Z p q p is a p adic locally analytic function then we can easily obtain a description of the p adic valuations of numbers f n, n. Sets of range uniqueness in a p adic field are examined. He spent the year 1974 75 and the spring semester 1978 in moscow, where he did research in p adic analysis and also translated yu. Pdf let k be an algebraically closed field of characteristic 0, complete with respect to an ultrametric absolute value. A modular form isnt really a function on the upper half plane. This paper studies padic comparison theorems in the general setting of rigidanalytic varieties. Sondows conjecture, convergents to e, and p adic analytic. We solve an open problem posed in 2, more precisely we show that those padic functions are locally analytic of. Sets of range uniqueness in a padic field are examined. For example, the padic lfunction of a cusp form fshould know the value lf. The adic of p adic comes from the ending found in words such as dyadic or triadic. Thus, the search for p adic incarnations of the classical zeta. In a wellknown paper, kubota and leopoldt 2 proved the existence of padic analytic functions that may be regarded as padic analogs of the classical dirichlet lseries.
For holomorphic modular forms, the basic analytic properties oftheirlfunctions were established by hecke. Thus, the search for padic incarnations of the classical zeta. Up to now, such comparison isomorphisms were studied for algebraic varieties over padic. On irreducible representations of compact padic analytic. Although, in the complex case, functions are more amenable to analysis of the above kind than gfunctions, the situation is completely reversed in the padic domain. Kubotaleopoldt padic lfunctions are, for the padic analysis, the functions corresponding to the complex variable lfunctions associated to dirichlet characters. Introduction to p adic analytic number theory download. In the latter half of the 20th century, this restricted viewpoint was enlarged through the foundational work of kubota and leopoldt and later by iwasawa who established much of the groundwork of a p adic analytic number theory. Teitelbaum in the early sixties, amice am1, am2 studied the space of kvalued, locally analytic functions on zzp and formulated a complete description of its dual, the ring of kvalued, locally qpanalytic distributions on zzp, when kis a complete sub. Also, an explicit formula is obtained for these functions, and a relationship between the values of the padic and classical lfunctions at positive integers is discussed.
Many interesting results on polynomial equations can be proved using the mysterious interactions between algebraic, complex analytic and padic analytic geometry. Let x be a connected a noid rigidanalytic variety over k. The blue plain line represents the graph of the function f. In a wellknown paper, kubota and leopoldt 2 proved the existence of p adic analytic functions that may be regarded as p adic analogs of the classical dirichlet lseries. Overconvergent modular symbols and padic lfunctions of. Hida, growth of hecke fields along a padic analytic family of modular forms, families of automorphic forms and the trace formula, pp.
They allow us to establish that the representation zeta functions of generic members of families of padic analytic propgroups obtained from a global, perfect. On irreducible representations of compact padic analytic groups. Artin conjecture for padic galois representations of. Construction of such padic lfunctions amounts to a problem of padic interpolation of special values of hecke lseries attached to grossencharac.
Applications of p adic analysis have mainly been in number theory, where it has a significant role in diophantine geometry and diophantine approximation. Igusas theorem on the rationality of the zeta function. For holomorphic modular forms, the basic analytic properties oftheirl functions were established by hecke. Exceptional values of padic analytic functions and derivatives. A course in padic analysis american mathematical society. In the latter half of the 20th century, this restricted viewpoint was enlarged through the foundational work of kubota and leopoldt and later by iwasawa who established much of the groundwork of a padic analytic number theory.
An application of strassmanns theorem keith conrad. One can also consider padic characters with values in the unit group of other padic rings such as o c p or the two dimensional local ring z pt. The padic lfunction of a modular form a padic lfunction is a padic object that interpolates special classical lvalues. However, this value is a complex number, and it doesnt make an awful lot of sense to talk about a function on a padic space. Delignes conjecture of special values of lfunctions 18 4. Cfunctions defined over an arbitrary number field, rather than the rational or quadratic fields.
Ultrametric absolute values and valuation functions. Construction of such p adic l functions amounts to a problem of p adic interpolation of special values of hecke lseries attached to grossencharac. Analytic constructions of padic lfunctions and eisenstein. They allow us to establish that the representation zeta functions of generic members of families of padic analytic prop groups obtained from a global, perfect lie lattice satisfy functional equations. In the course of writing those papers the author found it useful to adopt a particular point of view on locally analytic representation theory. The padic absolute value or padic norm of xis dened as jxj p p k if x6 0.
