In a similar way a seminorm p on e gives rise to a pseudo metric. This is an analogue of colemans analytic families in r. Browse other questions tagged functionalanalysis banachspaces norm or ask your own question. In this paper we define banach spaces of overconvergent halfintegral weight p adic modular forms and banach modules of families of overconvergent halfintegral weight p adic modular forms over admissible open subsets of weight space.
The main result is the construction of an explicit orthonormal basis of the f p tbanach spaces m r for large enough r. On padic families of automor phic forms springerlink. Completely continuous endomorphisms of p adic banach spaces by jeanpierre serre publ. For the banachmodule theory, see coleman padic banach spaces and families of modular forms. Springer at the meeting of september 26,1994 abstract in the set up of nonarchimedean functional analysis, banach spaces as well as. The authors would like to thank robert coleman and noriko yui for their. Panchishkin to dear jeanpierre serre for his eightieth birthday with admiration abstract. Every onetoone bounded linear operator from a banach space onto a banach space is an isomorphism. All vector spaces are assumed to be over the eld f.
Banach spaces from the metric we also get a topology notion of open set. The purpose of this paper however is to give an explicit construction of the parameter spaces in two of the simplest cases. A modular form isnt really a function on the upper half plane. The author begins with a discussion of weak topologies, weak compactness and isomorphisms of banach spaces before proceeding to the more detailed study of particular spaces. Special stanford algebraic geometry number theory seminar. The complementation property of linfinity in padic banach spaces. There exists an f p t banach space with an action of a compact operator u pwhose characteristic series is the reduction of p t. Twovariable padic lfunctions associated to hida deformations tadashi ochiai contents 1. Extended eigenvarieties for overconvergent cohomology. To achieve this was a real advance, and it led to the construction in cm98 of the eigencurve, a curve which is very likely of in nite degree over the weight space that parametrizes all padic eigenforms of nite.
X a nq n x na nq n lemma q is a weight 2 padic modular form. P adic banach spaces and families of modular forms robert f. Secondly, our trace is a trace of a compact operator acting on a complex of padic banach spaces although theirs is the usual classical trace computed by arthurselberg. Padic banach spaces and families of modular forms robert f. We formulate definitions of overconvergent padic automorphic forms for two more classes of reductive groups firstly for gl i over a number field, and secondly for d x, d a definite quaternion algebra over the rationals. For the banachmodule theory, see coleman padic banach spaces and families of modular forms, bgr, and buzzard eigenvarieties. Let p be a prime number, and let n be a positive integer coprime to p. Its a function on elliptic curves, or pairs consisting of an elliptic curve and a differential. If v is a normed space which is complete under its norm or rather, under the induced ultrametric dv,w kv. Math, 96 states that overconvergent modular forms of small slope are classical, and hence provides us with a way to identify elements of the above. Introduction let pbe a prime, cp the completion of an algebraic closure of the padicnumbers qp and ka.
Padic banach spaces and families of modular forms robert. Both spaces are equipped with a continuous hecke action for which u p 2 is moreover compact. Coleman, r a padic shimura isomorphism and padic periods of modular forms, contemporary mathematics 165, 2151 1994. It showcases research results in functional analysis over nonarchimedean valued complete fields. Throughout, f will denote either the real line r or the complex plane c. The book is intended to be used with graduate courses in banach space theory, so the prerequisites are a background in functional, complex and real analysis. This is a much more advanced topic, and one where the known results are quite fragmentary. Computations with classical and padic modular forms lms. P adic banach spaces and families of modular forms springerlink. Coleman conjectured that there should exist an fpxbanach space m. Dimension variation of classical and padic modular forms. Of course all these approaches have their own interest. As in the traditional setting, there are lots of simple examples of banach spaces, e. Function spaces a function space is a vector space whose \vectors are functions.
To achieve this was a real advance, and it led to the construction in cm98 of the eigencurve, a curve which is very likely of in nite degree over the weight space that parametrizes all p adic eigenforms of nite. The u p operator acts on this space as a compact operator, and cuts out a nitedimensional locus on which it acts with a bounded slope. Coleman, padic banach spaces and families of modular forms, invent. Our padic lfunctions are mellin transforms of certain measures with aluesv in a, where a ab denotes an a noid algebra associated with an a noid space b as in copb. Coleman november 22, 1954 march 24, 2014 was an american mathematician and professor at the university of california, berkeley. He is also known for introducing padic banach spaces into the study of modular forms. With barry mazur, he introduced the eigencurve and established some of its fundamental properties. In part a, which we entitle families of banach spaces, we show how serres padicbanachfredholmriesz theory s works in a family, i.
