P-adic analysis short course in recent work

Includes index

Includes bibliographical references

CHAPTER I. BASICS 1. History (very brief) 2. Basic concepts 3. Power series 4. Newton polygons CHAPTER II. p-ADIC C-FUNCTIONS, L-FUNCTIONS AND r-FUNCTIONS 1. Dirichlet L-series 2. p-adic measures 3. p-adic interpolation 4. p-adic Dirichlet L-functions 5. Leopoldt's formula for L (1,X) 6. The p-adic gamma function 7. The p-adic log gamma function 8. A formula for L'p(0,X) CHAPTER III. GAUSS SUMS AND THE p-ADIC GAMMA FUNCTION 1. Gauss and Jacobi sums 2. Fermat curves.3. L-series for algebraic varieties 4. Cohomology 5. p-adic cohomology 6. p-adic formula for Gauss sums 7. Stickleberger1s theorem CHAPTER IV. p-ADIC REGULATORS 1. Regulators and L-functions 2. Leopoldt's p-adic regulator 3. Gross's p-adic regulator 4. Gross's conjecture in the abelian over Q case APPENDIX 1. A theorem of Amice-Fresnel 2. The classical Stieltjes transform 3. The Shnirelman integral and the p-adic Stieltjes transfonsfor 4. p-adic spectral theorem

This introduction to recent work in p-adic analysis and number theory will make accessible to a relatively general audience the efforts of a number of mathematicians over the last five years. After reviewing the basics (the construction of p-adic numbers and the p-adic analog of the complex number field, power series and Newton polygons), the author develops the properties of p-adic Dirichlet L-series using p-adic measures and integration. p-adic gamma functions are introduced, and their relationship to L-series is explored. Analogies with the corresponding complex analytic case are stressed.