Author | Koblitz, Neal. author |
---|---|
Title | p-adic Numbers, p-adic Analysis, and Zeta-Functions [electronic resource] / by Neal Koblitz |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1984 |
Edition | Second Edition |
Connect to | http://dx.doi.org/10.1007/978-1-4612-1112-9 |
Descript | XII, 153 p. online resource |
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.ย