---
date: 2021-10-18 15:28
modification date: Monday 18th October 2021 15:28:05
title: arithmetic geometry
aliases: [arithmetic geometry]

---


---


- Tags 
	- #arithmetic-geometry 
- Refs:
	- [ ] <chrome-extension://ieepebpjnkhaiioojkepfniodjmjjihl/data/pdf.js/web/viewer.html?file=http%3A%2F%2Findividual.utoronto.ca%2Fgroechenig%2Farithmetic_techniques.pdf> #resources/notes/lectures #resources/full-courses 
	- <https://people.maths.ox.ac.uk/brantner/p-adic.pdf> #resources/notes 
	- p-adic Lie groups: 
		- <https://link.springer.com/book/10.1007/978-3-642-21147-8> #resources/books
		- Lazard (in French!!) 
		  <https://mathweb.ucsd.edu/~csorense/teaching/math299_LieB/lazard.pdf> #resources/papers
		- Banach space reps:
		  <https://mathweb.ucsd.edu/~csorense/teaching/math299_LieB/banach_iwasawa.pdf> #resources/course-notes 
- Links:
	- [rational point](Unsorted/rational%20points.md)
	- [Hodge decomposition](Unsorted/Hodge%20decomposition.md)
	- [p-adic Hodge theory](Unsorted/p-adic%20Hodge%20theory.md)
	- [spreading out](spreading%20out)
	- [Unsorted/section conjecture](Unsorted/section%20conjecture.md)
	- [spec Z as a curve](Unsorted/spec%20Z%20as%20a%20curve.md)
	- [Unsorted/number field function field analogy](Unsorted/number%20field%20function%20field%20analogy.md)
	- [Unsorted/roots of unity](Unsorted/roots%20of%20unity.md)
	- [Unsorted/separably generated](Unsorted/separably%20generated.md)
	- [Unsorted/local to global](Unsorted/local%20to%20global.md)
	- [Unsorted/Frobenius twist](Unsorted/Frobenius%20twist.md)


---

# arithmetic geometry

Idea: study [schemes](Unsorted/scheme.md) of [finite type](Unsorted/finite%20type.md) $X\in \Sch^\ft\slice{S}$ over $S = \spec \OO_K$ the [ring of integers](ring%20of%20integers.md) of a [number field](number%20field).

![](attachments/Pasted%20image%2020220424121918.png)


Standard setup:
![](attachments/Pasted%20image%2020220318194134.png)
![](attachments/Pasted%20image%2020220318201807.png)
![](attachments/Pasted%20image%2020220318201818.png)


# Main topics

- Finding [rational points](Unsorted/rational%20points.md) on [curves](curves.md). 
- Using [p-adic](Unsorted/p-adic.md), 
- The [Hasse-Minkowski theorem](Hasse-Minkowski%20theorem), 
- The [Riemann-Roch](Riemann-Roch.md) theorem for curves.  
- [Mordell's theorem](Mordell's%20theorem), 
- The [Weil conjectures](Weil%20conjectures.md), 
- [Jacobian varieties](Jacobian%20varieties),
- [Faltings theorem](Unsorted/Faltings%20theorem.md): $\size X(\QQ)< \infty$
- [Hodge theory MOC](Unsorted/Hodge%20theory%20MOC.md), more specifically [p-adic Hodge theory](Unsorted/p-adic%20Hodge%20theory.md)
- [rigid analytic geometry](rigid%20analytic%20geometry)
- [formal schemes](formal%20schemes)

## Rational Points

- [Faltings height](Faltings%20height)
- [Northcott property](Northcott%20property)
- [Jacobian](Unsorted/Jacobian.md)
- [Mordell conjecture](Mordell%20conjecture)
- [Vojta's inequality](Vojta's%20inequality)
- [Mumford's inequality](Mumford's%20inequality)
- [Mordell-Lang coonjecture](Mordell-Lang%20coonjecture)

## Cohomology

- [etale cohomology](Unsorted/etale.md)
- [Crystalline cohomology](Crystalline%20cohomology)
- [algebraic de Rham cohomology](Unsorted/algebraic%20de%20Rham%20cohomology.md)
- [prismatic cohomology](Unsorted/prismatic%20cohomology.md)
- [Hodge-Tate cohomology](Hodge-Tate%20cohomology)