p-adic Numbers, p-adic Analysis, and Zeta-Functions
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 Logi...
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| Format: | eBook |
| Language: | English |
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New York, NY
Springer New York
1984, 1984
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| Edition: | 2nd ed. 1984 |
| Series: | Graduate Texts in Mathematics
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 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.