Numbers as functions: the development of an idea in the Moscow school of algebraic geometry

TL;DR

This paper explores the historical development of arithmetical algebraic geometry in Moscow during the 1950s and 1960s, focusing on the analogy between numbers and functions and its mathematical and social implications.

Contribution

It provides a detailed historical analysis of the Moscow school's approach to numbers as functions, including key examples like reciprocity laws and Arakelov theory.

Findings

Analysis of the analogy between numbers and functions in Moscow school

Discussion of reciprocity laws and arithmetical surfaces

Insight into the social and educational context of the Moscow school

Abstract

This is expanded text of a lecture delivered by the author at the conference "Mat\'eriaux pour l'Histoire des Math\'ematiques au XX\`eme si\`ecle", which took place in Nice in January 1996. The task was to describe one area in the development of arithmetical algebraic geometry in Moscow during the 1950s and 1960s. We shall begin by explaining the meaning of the analogy between numbers and functions, starting with the simplest concepts. In the second part we study a nontrivial example: the explicit formula for the law of reciprocity. In the third part we shall become acquainted with certain aspects of the "social" life of the Moscow school, in particular, with certain seminars, lectures, and books. In the final part we shall examine another example of this analogy: arithmetical surfaces and Arakelov theory.