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The theory of p-adic and classic modular forms, and the study of arithmetic and p-adic L-functions has proved to be a fruitful area of mathematics over the last decade. Professor Hida has given courses on these topics in the USA, Japan, and in France, and in this book provides the reader with an elementary but detailed insight into the theory of L-functions. The presentation is self contained and concise, and the subject is approached using only basic tools from complex analysis and cohomology theory. Graduate students wishing to know more about L-functions will find that this book offers a unique introduction to this fascinating branch of mathematics.
L-functions. --- Eisenstein series. --- Series, Eisenstein --- Automorphic functions --- Functions, L --- -Number theory
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L-functions associated to automorphic forms encode all classical number theoretic information. They are akin to elementary particles in physics. This 2006 book provides an entirely self-contained introduction to the theory of L-functions in a style accessible to graduate students with a basic knowledge of classical analysis, complex variable theory, and algebra. Also within the volume are many new results not yet found in the literature. The exposition provides complete detailed proofs of results in an easy-to-read format using many examples and without the need to know and remember many complex definitions. The main themes of the book are first worked out for GL(2,R) and GL(3,R), and then for the general case of GL(n,R). In an appendix to the book, a set of Mathematica functions is presented, designed to allow the reader to explore the theory from a computational point of view.
L-functions. --- Automorphic forms. --- Automorphic functions --- Forms (Mathematics) --- Functions, L --- -Number theory
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Algebraic geometry --- L-functions --- Curves, Elliptic --- Iwasawa theory --- Class field theory --- Iwasawa's theorem --- Algebraic fields --- Functions, L --- -Number theory --- Elliptic curves --- Curves, Algebraic --- Algebraic number theory
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A groundbreaking contribution to number theory that unifies classical and modern resultsThis book develops a new theory of p-adic modular forms on modular curves, extending Katz's classical theory to the supersingular locus. The main novelty is to move to infinite level and extend coefficients to period sheaves coming from relative p-adic Hodge theory. This makes it possible to trivialize the Hodge bundle on the infinite-level modular curve by a "canonical differential" that restricts to the Katz canonical differential on the ordinary Igusa tower. Daniel Kriz defines generalized p-adic modular forms as sections of relative period sheaves transforming under the Galois group of the modular curve by weight characters. He introduces the fundamental de Rham period, measuring the position of the Hodge filtration in relative de Rham cohomology. This period can be viewed as a counterpart to Scholze's Hodge-Tate period, and the two periods satisfy a Legendre-type relation. Using these periods, Kriz constructs splittings of the Hodge filtration on the infinite-level modular curve, defining p-adic Maass-Shimura operators that act on generalized p-adic modular forms as weight-raising operators. Through analysis of the p-adic properties of these Maass-Shimura operators, he constructs new p-adic L-functions interpolating central critical Rankin-Selberg L-values, giving analogues of the p-adic L-functions of Katz, Bertolini-Darmon-Prasanna, and Liu-Zhang-Zhang for imaginary quadratic fields in which p is inert or ramified. These p-adic L-functions yield new p-adic Waldspurger formulas at special values.
