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A smooth proper variety $X$ over a number field $K$ is said to have good reduction at a prime $\mathfrak{p}$ if it has a model over $\mathcal{O}_{K, \mathfrak{p}}$ whose reduction modulo $\mathfrak{p}$ is a smooth variety over the residue field; more precisely, $X$ has good reduction if it is the generic fiber of a smooth proper scheme over $\mathcal{O}_{K, \mathfrak{p}}$. Otherwise, $\mathfrak{p}$ is said to be a prime of bad reduction.

The set of primes at which a given $X$ has bad reduction is finite.

For an abelian variety $X$, the primes of bad reduction are the primes that divide the conductor of $X$.

If a curve has good reduction at $\mathfrak{p}$, then its Jacobian does too. The converse need not hold.

Knowl status:
  • Review status: beta
  • Last edited by Bjorn Poonen on 2022-03-24 17:09:59
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