Elliptic Curves Hilbert Modular Forms And Galois Deformations

Author : Laurent Berger,Gebhard Böckle,Lassina Dembélé,Mladen Dimitrov,Tim Dokchitser,John Voight
Publisher : Springer Science & Business Media
Page : 257 pages
File Size : 42,7 Mb
Release : 2013-06-13
Category : Mathematics
ISBN : 9783034806183

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Elliptic Curves, Hilbert Modular Forms and Galois Deformations by Laurent Berger,Gebhard Böckle,Lassina Dembélé,Mladen Dimitrov,Tim Dokchitser,John Voight Pdf

The notes in this volume correspond to advanced courses held at the Centre de Recerca Matemàtica as part of the research program in Arithmetic Geometry in the 2009-2010 academic year. The notes by Laurent Berger provide an introduction to p-adic Galois representations and Fontaine rings, which are especially useful for describing many local deformation rings at p that arise naturally in Galois deformation theory. The notes by Gebhard Böckle offer a comprehensive course on Galois deformation theory, starting from the foundational results of Mazur and discussing in detail the theory of pseudo-representations and their deformations, local deformations at places l ≠ p and local deformations at p which are flat. In the last section,the results of Böckle and Kisin on presentations of global deformation rings over local ones are discussed. The notes by Mladen Dimitrov present the basics of the arithmetic theory of Hilbert modular forms and varieties, with an emphasis on the study of the images of the attached Galois representations, on modularity lifting theorems over totally real number fields, and on the cohomology of Hilbert modular varieties with integral coefficients. The notes by Lassina Dembélé and John Voight describe methods for performing explicit computations in spaces of Hilbert modular forms. These methods depend on the Jacquet-Langlands correspondence and on computations in spaces of quaternionic modular forms, both for the case of definite and indefinite quaternion algebras. Several examples are given, and applications to modularity of Galois representations are discussed. The notes by Tim Dokchitser describe the proof, obtained by the author in a joint project with Vladimir Dokchitser, of the parity conjecture for elliptic curves over number fields under the assumption of finiteness of the Tate-Shafarevich group. The statement of the Birch and Swinnerton-Dyer conjecture is included, as well as a detailed study of local and global root numbers of elliptic curves and their classification.

Author : L J P Kilford
Publisher : World Scientific Publishing Company
Page : 252 pages
File Size : 42,8 Mb
Release : 2015-03-12
Category : Mathematics
ISBN : 9781783265473

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Modular Forms by L J P Kilford Pdf

Modular Forms is a graduate student-level introduction to the classical theory of modular forms and computations involving modular forms, including modular functions and the theory of Hecke operators. It also includes applications of modular forms to various subjects, such as the theory of quadratic forms, the proof of Fermat's Last Theorem and the approximation of π. The text gives a balanced overview of both the theoretical and computational sides of its subject, allowing a variety of courses to be taught from it. This second edition has been revised and updated. New material on the future of modular forms as well as a chapter about longer-form projects for students has also been added.

Author : Gebhard Böckle,Gabor Wiese
Publisher : Springer Science & Business Media
Page : 376 pages
File Size : 54,9 Mb
Release : 2014-01-23
Category : Mathematics
ISBN : 9783319038476

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Computations with Modular Forms by Gebhard Böckle,Gabor Wiese Pdf

This volume contains original research articles, survey articles and lecture notes related to the Computations with Modular Forms 2011 Summer School and Conference, held at the University of Heidelberg. A key theme of the Conference and Summer School was the interplay between theory, algorithms and experiment. The 14 papers offer readers both, instructional courses on the latest algorithms for computing modular and automorphic forms, as well as original research articles reporting on the latest developments in the field. The three Summer School lectures provide an introduction to modern algorithms together with some theoretical background for computations of and with modular forms, including computing cohomology of arithmetic groups, algebraic automorphic forms, and overconvergent modular symbols. The 11 Conference papers cover a wide range of themes related to computations with modular forms, including lattice methods for algebraic modular forms on classical groups, a generalization of the Maeda conjecture, an efficient algorithm for special values of p-adic Rankin triple product L-functions, arithmetic aspects and experimental data of Bianchi groups, a theoretical study of the real Jacobian of modular curves, results on computing weight one modular forms, and more.

