Reduced minimal models and torsion

Publication date: 23 January 2023

Abstract: Let

E / m a t h b b Q

be an elliptic curve. The reduced minimal model of

E

is a global minimal model

y^{2} + a_{1} x y + a_{3} y = x^{3} + a_{2} x^{2} + a_{4} x + a_{6}

which satisfies the additional conditions that

a_{1}, a_{3} i n 0, 1

and

a_{2} i n 0, p m 1

. The reduced minimal model of

E

is unique, and in this article, we explicitly classify the reduced minimal model of an elliptic curve

E / m a t h b b Q

with a non-trivial torsion point. We obtain this classification by first showing that the reduced minimal model of

E

is uniquely determined by a congruence on

c_{6}

modulo

24

. We then apply this result to parameterized families of elliptic curves to deduce our main result. We also show that the reduction at

2

and

3

of

E

affects the reduced minimal model of

E

.

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