The Early Historical Research On The Arithmetization Of Algebraic Geometry

The arithmetization of algebraic geometry was important in the long history of algebraic geometry.It effectively linked algebraic geometry with algebraic number theory and topology,and promoted the formation and development of arithmetic-algebraic geometry.In the 1880 s,Kronecker,Dedekind and Weber initially explored the arithmetic terminology of algebraic geometry and took the first step in the arithmetization of algebraic geometry.At the end of the 19 th century,the Italian geometric school made an in-depth study on the theory of complex algebraic surfaces.On the basis of this study,Zariski created the arithmetic theory of algebraic varieties in the late 1930 s,which promoted the rapid development of the arithmetization of algebraic geometry.After Zariski,the proposition of the Weil conjecture in 1949 promoted the development of sheaf theory and cohomology theory.Finally,on the basis of previous work,Grothendieck discussed Abelian category and derived functor cohomology,established scheme theory,and completed the arithmetization of algebraic geometry.Therefore,studying the early history of the arithmetization of algebraic geometry will help people understand the ideological connotation and development history of the arithmetization of algebraic geometry.This dissertation is based on a large amount of original literature and research literature.By using the methods of literature research,chronicle,conceptual analysis,comparison and sociology,and taking the ideological development of the arithmetization of algebraic geometry as the clue,the ideological source,work and influence of the core mathematicians involved in the process of the arithmetization of algebraic geometry are discussed.The main results and conclusions are as follows:1.The beginning stage of the arithmetization of algebraic geometry is discussed,that is,the classical work of Kronecker,Dedekind and Weber on the arithmetization of algebraic geometry.In 1882,Kronecker defined algebraic quantity and algebraic divisor in “Grundz(?)ge einer arithmetischen Theorie der algebraischen Gr(?)ssen”,established the divisor theory,and introduced the modular system,which enabled algebraic geometry to open up the direction of the arithmetization based on divisor.In the same year,Dedekind and Weber introduced the terms “supplementary class” and“canonical class” in “Theorie der algebraischen Functionen einer Ver(?)nderlichen”,and proved Riemann-Roch theorem with them,which enabled algebraic geometry to open up the direction of the arithmetization based on ideal.2.The source of ideas,some work and influence of the arithmetization of algebraic geometry of Zariski are explored.On the basis of the work of the Italian school,Zariski began his research on the arithmetization of algebraic geometry.In1939,he introduced the concepts of “arithmetically normal variety” and “derived normal variety” in “Some Results in the Arithmetic Theory of Algebraic Varieties”,and established the arithmetic theory of algebraic varieties.In the same year,he also proved the theorem of reduction of singularities of algebraic surfaces by arithmetic method in “The Reduction of the Singularities of an Algebraic Surface”,which took a key step in the arithmetization of algebraic geometry.3.The background,content and influence of Weil conjecture,and the work of some mathematicians on sheaf theory and homology algebra are analyzed.Based on the Riemann zeta function and Riemann conjecture of algebraic curves in finite fields,Weil put forward the famous “Weil conjecture” in “Numbers of Solutions of Equations in Finite Fields” in 1949,which connected algebraic geometry with algebraic number theory and topology.Under the influence of Weil conjecture,Henri Cartan and Serre developed the theory of sheaf and cohomology.4.Grothendieck's some work on the arithmetization of algebraic geometry are studied.Under the influence of Serre,Grothendieck began to study homology algebra around 1955.Later,he put forward the concept of scheme in the first chapter “Le langage des schémas” of (?)léments De Géométrie Algébrique and established the theory of scheme in subsequent studies.5.The influence of the arithmetization of algebraic geometry is discussed.On the basis of the arithmetization of algebraic geometry by Grothendieck,Deligne proved Weil conjecture by cohomology theory,Faltings proved Model conjecture by scheme theory,and so on,which promoted the development of mathematics in the second half of the 20 th century.