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Arithmetic geometry. (Géométrie arithmétique.) (French) Zbl 0790.01017

Chabert, Jean-Luc (ed.) et al., Analyse diophantienne et géométrie algébrique. Exposés du Séminaire d’Histoire des Mathématiques de l’Institut Henri Poincaré. Paris: Université Pierre et Marie Curie, Lab. de Mathématiques Fondamentales, Cah. Sémin. Hist. Math., 2. Sér. 3, 79-84 (1993).
Referring to Hilbert’s 12th problem the author presents some elementary aspects of what is today called arithmetic geometry, a theory which shows the close relationship of number theory, algebraic geometry, and the theory of functions. In a very readable exposé, he presents the following highlights for “the educated layman”: analogies between \(\mathbb{Z}\) and \(\mathbb{C}[X]\); the theorems of Riemann-Roch and Minkowski (according to Weil); theory of height; Grothendieck schemes and Arakelov geometry.
For the entire collection see [Zbl 0782.00054].

MSC:

01A65 Development of contemporary mathematics
01A60 History of mathematics in the 20th century
14-03 History of algebraic geometry
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