The rear turn signals do not light. The front ones blink quickly. The bulbs are OK. In emergency flashing mode, all lights are flashing normally. Fuses checked in the cabin and under the hood. There are no errors. Maybe a relay. They write that it is in the cabin above the hood opening button. The bottom is not visible. How to get there
What could be the problem?

 

Did you check the wiring that goes to the bulb? Could be an issue with the connector.

 

in emergency flashing mode, all lights are flashing normally

 

I think you are on the right track with the relay.
Probably different relays for hazards and turn signals
Look for the relay cluster under the dash by the OBD connector

 

I was thinking a relay makes then blink fast

 

I took out the relay from under the panel. Disassembled. Inside are two relays. Both work (magnets and contacts close) both when turning on the emergency light and when cornering. But with the emergency light, the rear lights shine, but not when cornering. With the relay, everything is ok. Contacts close. Also, by the sound, I found a relay in the trunk) Dismantled it. There are four relays inside. Magnets in all work both during emergency and cornering. But when the emergency light bulbs shine and when cornering no. When emergency gang all four blink at normal intervals. When cornering, the rear ones do not shine and the front ones flash quickly. There are no errors on the computer.
Where to dig?

 

Contact your local parish priest and ask permission to dig up the remains of the spirit possessing your turn signal circuit...
But seriously, do you have these drawings from the RX-8 Electrical Shop Manual?
It shows a black box called the Flasher Control Module; could be an issue internal to that device.

 

in my trunk, under the skin, there is still a block of four relays through which the turn lamps are connected. I do not see this relay block in the diagrams

 

Do the wire colors match? What year is your car?
Pretty sure all those manuals are for 2004 model year.
The drawing is from the electrical manual from Mazda; downloaded from rotaryheads.com

 

2003. But in my trunk, under the skin, is another unit with four relays which is not shown in the diagram. Relay checked. Workers

 

2003? Are you sure you’re in the right forum?
Mazda stopped selling the RX-7 in 2002 and started selling the RX-8 in 2004.
Perhaps your car is a 2004 and the 4 relays in the trunk make-up the flasher control module.