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But superluminal transportation would not enable one to send messages from the future into the past. It would enable one only to send messages from the present into the future.

Here's something I've posted a time or two before, which I will repeat here in pretty much the original wording. "Physicist" has much more to say on this topic, so I'm pinging him in case he has any input.

Here's why an instant communications system creates time paradoxes:
Let's say you take off from earth in a ship that travels around 99% of lightspeed, fast enough for you to experience the effects of time dilation. And before you take off, you synchronize clocks with your buddy back home (synchronized clocks are a standard bit of equipment in these thought experiments). You've also got that mythical "subspace radio" which keeps you in instantaneous contact with earth. In due course, I shall demonstrate the impossibility of that device.
Okay, as you accelerate to 99% of lightspeed, less time is passing on ship than back home. You're not aging as rapidly as you would if you stayed home. This is apparent only in comparison with earth, however. On the ship you notice nothing odd. But your once-synchronized clocks are no longer in sync. Yours is now showing less time passing than the clock on earth (from earth's point of view). This is standard stuff, plain old special relativity.
After a while, let's say your ship clock is a day behind the earth clock (never earlier than launch date, however, as that would be quite impossible. The divergence commences after that point). Now you receive a message from earth: "Terrible news! President Hillary Clinton assasinated! USA in mourning." This message comes over your "subspace radio," so it's an instantaneous communication. However ... you have accelerated away from earth, so your clocks now register an earlier time than that on earth. If the message is sent on earth's July 4th, you receive it on your ships's July 3d. Okay. You're still with me, right? Get this next point, because it'll make your brain ache. Because of the relative velocity between the two reference frames, the sender's clock is later than the receiver's clock. This is always going to be true -- in all reference frames.
Now you get on the supspace radio and send your condolences back to earth: "So sorry to learn of the July 4th assasination of President Hillary Clinton." Ah, but here comes the catch. You are sending this message on your July 3d. Because of your acceleration away from earth, from your point of view -- which is just as valid as earth's viewpoint -- it's earlier on earth than it is on your ship. So your message travels instantly and gets to earth on July 2nd. Earth's July 2nd. And they now know of the event before it happened.
No, you say. Why would it get to earth earlier? Why? Because earth's message got to you earlier than it was sent, and it always gets where it's going earlier than it was sent. There are no privileged reference frames. From the point of view of your ship, earth accelerated away from you and is experiencing its own time dilation effect. It works both ways.
So your subspace radio allows messages to be sent into the past, which generates all kinds of paradoxes; and it is generally accepted that no such contrivance is possible. One of the advantages of a lightspeed limit is that it keeps the universe from going crazy.

But note that it takes large distances for this to have any significant "back in time" effect. So if there is a technique for exceeding lightspeed over a small distance, there's no major problem. We won't be receiving any messages from tomorrow.

If you're troubled by having Earth in the scenario, then consider two ships out in space somewhere, at rest relative to each other. They synchronize their clocks. Then they each accelerate in opposite directions, each moving very close to lightspeed, better than 99.5% of c, fast enough for each ship to experience significant time dilation. And they communicate with each other instantaneously. (In Heinlein's novel, he used twins who were telepathic for this.) Each ship has its clock slowed down with respect to the other (I think this is the result), so each one is sending messages "into the past," and they will be received at the other ship "earlier" than they were sent. As they exchange messages back and forth, each ship is receiving messages from the "future." Same paradox, but we leave earth out of it.