Rationality
Suppose X(t), Y (t) corresponds to e
P ∈ e
C.
We would like K(coeffs of X, Y ) = the field of definition of e
P.
" Rescalings t ← t′
= ct + O(t2
) typically destroy this!
If P is nonsingular, we can always have either X(t) = xP + t
or Y (t) = yP + t. But what if P is singular?
If X(t) = te
, Y (t) =
X
n⩾n0
antn
, can rescale t ← ζet (ζe
e = 1)
⇝ X(t) = te
, Y (t) =
X
n⩾n0
anζn
e tn
.
Nicolas Mascot Algebraic curves