Certain properties of p adic transcendental numbers are examined such as order and type of transcendence, with problems on p adic exponentials. Indeed, as yet, no natural p adic generalisation of the siegelshidlovsky. The starting point of this theory is in vladimir berkovichs book about spectral theory and nonarchimedean analytic geometry. We also classify subanalytic sets up to subanalytic bijection. Lectures on some aspects of padic analysis school of. Let fx be a power series with coefficients in qp that converges on zp and is not identically zero. Filtrations of padic analytic galois groups of local.
Locally analytic vectors in representations of padic groups 3 representation of g, then we say that this representation is ganalytic if for each w. Introduction let pdenote a xed prime number, z p the ring of padic integers, q p the eld of padic numbers, jjthe padic norm, and vthepadic. Hensel though they are foreshadowed in the work of his predecessor e. Exceptional values of padic analytic functions and derivatives article pdf available in complex variables and elliptic equations 5614. Iwasawa functions, padic measures, and distributions 1. Lazards problem for analytic functions inside a disk is explained. The functions zn, n a nonnegative integer, and ez are entire functions. Usually such a number is algebraic after normalization via the. Locally analytic vectors in representations of locally padic analytic groups matthew emerton northwestern university contents introduction 1 1. Exceptional values of p adic analytic functions and derivatives. Indeed, as yet, no natural padic generalisation of the siegelshidlovsky.
The p in p adic is a variable and may be replaced with a prime yielding, for instance, the 2adic numbers or another placeholder variable for expressions such as the. The key concepts are those of uniform groups, their lower pseries and their dimension. Lately padic numbers have attracted a great deal of attention in modern theoretical physics as a promising new approach for describing the nonarchimedean geometry of spacetime at small distances. These are the notes for a tutorial given by jack a. Strict topologies and duals in spaces of functions. Pdf analytic constructions of padic lfunctions and. This is applied to the differential equation yhy y,h analytic elements on d, analytic interpolation, injectivity, and to the padic fourier transform. Bh17 bergdall, john and hansen, david, on padic lfunctions for hilbert modular forms. It seems that hensels main motivation was the analogy between the ring of integers z, together with its. This site is like a library, use search box in the widget to get ebook that you want. Also, an explicit formula is obtained for these functions, and a relationship between the values of the p adic and classical l functions at positive integers is discussed. In section 3, we will build the background of elementary padic analysis and the theory of newton polygons, which will be used to bound the number of zeros of particular padic analytic functions whose zeros which lie in pz pcorrespond to the rational points on the curve.
Certain properties of padic transcendental numbers are examined such as order and type of transcendence, with problems on padic exponentials. Examples 1z is analytic except at z 0, so the function is singular at that point. Click download or read online button to get introduction to p adic analytic number theory book now. Pdf exceptional values of padic analytic functions and derivatives. Exceptional values of padic analytic functions and derivatives alain escassut, jacqueline ojeda to cite this version. Special padic analytic functions and fourier transforms. Pdf exceptional values of padic analytic functions and. We prove a conjecture of denef on parameterized padic analytic. If fz is analytic everywhere in the complex plane, it is called entire. Gwgiven by wextends to a wvalued rigid analytic function on g.
In section 4, we will continue expanding upon the theory of divisors to prove. The latter will be very useful in studying the variation of the lfunction when the character. In their study of diophantine approximation of the exponential function in connection with sondows conjecture, berndt et al. Continuous, analytic, and locally analytic vectors 45 4. The usual meaning taken for p adic analysis is the theory of p adic valued functions on spaces of interest.
Exceptional values of padic analytic functions and. Although, in the complex case, functions are more amenable to analysis of the above kind than g functions, the situation is completely reversed in the p adic domain. C functions defined over an arbitrary number field, rather than the rational or quadratic fields. We obtain the lfunctions as the mellin transforms in the sense of mazurswinnertondyer of a padic measure in two variables, whose moments are essentially. For general background information concerning padic numbers, analytic functions, padic functional analysis and related areas see 1, 4, 6 and 7, for example. Its a function on elliptic curves, or pairs consisting of an elliptic curve and a differential. The p adic l functions one encounters in life are certain p adic analytic or meromorphic functions that typically arise from spaces of power series, p adic measures, or p adic distributions. Today we know at least three distinct constructions of these functions. The padic lfunctions one encounters in life are certain padic analytic or meromorphic functions that. Lately p adic numbers have attracted a great deal of attention in modern theoretical physics as a promising new approach for describing the nonarchimedean geometry of spacetime at small distances.
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