This is almost certainly Red Planet Mars (1952)
Plot (from Wikipedia)
An American astronomer obtains images of Mars suggesting large-scale environmental changes are occurring at a pace that can only be accomplished by intelligent beings with advanced technology. At the same time a colleague claims to have been contacting Mars by radio, using technology stolen from the Nazis after World War II. He communicates first through an exchange of mathematical concepts, like the value of pi, and then through answers to specific questions about Martian life. The transmissions claim that Mars is a utopia, which has led to great technological advancement and the elimination of scarcity, but that there is no fear of nuclear war.
This revelation leads to political and economic chaos, especially in the Western hemisphere, and is said to have "done more to smash the democratic world (or Capitalist?) in the last four weeks than the Russians have been able to do in eleven years." The U.S. government imposes a news blackout and orders the transmissions to stop due to fears that the Soviet Union could pick up and decode their messages. This ends when the next message reveals that the Earth is condemned to the constant fear of nuclear war as a punishment for straying from the teachings of the Bible. Revolution sweeps the globe, including the Soviet Union, which is overthrown and replaced by a theocracy, which is met with celebration in America.
But doubts about the authenticity of the messages remain. An ex-Nazi who developed the original communication device prototype wants to announce that he has been duping the Americans with false messages from a secret Soviet-funded radio transmitter high in the Andes mountains of South America. He says that he transmitted the original messages supposedly from Mars, but that the United States government made up the religious messages, which he allowed because he wanted to see the destruction of the Soviet Union. The mystery thickens as it appears the messages may have continued even after the secret transmitter was destroyed in an avalanche, but the American transmitter is blown up before the message can be received.