differences between the complex case. Convergence of the cesarolike means on padic hilbert spaces.
Pdf rigged hilbert space of the free coherent states and p. Introduction let pbe a prime, cp the completion of an algebraic closure of the p adicnumbers qp and ka. Overconvergent cohomology of hilbert modular varieties and p. The goal of this paper is to construct plusminus padic lfunctions for hilbert modular forms for certain nonordinary primes p theorem 2. Hilbert modular forms and padic hodge theory cambridge core. In this talk, we introduce the well known hilbert schimdt and schatten p class operators on p adic hilbertspaces. In general, padic hilbert spaces do not posses many of the fundamental characteristics of hilbert spaces. The polynomial pd has integer coefficients, and is the minimal polynomial of the modular jvalue.
They are thus functions on babb where 0 p adic uniformization. This space is rather large and pathological from the viewpoint of classical algebraic geometry. 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. We prove that this rigged hilbert space is isomorphic to the space of generalized functions over a p adic disk. Introduction to p adic numbers an overview of ultrametric spaces and p adic numbers. Rigged hilbert space of free coherent states and p adic. Rigged hilbert space of the free coherent states and padic numbers article pdf available in theoretical and mathematical physics 52 may. From the beginnning it was clear that the straightforward generalization of schr odinger equation is not very interesting physically and gradually. On hilbert schmidt and schatten p class operators in p. Tgd framework padic quantum theory utilizes padic hilbert space. 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. There is a beautiful theory of representations over such and so there is no serious problem. In the p adic theory we study congruences between hilbert modular forms.
We also show that the trace class operators in p adic hilbert spaces contains the class of. In general, p adic uniformization replaces complex symmetric spaces xwith p adic analytic spaces xone in each dimension. Goren this paper is concerned with developing the theory of hilbert modular forms along the lines of the theory of elliptic modular forms. Hilbert spaces provide a userfriendly framework for the study of a wide range of subjects, from fourier analysis to quantum mechanics. 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. Invariant subspaces of p adic hilbert spaces request pdf.
To approach the topic let us recall the riemann hilbert correspondence over the complex numbers. 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. Mathematically it is natural to work over the p adic. Smooth padic analytic spaces are locally contractible.
In particular, we have a natural decomposition of the adic weight space w f w rig f w f. 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. Precisely, banach space x is isomorphic to a hilbert one if and only if fourier transform f. An important paper of zagier zag02 shows that the coe cients an2d. Characterizing hilbert spaces using fourier transform over. Halmos, a hilbert space problem book, springerverlag, new york. Approach with p adic valued wave functions has been introduced by vladimirov and volovich, 1 and developed by khrennikov 14, 15. Padic banach spaces and families of modular forms robert. Hensel though they are foreshadowed in the work of his predecessor e. In the padic theory we study congruences between hilbert modular forms. That space time geometry could be based on a padic or even a. On the padic lfunction of hilbert modular forms at. 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. For v pwe take them to be elements of the schwartzbruhat space, namely, functions which are locally constant and compactly supported.
They are thus functions on babb where 0 hilbert spaces 1. An introduction to hilbert spaces university of newcastle. Notice that the isomorphism 1 can be extendedtothe space of c. Hilbert space theory and applications in basic quantum mechanics. For the basic theory of toeplitz operators on the hardy space, the reader. This extends the result of panchishkin lecture notes in math. Hilbert space theory and applications in basic quantum. The development of hilbert space, and its subsequent popularity, were a result of both mathematical and physical necessity. Plusminus padic lfunctions for hilbert modular forms. We also show that the trace class operators in p adic hilbert spaces contains the class of completelycontinuousoperators,whichcontains the schatten class operators. 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.
P adic banach spaces and families of modular forms robert f. Representation of a quantum field hamiltonian in p adic. A hilbert space his a pre hilbert space which is complete with respect to the norm induced by the inner product. This is an analogy to riesz theorem on the selfduality for hilbert spaces. Mathematically it is natural to work over the padic. Pdf rigged hilbert space of the free coherent states and.
This applies to the study of congruences between special values of zeta functions of totally real elds. Coleman this paper is dedicated to bernard dwork who has been a friend and an inspiration for many years. The hamiltonians with singular potentials are realized as bounded symmetric operators in l2 space with respect to a p adic gaussian measure. Request pdf invariant subspaces of p adic hilbert spaces in this paper we study the invariant subspaces of padic hilbert spaces. Banachspace operators acting on semicircular elements. Nevertheless, in a recent preprint titled on torsion in the cohomology of locally symmetric varieties, scholze showed that agp1is a wellbehaved object. The historical events and individuals responsible for the topics we will cover make up an interesting story.
We investigate the rigged hilbert space of free coherent states. A subset cof a vector space xis said to be convex if for all. These values are algebraic integers, and their minimal polynomials are hilbert class polynomials. 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. A padic algorithm to compute the hilbert class polynomial. Another way to put it is that a hilbert space is a banach space where the norm arises from some inner product. 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. It can be told as a part of the history of mathematics or as an integral stage in the development of modern physics. Matti wurthen essen the proetale site of a rigid analytic space b101 52 wed. 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. A padic model of eternal in ation, a padic quantum. As examples we know that cnwith the usual inner product 3. A classical approach of constructing elliptic curves that can be used for cryptographic purposes relies on the theory of complex multiplication.
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