I downloaded and registered on openstreetmaps today . On my cell phone there are red markers with a white cross all over the map of Rome. Is it possible to remove them?
Clearly you just are browsing osm.org, not any map you downloaded. These are Notes, you can hide them by disabling their layer. (click the layers button)
There’s no “the” official OSM application by the way, only whatever people have made (like maps.me, OsmAnd, etc.) If you need more help, please be more precise, post names and screenshots what you are doing / using.
If you are using osmand on Android, touch the globe symbol top left. The on the pop out on the right there should be a slider next to “OSM notes”. Slide it to off.
If you are browsing the Web site, as Rico electric says, uncheck notes