L-functions. --- Number theory. --- p-adic analysis. --- MATHEMATICS / Number Theory. --- Analysis, p-adic --- Algebra --- Calculus --- Geometry, Algebraic --- Number study --- Numbers, Theory of --- Functions, L --- -Number theory
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511.6 --- Algebraic fields --- -Galois theory --- -L-functions --- -Functions, L --- -Number theory --- Equations, Theory of --- Group theory --- Number theory --- Algebraic number fields --- Algebraic numbers --- Fields, Algebraic --- Algebra, Abstract --- Algebraic number theory --- Rings (Algebra) --- Congresses --- Galois theory --- L-functions --- Congresses. --- -Algebraic number fields --- 511.6 Algebraic number fields --- -511.6 Algebraic number fields --- Functions, L --- Nombres, Théorie des --- Theorie des nombres --- Colloque
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Number theory --- Automorphic functions. --- Fonctions automorphes. --- L-functions. --- Fonctions L. --- Automorphic functions --- L-functions --- Functions, L --- -Number theory --- Fuchsian functions --- Functions, Automorphic --- Functions, Fuchsian --- Functions of several complex variables
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The main topics of the book are the critical values of Dirichlet L-functions and Hecke L-functions of an imaginary quadratic field, and various problems on elliptic modular forms. As to the values of Dirichlet L-functions, all previous papers and books reiterate a single old result with a single old method. After a review of elementary Fourier analysis, the author presents completely new results with new methods, though old results will also be proved. No advanced knowledge of number theory is required up to this point. As applications, new formulas for the second factor of the class number of a cyclotomic field will be given. The second half of the book assumes familiarity with basic knowledge of modular forms. However, all definitions and facts are clearly stated, and precise references are given. The notion of nearly holomorphic modular forms is introduced and applied to the determination of the critical values of Hecke L-functions of an imaginary quadratic field. Other notable features of the book are: (1) some new results on classical Eisenstein series; (2) the discussion of isomorphism classes of elliptic curves with complex multiplication in connection with their zeta function and periods; (3) a new class of holomorphic differential operators that send modular forms to those of a different weight. The book will be of interest to graduate students and researchers who are interested in special values of L-functions, class number formulae, arithmetic properties of modular forms (especially their values), and the arithmetic properties of Dirichlet series. It treats in detail, from an elementary viewpoint, the simplest cases of a fundamental area of ongoing research, the only prerequisite being a basic course in algebraic number theory.
Dirichlet series. --- L-functions. --- Forms, Modular. --- Modular forms --- Forms (Mathematics) --- Functions, L --- -Number theory --- Series, Dirichlet --- Series --- Number theory. --- Geometry, algebraic. --- Number Theory. --- Algebraic Geometry. --- Algebraic geometry --- Geometry --- Number study --- Numbers, Theory of --- Algebra --- Algebraic geometry.
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Functions, Zeta. --- L-functions. --- Random matrices. --- Limit theorems (Probability theory) --- Monodromy groups. --- Group theory --- Probabilities --- Matrices, Random --- Matrices --- Functions, L --- -Number theory --- Zeta functions --- Functions, Zeta --- L-functions --- Monodromy groups --- Random matrices
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The seminal formula of Gross and Zagier relating heights of Heegner points to derivatives of the associated Rankin L-series has led to many generalisations and extensions in a variety of different directions, spawning a fertile area of study that remains active to this day. This volume, based on a workshop on Special Values of Rankin L-series held at the MSRI in December 2001, is a collection of thirteen articles written by many of the leading contributors in the field, having the Gross-Zagier formula and its avatars as a common unifying theme. It serves as a valuable reference for mathematicians wishing to become further acquainted with the theory of complex multiplication, automorphic forms, the Rankin-Selberg method, arithmetic intersection theory, Iwasawa theory, and other topics related to the Gross-Zagier formula.
Curves, Elliptic --- L-functions --- 512.75 --- Functions, L --- -Number theory --- Elliptic curves --- Curves, Algebraic --- 512.75 Arithmetic problems of algebraic varieties. Rationality questions. Zeta-functions --- Arithmetic problems of algebraic varieties. Rationality questions. Zeta-functions --- Curves, Elliptic. --- L-functions.
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This volume is an outgrowth of the LMS Durham Symposium on L-functions, held in July 1989. The symposium consisted of several short courses, aimed at presenting rigorous but non-technical expositions of the latest research areas, and a number of individual lectures on specific topics. The contributors are all outstanding figures in the area of algebraic number theory and this volume will be of lasting value to students and researchers working in the area.
L-functions --- Algebraic number theory --- Functions, L --- -Number theory --- 511.3 --- 511.3 Analytical, additive and other number-theory problems. Diophantine approximations --- Analytical, additive and other number-theory problems. Diophantine approximations --- Congresses --- L-functions - Congresses --- Algebraic number theory - Congresses
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