p-adic Numbers, p-adic Analysis, and Zeta-Functions
2nd Edition
Published on 4. October 2012
Book
Paperback/Softback
XII, 153 pages
978-1-4612-7014-0 (ISBN)
Description
Neal Koblitz was a student of Nicholas M. Katz, under whom he received his Ph.D. in mathematics at Princeton in 1974. He spent the year 1974 -75 and the spring semester 1978 in Moscow, where he did research in p -adic analysis and also translated Yu. I. Manin's "Course in Mathematical Logic" (GTM 53). He taught at Harvard from 1975 to 1979, and since 1979 has been at the University of Washington in Seattle. He has published papers in number theory, algebraic geometry, and p-adic analysis, and he is the author of "p-adic Analysis: A Short Course on Recent Work" (Cambridge University Press and GTM 97: "Introduction to Elliptic Curves and Modular Forms (Springer-Verlag).
Reviews / Votes
From the reviews of the second edition:
"In the second edition of this text, Koblitz presents a wide-ranging introduction to the theory of p-adic numbers and functions. . there are some really nice exercises that allow the reader to explore the material. . And with the exercises, the book would make a good textbook for a graduate course, provided the students have a decent background in analysis and number theory." (Donald L. Vestal, The Mathematical Association of America, April, 2011)
More details
Series
Edition
Second Edition 1984
Language
English
Place of publication
New York
United States
Target group
Primary & secondary/elementary & high school
Graduate
Illustrations
XII, 153 p.
Dimensions
Height: 235 mm
Width: 155 mm
Thickness: 10 mm
Weight
265 gr
ISBN-13
978-1-4612-7014-0 (9781461270140)
DOI
10.1007/978-1-4612-1112-9
Schweitzer Classification
Other editions
Additional editions
Neal Koblitz
p-adic Numbers, p-adic Analysis, and Zeta-Functions
E-Book
12/2012
2nd Edition
Springer
€60.98
Available for download
Neal Koblitz
p-adic Numbers, p-adic Analysis, and Zeta-Functions
Book
07/1984
2nd Edition
Springer
€85.59
Shipment within 5-7 days
Person
Neal Koblitz is a Professor of Mathematics at the University of Washington in the Department of Mathematics. He is also an adjunct professor with the Centre for Applied Cryptographic Research at the University of Waterloo. He is the creator of hyperelliptic curve cryptography and the independent co-creator of elliptic curve cryptography. Professor Koblitz received his undergraduate degree from Harvard University, where he was a Putnam Fellow, in 1969. He received his Ph.D. from Princeton University in 1974 under the direction of Nickolas Katz.
Content
I p-adic numbers.- 1. Basic concepts.- 2. Metrics on the rational numbers.- Exercises.- 3. Review of building up the complex numbers.- 4. The field of p-adic numbers.- 5. Arithmetic in ?p.- Exercises.- II p-adic interpolation of the Riemann zeta-function.- 1. A formula for ?(2k).- 2. p-adic interpolation of the function f(s) = as.- Exercises.- 3. p-adic distributions.- Exercises.- 4. Bernoulli distributions.- 5. Measures and integration.- Exercises.- 6. The p-adic ?-function as a Mellin-Mazur transform.- 7. A brief survey (no proofs).- Exercises.- III Building up ?.- 1. Finite fields.- Exercises.- 2. Extension of norms.- Exercises.- 3. The algebraic closure of ?p.- 4. ?.- Exercises.- IV p-adic power series.- 1. Elementary functions.- Exercises.- 2. The logarithm, gamma and Artin-Hasse exponential functions.- Exercises.- 3. Newton polygons for polynomials.- 4. Newton polygons for power series.- Exercises.- V Rationality of the zeta-function of a set of equations over a finite field.- 1. Hypersurfaces and their zeta-functions.- Exercises.- 2. Characters and their lifting.- 3. A linear map on the vector space of power series.- 4. p-adic analytic expression for the zeta-function.- Exercises.- 5. The end of the proof.- Answers and Hints for the Exercises.