"Modern pop and rock music has been greatly influenced by jazz."
The sentence actually omits elements (for brevity) which we take for granted.
The full sentence would be:
Both modern pop music and modern rock music have been greatly influenced by jazz.
Strip it down a bit:
Both pop and rock music have been greatly influenced...
...so that the verb corresponds with the plural "both". Then take away the "both" and leave the plural form of the verb!