Arakelov theory

E790514

mathematical theory theory in arithmetic geometry

Arakelov theory is a framework in arithmetic geometry that extends intersection theory to arithmetic surfaces by incorporating both finite and infinite places, enabling analytic tools to study Diophantine problems.

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Statements (49)

Predicate	Object
instanceOf	mathematical theory ⓘ theory in arithmetic geometry ⓘ
aimsToSolve	Diophantine problems ⓘ problems in Diophantine geometry ⓘ
appliesTo	arithmetic surfaces ⓘ schemes over the spectrum of the ring of integers of a number field ⓘ
developedBy	Suren Arakelov NERFINISHED ⓘ
field	arithmetic geometry ⓘ
furtherDevelopedBy	Christophe Soulé NERFINISHED ⓘ Gerd Faltings NERFINISHED ⓘ Henri Gillet NERFINISHED ⓘ Jean-Benoît Bost NERFINISHED ⓘ Shou-Wu Zhang NERFINISHED ⓘ
generalizationOf	classical intersection theory on algebraic surfaces ⓘ
hasVariant	adelic Arakelov theory ⓘ higher-dimensional Arakelov theory NERFINISHED ⓘ
mainConcept	Arakelov Chow group NERFINISHED ⓘ Arakelov class group NERFINISHED ⓘ Arakelov divisor NERFINISHED ⓘ Green function ⓘ adelic metrized line bundle ⓘ arithmetic intersection number ⓘ arithmetic surface ⓘ height function ⓘ hermitian line bundle ⓘ intersection theory ⓘ
namedAfter	Suren Arakelov NERFINISHED ⓘ
provides	arithmetic Riemann–Roch theorems NERFINISHED ⓘ arithmetic analogues of classical geometric formulas ⓘ framework for heights of algebraic points ⓘ intersection theory including archimedean contributions ⓘ
relatedTo	Beilinson–Bloch conjectures NERFINISHED ⓘ Diophantine approximation NERFINISHED ⓘ Faltings’s theorem NERFINISHED ⓘ Mordell conjecture NERFINISHED ⓘ Néron–Tate height NERFINISHED ⓘ equidistribution of small points ⓘ height theory ⓘ
timePeriod	1970s ⓘ
usesConcept	Dirichlet energy NERFINISHED ⓘ Green’s function on a Riemann surface ⓘ Riemann surface NERFINISHED ⓘ archimedean places ⓘ complex analytic geometry ⓘ finite places of a number field ⓘ harmonic analysis ⓘ infinite places of a number field ⓘ non-archimedean places ⓘ potential theory ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Diophantine geometry → usesMethod → Arakelov theory ⓘ