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Why do we need the 'minimal' Weierstrass equation of an elliptic curve in order to study it's different reduction types (good and bad) ?

What happens if we don't start with a minimal Weierstrass equation ?

Andrew
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  • It depends on what you mean with "needing" a minimal model. It just happens that if you have a minimal model then you can explicitly relate the good and bad reduction to the discriminant of your Weierstrass equation, for example. Otherwise, there might be primes of good reduction that divide the discriminant, and this is more confusing. – Ferra Oct 26 '16 at 09:47

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Consider the elliptic curve $E: y^2=x^3+15625$. The discriminant of this curve is $-105468750000=2^4\cdot 3^3\cdot 5^{12}$. Thus, if we reduce the given model modulo $5$, we obtain $y^2\equiv x^3 \bmod 5$ which is a singular curve over $\mathbb{F}_5$.

This "bad reduction" at $p=5$ is unnecessary (or removable, in the sense of removable discontinuities in Calculus) and it appeared because we didn't choose a better model for the curve. Since $15625=5^6$, it turns out that our curve $E$ is isomorphic over $\mathbb{Q}$ to the curve $E': y^2=x^3+1$ with discriminant $-2^4\cdot 3^3$ that has good reduction at $p=5$.

Choosing a minimal model from the beginning is desired so that all removable primes of bad reduction are already not present, and only essential primes of bad reduction remain.

Álvaro Lozano-Robledo
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  • is it obvious how the discriminant of the curve evolves under changes of variable $(x,y) = (cx'+d,ey'+f)$, and is it the correct way to think to this minimal model ? – reuns Oct 28 '16 at 01:06
  • You can find the formulas for how the discriminant changes with changes of variables in Ch 3 of Silverman's "The Arithmetic of Elliptic Curves" for example. – Álvaro Lozano-Robledo Oct 28 '16 at 01:45