Author : Zhibin Liang,Chandrakant Aribam
Publisher : Springer
Page : 95 pages
File Size : 53,6 Mb
Release : 2019-05-22
Category : Mathematics
ISBN : 9789811366642

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The Computational and Theoretical Aspects of Elliptic Curves by Zhibin Liang,Chandrakant Aribam Pdf

This volume presents a collection of results related to the BSD conjecture, based on the first two India-China conferences on this topic. It provides an overview of the conjecture and a few special cases where the conjecture is proved. The broad theme of the two conferences was “Theoretical and Computational Aspects of the Birch and Swinnerton-Dyer Conjecture”. The first was held at Beijing International Centre for Mathematical Research (BICMR) in December 2014 and the second was held at the International Centre for Theoretical Sciences (ICTS), Bangalore, India in December 2016. Providing a broad overview of the subject, the book is a valuable resource for young researchers wishing to work in this area. The articles have an extensive list of references to enable diligent researchers to gain an idea of the current state of art on this conjecture.

Author : Neal I. Koblitz
Publisher : Springer Science & Business Media
Page : 262 pages
File Size : 42,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461209096

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Introduction to Elliptic Curves and Modular Forms by Neal I. Koblitz Pdf

The theory of elliptic curves and modular forms provides a fruitful meeting ground for such diverse areas as number theory, complex analysis, algebraic geometry, and representation theory. This book starts out with a problem from elementary number theory and proceeds to lead its reader into the modern theory, covering such topics as the Hasse-Weil L-function and the conjecture of Birch and Swinnerton-Dyer. This new edition details the current state of knowledge of elliptic curves.

Author : Jennifer S. Balakrishnan,Noam Elkies,Brendan Hassett,Bjorn Poonen,Andrew V. Sutherland,John Voight
Publisher : Springer Nature
Page : 587 pages
File Size : 46,7 Mb
Release : 2022-03-15
Category : Mathematics
ISBN : 9783030809140

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Arithmetic Geometry, Number Theory, and Computation by Jennifer S. Balakrishnan,Noam Elkies,Brendan Hassett,Bjorn Poonen,Andrew V. Sutherland,John Voight Pdf

This volume contains articles related to the work of the Simons Collaboration “Arithmetic Geometry, Number Theory, and Computation.” The papers present mathematical results and algorithms necessary for the development of large-scale databases like the L-functions and Modular Forms Database (LMFDB). The authors aim to develop systematic tools for analyzing Diophantine properties of curves, surfaces, and abelian varieties over number fields and finite fields. The articles also explore examples important for future research. Specific topics include● algebraic varieties over finite fields● the Chabauty-Coleman method● modular forms● rational points on curves of small genus● S-unit equations and integral points.

Author : Jean-Michel Morel,Bernard Teissier
Publisher : Springer Nature
Page : 629 pages
File Size : 55,8 Mb
Release : 2023-06-14
Category : Mathematics
ISBN : 9783031122446

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Mathematics Going Forward by Jean-Michel Morel,Bernard Teissier Pdf

This volume is an original collection of articles by 44 leading mathematicians on the theme of the future of the discipline. The contributions range from musings on the future of specific fields, to analyses of the history of the discipline, to discussions of open problems and conjectures, including first solutions of unresolved problems. Interestingly, the topics do not cover all of mathematics, but only those deemed most worthy to reflect on for future generations. These topics encompass the most active parts of pure and applied mathematics, including algebraic geometry, probability, logic, optimization, finance, topology, partial differential equations, category theory, number theory, differential geometry, dynamical systems, artificial intelligence, theory of groups, mathematical physics and statistics.

Author : John Cremona,Joan-Carles Lario,Jordi Quer,Kenneth Ribet
Publisher : Birkhäuser
Page : 291 pages
File Size : 51,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034879194

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Modular Curves and Abelian Varieties by John Cremona,Joan-Carles Lario,Jordi Quer,Kenneth Ribet Pdf

This book presents lectures from a conference on "Modular Curves and Abelian Varieties'' at the Centre de Recerca Matemtica (Bellaterra, Barcelona). The articles in this volume present the latest achievements in this extremely active field and will be of interest both to specialists and to students and researchers. Many contributions focus on generalizations of the Shimura-Taniyama conjecture to varieties such as elliptic Q-curves and Abelian varieties of GL_2-type. The book also includes several key articles in the subject that do not correspond to conference lectures.

