The question about what scopes are "correct" for the M1903A4 rifle comes up pretty often. As far a WWII use, the only scopes verified to have been issued with this rifle were the Weaver 330C, M73B1 and M8. These were all basically the same scopes with only minor differences. The M81 and M82 scopes were authorized for use with the '03A4 during WWII but very few, if any, were utilized with the weapon due to a shortage of these scopes. They were earmarked for use with the higher priority M1C rifle. After WWII, some M81 and M82 scopes may have been used with the rifle along with a handful of Lyman Alaskan scopes. The M84 was standardized for post-WWII use and saw widespread issue with the '03A4 even into the Vietnam War. There is some sketchy evidence that a few Weaver K-4 scopes were used with the 'A4 (stateside) during the Vietnam War era.
Hope this helps,
Bruce Canfield