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- smooth
- projective (schemes)
- geometrically integral
- geometrically irreducible
- model of a scheme
good reduction
- In terms of equations: for $E_{/ {{\mathbf{Q}}}} $ an elliptic curve \(y^2=x^3+ax + b\) of discriminants \(\Delta(E) = -16(4a^3+27b^2)\), \(E\) has good reduction at \(p\) for any prime \(p\nmid \Delta(E)\).
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A variety $X\in{\mathsf{Var}}_{/ {k}} $ has good reduction if there exists a smooth proper \({\mathcal{O}}_k\) scheme \({\mathcal{X}}\) whose generic fiber \({\mathcal{X}}_k\) is \(k{\hbox{-}}\)isomorphic to \(X\).
- Assume \(k\) is a characteristic zero field with a complete discrete valuation and $X_{/ {k}} $ is smooth and proper.
- Alternatively: if \(k\in \mathsf{Field}\) is complete wrt a diiscrete valuation \(v: k^{\times}\to {\mathbf{Z}}\), let $X\in\mathsf{smProj}{\mathsf{Var}}{/ {k}} $ with \(\dim X = d\). Then \(X\) has good reduction if there exists a smooth model of \(X\) over the valuation ring \({\mathcal{O}}\subseteq k\), i.e. there exists homogeneous polynomials cutting out a \(k{\hbox{-}}\)variety in some ${\mathbf{P}}^N{/ {k}} $ which is isomorphic to \(X\) whose coefficients are in the valuation ring of \(v\) where reducing all of them modulo the maximal ideal \({\mathfrak{m}}\) gives equations defining a smooth variety of dimension \(d\) over the residue field \(\kappa\). - If \(k\) is not complete (e.g. a number field), say \(X\) has good reduction at \(v\) if $X_{k_v}\in{\mathsf{Var}}_{/ {k_v}} $ has good reduction. - Nice varieties: smooth, projective, geometrically integral or geometrically irreducible.
Special cases: curves and abelian varieties
Notes
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Consequence: if $E_{/ {{ {\mathbf{Q}}p }}} $ has good reduction, there is a unique elliptic scheme ${\mathcal{E}}{/ {{ {\mathbf{Z}}_{\widehat{p}} }}} $ with \({\mathcal{E}}_{{ {\mathbf{Q}}_p }} = E\) and a structure map \({\mathcal{E}}\to \operatorname{Spec}{ {\mathbf{Z}}_{\widehat{p}} }\).
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Neron-Ogg-Shaferevich: for an elliptic curve, \(X\) has good reduction iff the \(G_k\) representation on \(H_\text{ét}^1(X; { {\mathbf{Z}}_{\widehat{\ell}} })\) is an unramified Galois representation, i.e the action of inertia \(I_k\) is trivial.
Crystalline representations
For elliptic curves, a first incarnation of p-adic Hodge theory:
