---
date: 2022-05-03 12:06
modification date: Tuesday 3rd May 2022 12:06:13
title: "arithmetic intersection theory"
aliases: [arithmetic intersection theory]
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# arithmetic intersection theory

Manin proves the [Mordell conjecture](Unsorted/Faltings%20theorem.md) using the [Gauss-Manin connection](Unsorted/Gauss-Manin.md), generalized to [Areklov theory](Areklov%20theory). Uses [Green functions](Green%20functions) on curves. Used in [Faltings theorem](Unsorted/Faltings%20theorem.md). Imports tools like the [adjunction formula](Unsorted/adjunction%20formula.md), the [Hodge index theorem](Unsorted/Hodge%20index%20theorem.md), [Riemann-Roch](Unsorted/Riemann-Roch.md).

New directions: arithmetic in [algebraic K theory](Unsorted/K-theory.md), [higher Chow groups](higher%20Chow%20groups).
![](attachments/Pasted%20image%2020220503120615.png)