Number Theory: Volume II: Analytic and Modern ToolsThis book deals with several aspects of what is now called "explicit number theory." The central theme is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three of its most basic aspects. The local aspect, global aspect, and the third aspect is the theory of zeta and L-functions. This last aspect can be considered as a unifying theme for the whole subject. |
Contents
| 3 | |
| 11 | |
2 | 42 |
4 | 73 |
Basic Algebraic Number Theory | 101 |
Dirichlet Series and LFunctions | 151 |
4 | 181 |
padic Gamma and LFunctions | 275 |
Rational Points on HigherGenus Curves | 441 |
The SuperFermat Equation | 463 |
Elliptic Curves | 465 |
The Modular Approach to Diophantine Equations 495 | 494 |
Diophantine Aspects of Elliptic Curves | 517 |
Catalans Equation | 529 |
| 561 | |
| 571 | |
Quadratic Forms and LocalGlobal Principles | 285 |
Diophantine Equations | 335 |
Applications of Linear Forms in Logarithms | 411 |
| 578 | |
| 585 | |
Other editions - View all
Common terms and phrases
addition algebraic analytic apply assume Bernoulli bound Chapter character character modulo choose clear complex compute conductor congruence consider constant continuous convergence coprime Corollary corresponding course deduce defined definition denote derivative Dirichlet dividing easy equal equivalent essentially estimate Euler example Exercise exists expansion fact factor fields first follows formula Fourier functional equation gamma function given gives hand hence ideal immediate implies important instance integral L-functions Lemma logarithm method multiplicative nontrivial notation Note obtain p-adic particular points pole polynomials positive power series prime Proof properties Proposition prove radius of convergence rational reader reason Recall Remarks replace respectively result root seen side simple solutions summation tends Theorem theory transform trivial usual valid values variable write zero zeta function