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A curve defined over the rational numbers is said to be locally solvable if the equation(s) defining the curve have a solution over $\R$ and over $\Q_p$ for every prime $p$. For curves over number fields we require solutions over every completion of the number field.

If the curve is non-singular (or even merely geometrically irreducible), then there are always at most finitely many primes $p$ such that the curve has no point over $\Q_p$.

Curves with a rational point are obviously always locally solvable, but the converse need not hold (unless the curve has genus zero). For example, the genus 1 curve $3x^3+4y^3+5z^3=0$ over $\Q$ is locally solvable but has no rational points (this example is due to Selmer).

Knowl status:
  • Review status: reviewed
  • Last edited by Jennifer Paulhus on 2019-04-20 15:18:35
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