You can use any programming language and technology to implement the authentication endpoint as long as it can expose an HTTP endpoint so it can be accessed by visitors using a browser.
Below are a few examples for inspiration. Note that the examples below are not meant to be used in a production environment, but may make it easier to understand what the authentication endpoint should do at a high level.
The code examples are written in NodeJS to make it easy for you to run locally. In order to run them, you will need to install NodeJS on your machine.
Explore these examples: