I am not that deep in the maths et al. but the problem I believe is not the weapon damage die, but the fixed bonuses you receive and which
a) increase over the levels
b) get multiplied on a critical hit too.
With Improved Crit, you would have a weapon that crits on 15+ and does three-times damage.
This is probably severly overpowered.
The Falcata "only" has 19-20/x3 and EVERYONE says this is broken (mainly because no other weapon is THAT good).
That being sad, a weapon with 1d4, 19-20/x3 "might"(?) be acceptable.
Dracodruid said it pretty well so I am just going to try to explain why.
A note though overpowered compared to the weapons available is what I am saying.
Well for simplicity's sake let's say that everything 15+ hits.
Now after some levels the damage die doesn't really matter at all apart from spesific builds size increase would be one way to do utilize it.
So let's take a look shall we. I will post the damage total of rolls with 15-20 asuming everything hits. Also asuming all critical's will be confirmed. Placing examples of a how to get the crit modifier and threat range. I am also ignoring with examples class abilities that change those since I canät be bothered to look everything up.
20x2= 6 hits with 1 critical x7 (Club)
20x3= 6 hits with 1 critical x8 (Great axe)
20x4= 6 hits with 1 critical x9 (Scythe)
19-20x2= 6 hits with 2 critical x8 (Dagger)
19-20x3= 6 hits with 2 critical x10 (Falcata)
19-20x4= 6 hits with 2 critical x12 (Scythe with improved critical)
18-20x2= 6 hits with 3 critical x9 (Scimitar)
18-20x3= 6 hits with 3 critical x12 (None to my knowledge)
18-20x4= 6 hits with 3 critical x15 (None to my knowledge)
17-20x2= 6 hits with 4 critical x10 (dagger with improved critical)
17-20x3= 6 hits with 4 critical x14 (Falcata with improved critical)
17-20x4= 6 hits with 4 critical x18 (None to my knowledge)
15-20x2= 6 hits all critical x12 (Scimitar with improved critical)
15-20x3= 6 hits all critical x18 (your suggestion with improved critical
15-20x4= 6 hits all critical x24 (None)
If we take your suggestion of 18-20x3 it's x12 which is as good as Falcion and scythe with improved critical which are the best martial weapons in this sense. And when we take improved critical with your suggestion it's x18 which is better than anything Falcata which is the best weapon in this sense is only x14.
I hope that helped to understand how it works. Now it's ultimately up to GM what to allow, but my meaning was to give the information to base the decision on.
I'd say it is strong but as a light weapon (can't get the higher bonus from PA) that is not a dagger or similar and has no additional special abilities it is ok for exotic.
Why not a dagger? There are some specials for knifes or daggers like the bonus damage via river rat or the higher sneak damage via knife master.
Instead of "Dagger" you could also say "Longsword" in the list above!
When they start calling the weapon by it's actual name and not confuse it with longsword I might :P
Nah in serious I just went with what first came to mind.
And yes I would agree that it definatly needs to be exotic weapon. What it would be regardless if it's 19-20X3 or 18-20X3 1d4. Well x3 means that it can inflict serious wounds and the threath range means it does it fairly often while 1d4 suggest that if it doesn't it doesn't do that much. Only thing that comes to mind is kris and I mean the one's with the pistol grip not the showy ritual knives. Also I do not know if it has a name and it most likely was never a real weapon but anyways. A dagger that had two extra blades coming from the main blade pointing towards the user in 45 degree angle, but they went in to "holes" in the main blade but would come back out when pulling the blade out of the victim. Hope that is somewhat clear explanation.