Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories (Encyclopaedia of Mathematical Sciences Book 49) 2nd Edition by Yu. I. Manin (PDF)

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This edition has been called ‘startlingly up-to-date’, and in this corrected second printing you can be sure that it’s even more contemporaneous. It surveys from a unified point of view both the modern state and the trends of continuing development in various branches of number theory. Illuminated by elementary problems, the central ideas of modern theories are laid bare. Some topics covered include non-Abelian generalizations of class field theory, recursive computability and Diophantine equations, zeta- and L-functions. This substantially revised and expanded new edition contains several new sections, such as Wiles’ proof of Fermat’s Last Theorem, and relevant techniques coming from a synthesis of various theories.

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⭐In the branch of arithmetic geometry called Arakelov theory one begins with a projective curve X over the rational numbers Q which is described by equations with integer coefficients and from this one obtains an arithmetic surface X(Z). X is then viewed as a generic fiber of the projection of X(Z) to Spec Z (Z being the integers). If Spec Z is viewed as an arithmetic curve then the primes have a finite height. To deal with “arithmetic infinity” of Spec Z, Arakelov defined an analog of a p-adic completion of X(Z) by viewing Q as embedded in C (the complex numbers). The analog was obtained by putting a Hermitian structure on XC. This allowed Green’s functions to be defined which play the role of intersection indices of arithmetic curves at the infinite fiber. Along these lines the authors point to a missing element in Arakelov geometry, namely an explicit description of the ‘fiber at infinity’. Their remedy for this is very involved and is hard to follow without consulting the references, as is the case for most of the topics in this book, with the exception of the those in the first few chapters.It is for this reason that this book should be viewed as more of an introduction to the literature on number theory, and not as a self-contained overview of some the more exciting topics in number theory and arithmetic geometry that have taken place in the last two decades. This is not to say that there are a few places in the book where the authors give the reader some deep insights into difficult concepts, and these discussions make the book worth reading. Some of these include the following:1. The contrast of Z/NZ with Z in terms of the number of invertible elements in each, which is used to motivate Euler’s function. 2. The (brief) discussion on finding a formula that generates the prime numbers. 3. The Tate-Shafarevich group as being a measure of the degree to which a Diophantine equation fails to satisfy the Minkowski-Hasse principle. 4. The very insightful and therefore helpful discussion on a “geometric realization” of algebraic numbers. 5. The statement that valuations and absolute values are an alternative approach to the theory of divisibility.The authors motivate brilliantly the notion of an idele and adele and the origin of the class of Krull rings (which is wider than the class of Dedekind rings). 6. The origin of the need for Spec(A) for a commutative ring A, namely the widening of the ring of functions to include nilpotent elements. 7. The statement that specializations are used to measure the depths of Spec(A), i.e. the number of specializations of the generic point that are needed to reach a given point. This leads to a definition of dimension for both varieties and schemes. 8. The statement that integral extensions of rings can be viewed as coverings of complex varieties (essentially Riemann surfaces). 9. The motivation for Arakelov geometry as a method for controlling the size of rational points via Hermitian metrizations of objects assigned to algebraic varieties. 10. The detailed discussion of Deninger’s Archimedean cohomology.But there are many places in the book where the authors insufficiently motivate topics of great interest. Some of these include: 1. No in-depth discussion of the Tate field. Readers will have to know before reading the book why this field is important. 2. Why Pontryagin duality gives compactness for the quotient group of the ring of adeles by a global field. 3. The lack of motivation for the geometric Frobenius operator and its role in etale cohomology, the latter of which is also not explained in any detail. 4. No details behind the need for K-theory to study homotopy quotients nor any motivation for the use of the Pimser-Voiculescu exact sequence.As is clear from both the title and the content, Part III is mostly about the authors’ own work and those it depended on, as well as other collaborators. This is the most difficult part of the book, and for the reviewer it took an exceedingly long time to digest. Nearly all of the references had to be consulted, and even then the understanding of the relevant concepts was not a trivial exercise. An improved version of the book, or possibly a third edition could possibly flesh out some of the details.Note: this book was read and studied between Oct 2007 and September 2011.

⭐This Book is a cornerstone in Arithmetic Geometry.It is the first time in a single Book so differentarguments find a common place.Let me say that the idea of dividing the work intothree parts,depending on the approach, is entirelynew. In fact,Part 1 starts with elementary theory & applications(primes,diophantine equations& approx)Part 2 gives an account of recent ideas and theory(ch.3:Logic & Recursion, with a sketch of proof ofMatiyasevic’s Theorem;ch.4:Algebraic NumberTheory;ch.5:Arithmetic of Algebraic Varieties;ch.6: dealswith Zeta functions and modular forms;ch.7:gives apicture, complete indeed, of Wiles’proof of FermatLast Theorem)Part 3 gives “Analogies and Visions”,i.e. the linkbetween numbers fields and function fields(usuallythis analogy is only admitted, but never explainedin other books) and other analogies involving manyrecent arguments in Arithmetic Geometry (such as :Schottky uniformization, Arakelov Geometry, Zetas,Dynamics and Cohomology).

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Free Download Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories (Encyclopaedia of Mathematical Sciences Book 49) 2nd Edition in PDF format
Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories (Encyclopaedia of Mathematical Sciences Book 49) 2nd Edition PDF Free Download
Download Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories (Encyclopaedia of Mathematical Sciences Book 49) 2nd Edition 2006 PDF Free
Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories (Encyclopaedia of Mathematical Sciences Book 49) 2nd Edition 2006 PDF Free Download
Download Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories (Encyclopaedia of Mathematical Sciences Book 49) 2nd Edition PDF
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