Tachyons allow messages to be sent to earlier times according to the equations of Einstein’s special relativity. A device for doing this has been called a “tachyonic antitelephone.” The device operates with the aid of our distant friend Prot in a spaceship near the Earth-like planet K-PAX, which is a distance L = 1,000 light years away in the Lyra constellation. We modulate our tachyon beam to encode a message sent to Prot’s ship which is traveling away from Earth at a speed v < c. Prot then immediately retransmits our message back to Earth, again using tachyons of FTL speed a > c. Using the standard relativity equations, it can be easily shown that the return message arrives back at Earth in a time T, which depends on the variables v, L and a. The mathematical details can be found at the antitelephone Wikipedia web site linked above. The important point is that when Prot’s ship speed v is sufficiently large (but less than lightspeed) then the time T will be negative, meaning that we receive the return signal from Prot before we transmitted the original signal to him, and our signal has therefore reached some past time on Earth. How far back in time T would be depends on the distance and speed of Prot’s ship, and the speed of the tachyons. Since K-PAX is 1000 light-years away, it turns out we can send messages back in time at most 1000 years.
K-PAX and Prot, as I suspect you realize, are fictitious, and based on the movie K-PAX, a 2001 American-German science fiction-mystery film based on Gene Brewer’s 1995 novel of the same name. However, the above description about sending messages back in time using tachyons is accurate, assuming (a) tachyons exist, (b) they can be used to send messages, and (c) they satisfy the equations of relativity in all reference frames. In the movie Prot has the following dialog with the psychiatrist who at one point is trying to poke holes in his story about coming to Earth at light speed.
Dr. Mark Powell: What if I were to tell you that according to a man who lived on our planet, named Einstein, that nothing can travel faster than the speed of light?
Prot: I would say that you misread Einstein, Dr. Powell. May I call you Mark? You see Mark, what Einstein actually said was that nothing can accelerate to the speed of light because its mass would become infinite. Einstein said nothing about entities already traveling at the speed of light or faster, at tachyon speeds.
Here Prot brings in the idea of tachyons, but fails to explain how he travelled at light speed without changing his rest mass to zero (like a photon).