TITLE: Why 1728 in $j$-invariant?
QUESTION [17 upvotes]: The $j$-invariant for elliptic curves has a $1728$ in it. According to Hartshorne, this is supposedly for characteristic-$2$ and $3$ reasons, despite appearances to the contrary.
Indeed, it is unfathomable why it would help in char $2$ and $3$ when it would vanish.
For that matter, the functions $g_2$, $g_3$ and $\Delta$ too have these constants. Is there a ``good reason'' why these exist, other than historical reasons? And why are they kept on when we pass to abstract algebraic geometry, when $1728$ only seems to do harm?

REPLY [15 votes]: Indeed, it is unfathomable why it would help in char 2 and 3 when it would vanish.

It doesn't vanish, in characteristic 2 or 3! Instead, it cancels out the coefficients that would make it otherwise impossible to define the $j$-invariant.
Also, note that the $j$-invariant doesn't have a 1728 in it: the 1728 only appears in the specific formula for the $j$-invariant of a curve expressed in short Weierstrass form.
In characteristic $2$ and $3$, you can't write elliptic curves in short Weierstrass form!
An instructive exercise (it helped me) is to take the forms that you need to use for characteristic $2$ (or $3$), and use the same form to express a rational elliptic curve. Then, compute it's $j$-invariant (e.g. by converting the curve to short Weierstrass form). You'll see the $2$'s or $3$'s cancel out.