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Elementary theory of L-functions and Eisenstein series
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ISBN: 1316087069 110736678X 0511623690 1107368170 0511882432 1107361877 1299409040 1107364329 9781107361874 9780511882432 0521434114 9780521434119 0521435692 9780521435697 9780511623691 9781316087060 9781107368170 9781299409040 9781107364325 Year: 1993 Publisher: Cambridge [England] New York Cambridge University Press

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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.

Automorphic forms and L-functions for the group GL(n, R)
Author:
ISBN: 9780511542923 9780521837712 9781107565029 9780511221019 0511221010 0511219709 9780511219702 0511220715 9780511220715 9780511219023 0511219024 0521837715 1107161797 9781107161795 1107565022 1280567589 9781280567582 9786610567584 6610567581 0511317050 9780511317057 0511542925 Year: 2006 Publisher: Cambridge, UK New York

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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.


Book
Iwasawa theory of elliptic curves with complex multiplication : p-adic L functions.
Author:
ISBN: 012210255X 9780122102554 Year: 1987 Volume: 3 Publisher: Boston Academic press

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Book
Supersingular p-adic L-functions, Maass-Shimura operators and Waldspurger formulas
Author:
ISBN: 0691225737 Year: 2021 Publisher: Princeton, New Jersey : Princeton University Press,

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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.


Book
Algebraic number fields (L-functions and galois properties): proceedings of a symposium organised by the London Mathematical Society with the support of the Science Research Council and the Royal Society
Author:
ISBN: 0122689607 Year: 1977 Publisher: London Academic press

Degree 16 standard L-function of GSp(2) x GSp(2)
Author:
ISSN: 00659266 ISBN: 0821804766 Year: 1996 Publisher: Providence, R.I. American Mathematical Society

Elementary Dirichlet Series and Modular Forms
Author:
ISBN: 0387724745 0387724737 1441924787 Year: 2007 Publisher: New York, NY : Springer New York : Imprint: Springer,

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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.

Random matrices, Frobenius eigenvalues, and monodromy
Authors: ---
ISSN: 00659258 ISBN: 0821810170 9780821810170 Year: 1999 Volume: 45 Publisher: Providence, R.I. American Mathematical Society

Heegner points and Rankin L-series
Authors: ---
ISBN: 052183659X 0511211899 9780511211898 9780521836593 0511217269 9780511217265 0511215479 9780511215476 9780511756375 0511756372 9780521158206 0521158206 1139883186 1280540664 9786610540662 0511315864 0511213662 9781139883184 9781280540660 6610540667 9780511315862 9780511213663 Year: 2004 Volume: 49 Publisher: Cambridge, UK New York, NY, USA Cambridge University Press

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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.

L-functions and arithmetic
Authors: ---
ISBN: 1139884549 1107366518 110737121X 1107361605 110736888X 1299404286 1107364051 0511526059 9781107361607 9780511526053 0521386195 9780521386197 Year: 1991 Volume: 153 Publisher: London Cambridge university press

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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.

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