Samsung Galaxy S3 black screen and automatic reboots
I gave up and did a factory reset. Since then, the phone has been working for 5 days in a row... so it seems like at least the reboots are gone (I hope not to jinx it...).

I've also noticed that since the reset, the battery lasts at least twice longer than before. I've fewer installed apps, which should lead to a slight battery life improvement, but this isn't slight, it's the double. Before the reset, at night (no data connections, no WiFi), the phone consumed about 1% of its battery per hour. Now it's consuming less than 1% in the whole night. Obviously there was something wrong before the factory reset. I hope everything is fine now...



In case it's helpful for anybody else, this what the Samsung Customer Service told me to do:


  
  Start the phone in Secure Mode for a couple of days. That means that only preinstalled apps will work. So if that solves the problems, there is an app causing them. To enable Secure Mode, you have to reboot the phone, and press the options button (the left one) when the Samsung logo appears; it should vibrate, and then you will see "Secure Mode" at the left bottom corner. To disable Secure Mode, turn off the phone and extract the battery for a few seconds.
  --- For Verizon GSIII phones: after you reboot the phone, press and hold the volume down button  when the Samsung logo appears. Directions also state that you may need to hold the button for up to 45 seconds before you will see "Safe Mode" in lower left-hand corner. http://support.verizonwireless.com/support/devices/knowledge_base.html/59242/
  Perform a factory reset (if the phone still has problems running in Secure Mode). To perform it, go to Menu > Settings > Backup &amp; reset > Factory data reset. It will revert the phone to its original factory status, therefore, you will lose all the data and applications stored in your phone's memory (it will not affect the SIM card, nor the microSD if you have one). Be sure to backup everything you don't want to lose! You can use Samsung KIES for that backup.
  If the phone still has problems after the factory reset, it should be checked by Samsung Technical Service.