Last modified date: <%+ tp.file.last_modified_date() %>
- Tags: - #AG #arithmetic-geometry - Refs: - #todo/add-references - Links: - scheme - elliptic curve - algebraic curve - good reduction - semistable reduction
curves
Definitions
- Variety: an integral separated scheme of finite type over \(k\).
- Dimension: dimension as a Noetherian topological space.
- Curve: a complete variety (proper over \(k\)) of dimension 1.
- Singular: for \({\mathfrak{m}}{~\trianglelefteq~}{\mathcal{O}}_{X, x}\) the maximal ideal of the local ring at a closed point \(x\), \(X\) is singular if \(\dim_{\kappa(x)} {\mathfrak{m}}/{\mathfrak{m}}^2 \neq 1\) where \(\kappa(x)\) is the residue field at \(x\).
-
valuation : for \(f \in {\mathcal{O}}_{X, x}\), \(v_p(f)\) is the largest \(n\) such that \(f\in {\mathfrak{m}}^n\).
- Zero: \(v_p(f) > 0\)
- Pole: \(v_p(f) < 0\)
- Nonvanishing: \(v_p(f) = 0\).
- Unsorted/function field : the local ring \(K(X) \coloneqq{\mathcal{O}}_{X, \tilde x}\) for \(\tilde x\) the generic point.
- Degree: for \(f:X\to Y\), the degree of the field extension \([K(X) : f^* K(Y)]\).
- Ramification index : \(e_f(x)\) defined as the largest \(n\) such that \(f^* {\mathfrak{m}}_{f(x)} \subseteq {\mathfrak{m}}_x^n\).
-
ramified : \(e_f(x) > 1\).
- Alternatively: at a closed point \(x\), \(f^* {\mathfrak{m}}_{f(x)} = {\mathfrak{m}}_x\) and the extension \({\mathcal{O}}_x/{\mathfrak{m}}_x\) is a finite separable extension of \({\mathcal{O}}_{f(x)} / {\mathfrak{m}}_{f(x)}\).
- Structure morphism: for a scheme \(X\) over \(k\), the map \(S: X\to \operatorname{Spec}k\)
- geometric point : a section to the structure map, \(s: \operatorname{Spec}k \to X\) so that \(\operatorname{Spec}k \xrightarrow{s} X \xrightarrow{S} \operatorname{Spec}k\) is the identity on \(\operatorname{Spec}k\)
- elliptic curve : genus 1. Coincides with \(y^2 = x^3 + Ax + B\). Exists as a pointed scheme \((E, O)\)
- isogeny : a pointed map \((E_1, O_1) \xrightarrow{f} (E_2, O_2)\), so \(f(O_1) = O_2\).
- supersingular : \(\ker\qty{E \xrightarrow{\cdot p} E } = 0\) where \(\operatorname{ch}k = p > 0\).
- Ordinary: \(\ker\qty{E\xrightarrow{\cdot p} E } = {\mathbf{Z}}/p\)
- etale morphism : flat and unramified. Supposed to look like a local homeomorphism in a covering space.
Genus formula
Examples
Genus 0
Genus 1
Genus 2
Genus 3
Genus 4
Genus 5
Homology
Examples