Pages 623697 from volume 181 2015, issue 2 by fabrizio andreatta, adrian iovita, vincent pilloni. Coleman, r a padic shimura isomorphism and padic periods of. Since coleman and mazurs original work 10, there have been. The theta operator and the weight 2 eisenstein series f q. There are more choices, but none of them is natural, to my knowledge, or preferred. P adic banach spaces and families of modular forms. A padic banach space b is a q pvector space with a lattice b0z pmodule separated and complete for the padic topology, i. Coleman is also known for introducing p adic banach spaces into the study of modular forms and discovering important classicality criteria for overconvergent p adic modular forms. Coleman, p adic banach spaces and families of modular forms, invent. He worked primarily in number theory, with specific interests in padic analysis and arithmetic geometry. Hence we get a map from classical modular forms of level p over k0 to. The product of a finite number of banach spaces can easily made into a banach space by, e. A presentation of results in p adic banach spaces, spaces over fields with an infinite rank valuation, frechet and locally convex spaces with schauder bases, function spaces, p adic harmonic analysis, and related areas. Introduction in the construction of eigenvarieties, the space of overconvergent modular forms arises as an in nitedimensional p adic banach space.
For the banach module theory, see coleman padic banach spaces and families of modular forms. Coleman is also known for introducing padic banach spaces into the study of modular forms and discovering important classicality criteria for overconvergent padic modular. X y be a surjective continuous linear operator, then t is an open map. On padic families of hilbert cusp forms of finite slope core. We consider the padic weight space xcontaining all k j. Pdf higher hida and coleman theories on the modular curve. Introduction let pbe a prime, cp the completion of an algebraic closure of the p adicnumbers qp and ka. In many ways padic analysis is less subtle than classical analysis, since the ultrametric inequality means, for example, that convergence of infinite series of padic numbers is much simpler. The purpose of this paper however is to give an explicit construction of the parameter spaces in two of the simplest. On a padic interpolation of the kohnenzagier formula and. Topological vector spaces over p adic fields show distinctive features. In part a, which we entitle families of banach spaces, we show how serres padicbanachfredholm riesz theory s works in a family, i.
Proceedings of the international conference held in trento, italy, may 29june 2, 1989. Background from p adic functional analysis throughout this course kis a nite extension of the eld q p of p adic numbers, jjdenotes the absolute value on k, o kthe ring of integers. Pdf dimension variation of classical and padic modular forms. I will present a new method for proving a generalization of colemans theorem which. Throughout this course k is a finite extension of the field qp of padic numbers, denotes the absolute. In this paper we define banach spaces of overconvergent halfintegral weight padic modular forms and banach modules of families of overconvergent halfintegral weight padic modular forms over admissible open subsets of weight space. Schikhof mathematical institute, university ofnijmegen, toernooiveld, 6525 ed nijmegen, the netherlands communicated by prof.
Let pn denote the partition function, and let q124 q 1 n1 1 q n denote the dedekind function. Coleman and mazur have constructed eigencurves, geometric objects parametrising certain overconvergent padic modular forms. Exercise 1 sequence spaces p are banach spaces given a sequence x 1. Arithmetic properties of fredholm series for padic modular forms. Coleman this paper is dedicated to bernard dwork who has been a friend and an inspiration for many years. Cpsi coleman, r a p adic shimura isomorphism and p adic periods of modular forms, contemporary mathematics 165, 2151 1994. Completely continuous endomorphisms of adic banach spaces. Browse other questions tagged functionalanalysis banach spaces norm or ask your own question. Q is even, we obtain padic analytic families of hilbert eigenforms having fixed finite slope parametrized by weights. There exists an f ptbanach space with an action of a compact operator u pwhose characteristic series is the reduction of pt. Each of m y n and s y n is an lbspace a compact inductive limit of banach spaces and there is the usual hecke action where, in.
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