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
Bernoulli Polynomials and the Gamma Function | 3 |
Quadratic Forms and LocalGlobal Principles | 5 |
Tools | 11 |
Basic Algebraic Number Theory | 101 |
Dirichlet Series and LFunctions | 151 |
padic Fields | 183 |
padic Gamma and LFunctions | 275 |
Applications of Linear Forms in Logarithms | 411 |
Diophantine Equations | 496 |
7 | 497 |
3 | 514 |
Diophantine Aspects of Elliptic Curves | 517 |
Catalans Equation | 529 |
Bibliography | 561 |
| 566 | |
| 571 | |
Introduction | 441 |
The SuperFermat Equation | 464 |
Elliptic Curves | 465 |
The Modular Approach to Diophantine Equations | 494 |
| 578 | |
| 580 | |
| 585 | |
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Common terms and phrases
algebraic integer algebraic number apply assume Bernoulli numbers Bernoulli polynomials bound Chapter character modulo Cl(K coefficients complex compute conductor congruence conjecture coprime integers Corollary corresponding deduce defined over Q definition denote Diophantine equations Dirichlet character divisor elliptic curve equal equation Y2 Euler Euler-MacLaurin Euler-MacLaurin formula Exercise exists fact finite follows Fourier Frey curve functional equation gamma function given gives hence Hurwitz zeta function integral Jacobian L-functions Lemma log(N logarithms Lp(x method modular modularity theorem modulo multiplicative newform nontrivial nonzero notation Note number field number theory obtain odd prime opposite parity p-adic pairwise coprime parametrizations particular power series prime number primitive character Proof rational points Recall result Ribet's root of unity Section solutions summation formula theorem trivial words zero zeta function