Arithmetic geometry

In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory.

Quotes

  • Like Moses, André Weil caught sight of the Promised Land, but unlike Moses, he was unable to cross the Red Sea on dry land, nor did he have an adequate vessel. For his own work, he had already reconstructed algebraic geometry on a purely algebraic basis, in which the notion of a “field” is predominant. To create the required arithmetic geometry, it is necessary to replace the algebraic notion of a field by that of a commutative ring, and above all to invent an adaptation of homological algebra able to tame the problems of arithmetic geometry. André Weil himself was not ignorant of these techniques nor of these problems, and his contributions are numerous and important (adeles, the so-called Tamagawa number, class field theory, deformation of discrete subgroups of symmetries). But André Weil was suspicious of “big machinery” and never learned to feel familiar with sheaves, homological algebra or categories, contrarily to Grothendieck, who embraced them wholeheartedly.
    • Pierre Cartier. A country of which nothing is known but the name Grothendieck and “motives”. 
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