Number Theory III: Diophantine Geometry

di Serge Lang (Autore)

Serge Lang (Curatore)

Springer-Verlag Berlin and Heidelberg GmbH & Co. KG, 1991

In 1988 Shafarevich asked me to write a volume for the Encyclopaedia of Mathematical Sciences on Diophantine Geometry. I said yes, and here is the volume. By definition, diophantine problems concern the solutions of equations in integers, or rational numbers, or various generalizations, such as finitely generated rings over Z or finitely generated fields over Q. The word Geometry is tacked on to suggest geometric methods. This means that the present volume is not elementary. For a survey of some basic problems with a much more elementary approach, see [La 9Oc]. The field of diophantine geometry is now moving quite rapidly. Out­ standing conjectures ranging from decades back are being proved. I have tried to give the book some sort of coherence and permanence by em­ phasizing structural conjectures as much as results, so that one has a clear picture of the field. On the whole, I omit proofs, according to the boundary conditions of the encyclopedia. On some occasions I do give some ideasfor the proofs when these are especially important. In any case, a lengthy bibliography refers to papers and books where proofs may be found. I have also followed Shafarevich's suggestion to give examples, and I have especially chosen these examples which show how some classical problems do or do not get solved by contemporary in­ sights. Fermat's last theorem occupies an intermediate position. Al­ though it is not proved, it is not an isolated problem any more.

• Autore: Serge Lang
• Curatore: Serge Lang
• Editore: Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
• Collana: Encyclopaedia of Mathematical Sciences
• Anno: 1991
• Rilegatura: Hardback
• Pagine: 296 p.
• Dimensioni: 235 x 155 mm
• EAN: 9783540530046

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