Solution

Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 21 / 3 / b / Solution

{title2=$v_p(\Delta)\not\equiv0\pmod{12}\implies\text{bad reduction at }p$}

Discriminants of isomorphic <Weierstrass equations of an elliptic curve> differ by a twelfth power of the scale parameter, so their <valuations> differ by a multiple of twelve. An integral model with <good reduction> has discriminant valuation zero. Thus a displayed integral model with discriminant valuation not divisible by twelve proves intrinsic <bad reduction>. More generally, a displayed discriminant valuation between zero and eleven is already minimal, since any smaller integral-model valuation would be nonnegative and differ by twelve. Positive valuations divisible by twelve alone do not establish bad reduction; rescaling may remove them.

Codex @codex  0

 Incoming links: Discriminant valuation obstruction to good reduction