Tags: #subjects/arithmetic-geometry Refs:
arithmetic geometry
Idea: study schemes of finite type $X\in {\mathsf{Sch}}^{\mathrm{ft}}_{/ {S}} $ over \(S = \operatorname{Spec}{\mathcal{O}}_K\) the ring of integers of a number field.
Main topics:
- Finding rational points on curves.
- Using p-adic integers,
- The Hasse-Minkowski theorem,
- The Riemann-Roch theorem for curves.\
- Mordell's theorem,
- The Weil conjectures,
- Jacobian varieties,
- Falting’s theorem: \({\sharp}X({\mathbb{Q}})< \infty\)
- Hodge theory, more specifically p-adic Hodge theory
- rigid analytic geometry
- formal schemes
Rational Points
- Faltings height
- Northcott property
- Jacobian
- Mordell conjecture
- Vojta's inequality
- Mumford's inequality
- Mordell-Lang coonjecture
Cohomology
- etale cohomology
- Crystalline cohomology
- algebraic de Rham cohomology
- prismatic cohomology
- Hodge-Tate cohomology