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| Author | : John William Scott Cassels |
| Publisher | : Cambridge University Press |
| Total Pages | : 148 |
| Release | : 1991-11-21 |
| Genre | : Mathematics |
| ISBN | : 9780521425308 |
A self-contained introductory text for beginning graduate students that is contemporary in approach without ignoring historical matters.
| Author | : J. W. S. Cassels |
| Publisher | : Cambridge University Press |
| Total Pages | : 0 |
| Release | : 1991-11-21 |
| Genre | : Mathematics |
| ISBN | : 9780521425308 |
The study of special cases of elliptic curves goes back to Diophantos and Fermat, and today it is still one of the liveliest centers of research in number theory. This book, addressed to beginning graduate students, introduces basic theory from a contemporary viewpoint but with an eye to the historical background. The central portion deals with curves over the rationals: the Mordell-Wei finite basis theorem, points of finite order (Nagell-Lutz), etc. The treatment is structured by the local-global standpoint and culminates in the description of the Tate-Shafarevich group as the obstruction to a Hasse principle. In an introductory section the Hasse principle for conics is discussed. The book closes with sections on the theory over finite fields (the "Riemann hypothesis for function fields") and recently developed uses of elliptic curves for factoring large integers. Prerequisites are kept to a minimum; an acquaintance with the fundamentals of Galois theory is assumed, but no knowledge either of algebraic number theory or algebraic geometry is needed. The p-adic numbers are introduced from scratch. Many examples and exercises are included for the reader, and those new to elliptic curves, whether they are graduate students or specialists from other fields, will find this a valuable introduction.
| Author | : John William Scott Cassels |
| Publisher | : |
| Total Pages | : 137 |
| Release | : 1991 |
| Genre | : Curves, Elliptic |
| ISBN | : 9781316086995 |
The study of (special cases of) elliptic curves goes back to Diophantos and Fermat, and today it is still one of the liveliest centres of research in number theory. This book, which is addressed to beginning graduate students, introduces basic theory from a contemporary viewpoint but with an eye to the historical background. The central portion deals with curves over the rationals: the Mordell-Weil finite basis theorem, points of finite order (Nagell-Lutz) etc. The treatment is structured by the local-global standpoint and culminates in the description of the Tate-Shafarevich group as the obstruction to a Hasse principle. In an introductory section the Hasse principle for conics is discussed. The book closes with sections on the theory over finite fields (the 'Riemann hypothesis for function fields') and recently developed uses of elliptic curves for factoring large integers. Prerequisites are kept to a minimum; an acquaintance with the fundamentals of Galois theory is assumed, but no knowledge either of algebraic number theory or algebraic geometry is needed. The p-adic numbers are introduced from scratch, as is the little that is needed on Galois cohomology. Many examples and exercises are included for the reader. For those new to elliptic curves, whether they are graduate students or specialists from other fields, this will be a fine introductory text.
| Author | : Miles Reid |
| Publisher | : Cambridge University Press |
| Total Pages | : 312 |
| Release | : 2003 |
| Genre | : Mathematics |
| ISBN | : 9780521545181 |
This volume honors Sir Peter Swinnerton-Dyer's mathematical career spanning more than 60 years' of amazing creativity in number theory and algebraic geometry.
| Author | : Sergio Cecotti |
| Publisher | : Springer Nature |
| Total Pages | : 846 |
| Release | : 2023-11-07 |
| Genre | : Science |
| ISBN | : 3031365305 |
Graduate students typically enter into courses on string theory having little to no familiarity with the mathematical background so crucial to the discipline. As such, this book, based on lecture notes, edited and expanded, from the graduate course taught by the author at SISSA and BIMSA, places particular emphasis on said mathematical background. The target audience for the book includes students of both theoretical physics and mathematics. This explains the book’s "strange" style: on the one hand, it is highly didactic and explicit, with a host of examples for the physicists, but, in addition, there are also almost 100 separate technical boxes, appendices, and starred sections, in which matters discussed in the main text are put into a broader mathematical perspective, while deeper and more rigorous points of view (particularly those from the modern era) are presented. The boxes also serve to further shore up the reader’s understanding of the underlying math. In writing this book, the author’s goal was not to achieve any sort of definitive conciseness, opting instead for clarity and "completeness". To this end, several arguments are presented more than once from different viewpoints and in varying contexts.
| Author | : J. W. S. Cassels |
| Publisher | : |
| Total Pages | : 146 |
| Release | : 1991 |
| Genre | : Curves, Elliptic |
| ISBN | : 9781107094505 |
A self-contained introductory text for beginning graduate students that is contemporary in approach without ignoring historical matters.
| Author | : Gérard Laumon |
| Publisher | : Cambridge University Press |
| Total Pages | : 362 |
| Release | : 1996 |
| Genre | : Mathematics |
| ISBN | : 0521470609 |
Originally published in 1995, Cohomology of Drinfeld Modular Varieties aimed to provide an introduction, in two volumes, both to this subject and to the Langlands correspondence for function fields. These varieties are the analogues for function fields of the Shimura varieties over number fields. The Langlands correspondence is a conjectured link between automorphic forms and Galois representations over a global field. By analogy with the number-theoretic case, one expects to establish the conjecture for function fields by studying the cohomology of Drinfeld modular varieties, which has been done by Drinfeld himself for the rank two case. The present volume is devoted to the geometry of these varieties, and to the local harmonic analysis needed to compute their cohomology. Though the author considers only the simpler case of function rather than number fields, many important features of the number field case can be illustrated.
| Author | : Hugh C. Williams |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 412 |
| Release | : |
| Genre | : Science |
| ISBN | : 9780821887592 |
This volume consists of a selection of papers based on presentations made at the international conference on number theory held in honor of Hugh Williams' sixtieth birthday. The papers address topics in the areas of computational and explicit number theory and its applications. The material is suitable for graduate students and researchers interested in number theory.
| Author | : Reinhard Schertz |
| Publisher | : Cambridge University Press |
| Total Pages | : |
| Release | : 2010-04-29 |
| Genre | : Mathematics |
| ISBN | : 1139486837 |
This is a self-contained 2010 account of the state of the art in classical complex multiplication that includes recent results on rings of integers and applications to cryptography using elliptic curves. The author is exhaustive in his treatment, giving a thorough development of the theory of elliptic functions, modular functions and quadratic number fields and providing a concise summary of the results from class field theory. The main results are accompanied by numerical examples, equipping any reader with all the tools and formulas they need. Topics covered include: the construction of class fields over quadratic imaginary number fields by singular values of the modular invariant j and Weber's tau-function; explicit construction of rings of integers in ray class fields and Galois module structure; the construction of cryptographically relevant elliptic curves over finite fields; proof of Berwick's congruences using division values of the Weierstrass p-function; relations between elliptic units and class numbers.
| Author | : Haruzo Hida |
| Publisher | : Oxford University Press |
| Total Pages | : 417 |
| Release | : 2006-06-15 |
| Genre | : Mathematics |
| ISBN | : 019857102X |
Describing the applications found for the Wiles and Taylor technique, this book generalizes the deformation theoretic techniques of Wiles-Taylor to Hilbert modular forms (following Fujiwara's treatment), and also discusses applications found by the author.
