I bought this rifle today as a 1903A1. It's marked : U.S. Remington model 1903, s/n 3355333.
Not being familiar with the 1903 family, I wonder whether it is an A1 or a 1903 in a "C" stock.

Also, the tangent rear sight looks quite complex with 3 notches and a peep sight + a windage ladder. Could someone explain how to use it and tell me where to find the year of manufacture ?

I suggest reposting on the "Ask the Gunwriters" forum. More people in a position to answer your question should see your post there.

The M-1903A1 was a modified 1903 adapted to use the Pedersen device which turned it into a .30 caliber semi auto.The easiest way to identify one is the oblong ejection port cut int the left bolt raceway.Your picture seems to be an M-1903 in the later and better style stock.

The A1 was issued in 1929. This Remington is a 1942/43 rifle with an A1 rear sight, probably one of the Rock Islands completed by Remington when they took over production of the 1903 during WWII.

Andre, see my original detailed answer to you in the other web site.

Thanks to your combined info, I was able to do some research. My rifle seems to be a '42 1903 Modified. A bigger surprise came when I took it to the range for the 1st. time. I shot at 50 m to verify the sights with peep hole set at 100 (BTW, that hole is so small I see it growing hair when sighting for more than a few seconds - or then my eyes are no longer what they used to be...). Anyway, I was amazed by a 5 shot group, 1" high - 1.5 left, measuring .9". Looks like it can shoot ! Meanwhile, I've moved the front sight a wee bit to the left and will try the 100 m range this w-e.