You need the version. Do not use Nano or Stock; the S3’s system partition is too small. Link: https://sourceforge.net/projects/opengapps/files/arm/20220215/open_gapps-arm-10.0-pico-20220215.zip
Look for the Unofficial LineageOS 17.1 for i9300 . (You may need to search XDA Developers forum for active download links, as direct alpha links often die).
This process completely wipes your internal storage. Save your photos, contacts, and messages externally. samsung s3 custom rom android 10 link
Select ARM -> Android 10.0 -> Pico or Nano .
As this is an unofficial "Alpha" quality build, it has a few known bugs. Developers report that GPS, Bluetooth, and root access (Magisk installation does not work) are not functioning. Users have also reported occasional screen glitches and sensor reliability issues. You need the version
Connect your phone to your PC via a USB cable. The ID:COM box in Odin should turn blue, indicating a secure connection.
You need Team Win Recovery Project (TWRP) version 3.3 or higher to handle Android 10 zip files. Official TWRP for Samsung Galaxy S3 (i9300) (You may need to search XDA Developers forum
Release the buttons once the Samsung logo appears and the blue recovery text shows in the top-left corner. Step 3: Wipe the Old System
Always read the last few pages of the XDA thread before downloading to check for any newly reported bugs regarding camera stability, RIL (cellular network calls), or Wi-Fi connectivity.