Author : Tadashi Ochiai
Publisher : American Mathematical Society
Page : 228 pages
File Size : 48,9 Mb
Release : 2024-04-25
Category : Mathematics
ISBN : 9781470456733

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Iwasawa Theory and Its Perspective, Volume 2 by Tadashi Ochiai Pdf

Iwasawa theory began in the late 1950s with a series of papers by Kenkichi Iwasawa on ideal class groups in the cyclotomic tower of number fields and their relation to $p$-adic $L$-functions. The theory was later generalized by putting it in the context of elliptic curves and modular forms. The main motivation for writing this book was the need for a total perspective of Iwasawa theory that includes the new trends of generalized Iwasawa theory. Another motivation is to update the classical theory for class groups, taking into account the changed point of view on Iwasawa theory. The goal of this second part of the three-part publication is to explain various aspects of the cyclotomic Iwasawa theory of $p$-adic Galois representations.

Author : John Coates,Shing-Tung Yau
Publisher : International Press of Boston
Page : 360 pages
File Size : 54,9 Mb
Release : 1997
Category : Mathematics
ISBN : UOM:39015043823981

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Elliptic Curves, Modular Forms & Fermat's Last Theorem by John Coates,Shing-Tung Yau Pdf

These proceedings are based on a conference at the Chinese University of Hong Kong, held in response to Andrew Wile's conjecture that every elliptic curve over Q is modular. The survey article describing Wile's work is included as the first article in the present edition.

Author : Pantelimon Stănică,Sihem Mesnager,Sumit Kumar Debnath
Publisher : Springer Nature
Page : 155 pages
File Size : 48,8 Mb
Release : 2021-11-09
Category : Computers
ISBN : 9783030905538

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Security and Privacy by Pantelimon Stănică,Sihem Mesnager,Sumit Kumar Debnath Pdf

This book constitutes the refereed proceedings of the Second International Conference, ICSP 2021, held in Jamshedpur, India, in November 2021. The 10 full papers were carefully reviewed and selected from 44 submissions. The contributions are organized in the following blocks: ​Cryptanalysis and other attacks; Symmetric cryptography and hash functions; Mathematical foundations of cryptography; Embedded systems security; Security in hardware; Authentication, Key management, Public key (asymmetric) techniques, and Information-theoretic techniques.

Author : John Voight
Publisher : Springer Nature
Page : 877 pages
File Size : 51,5 Mb
Release : 2021-06-28
Category : Mathematics
ISBN : 9783030566944

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Quaternion Algebras by John Voight Pdf

This open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike. Divided into five parts, the book begins with a basic introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. An in-depth exploration of the arithmetic of quaternion algebras and orders follows. The third part considers analytic aspects, starting with zeta functions and then passing to an idelic approach, offering a pathway from local to global that includes strong approximation. Applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants. Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces. Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields. Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored. Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment. Readers are assumed to have some familiarity with algebraic number theory and commutative algebra, as well as the fundamentals of linear algebra, topology, and complex analysis. More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout.

Author : Fred Diamond,Jerry Shurman
Publisher : Springer Science & Business Media
Page : 448 pages
File Size : 51,6 Mb
Release : 2006-03-30
Category : Mathematics
ISBN : 9780387272269

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A First Course in Modular Forms by Fred Diamond,Jerry Shurman Pdf

This book introduces the theory of modular forms, from which all rational elliptic curves arise, with an eye toward the Modularity Theorem. Discussion covers elliptic curves as complex tori and as algebraic curves; modular curves as Riemann surfaces and as algebraic curves; Hecke operators and Atkin-Lehner theory; Hecke eigenforms and their arithmetic properties; the Jacobians of modular curves and the Abelian varieties associated to Hecke eigenforms. As it presents these ideas, the book states the Modularity Theorem in various forms, relating them to each other and touching on their applications to number theory. The authors assume no background in algebraic number theory and algebraic geometry. Exercises are included.

Author : Proceedings of the National Academy of Sciences
Publisher : National Academies Press
Page : 52 pages
File Size : 42,7 Mb
Release : 1998-01-01
Category : Reference
ISBN : 0309058759

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Elliptic Curves and Modular Forms by Proceedings of the National Academy of Sciences Pdf

Author : J. E. Cremona
Publisher : CUP Archive
Page : 388 pages
File Size : 50,8 Mb
Release : 1997-05-15
Category : Mathematics
ISBN : 0521598206

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Algorithms for Modular Elliptic Curves Full Canadian Binding by J. E. Cremona Pdf

This book presents an extensive set of tables giving information about elliptic curves.