Q:

Use of sequences like "In modern's US"

Is it correct to use possessive case for referring to the time in consideration, like in

in today's US
in modern's US
in last century's England

etc?

A:

Possessives in English can generally only be used to modify nouns (strictly speaking, noun phrases).   In your examples, two are nouns but one is not:

in today's US = in the US of today (OK)
in modern's US = *in the US of modern (not OK)
in last century's England = in the England of the last century (OK, but note we need to add the definite article)

