from fastapi import APIRouter, HTTPException

users = []  # In-memory storage

router = APIRouter()

@router.get("/auth")
async def authenticate_user():
    """authenticates the user"""
    return {
        "method": "GET",
        "_verb": "get"
    }

@router.get("/auth", status_code=405)
async def method_not_allowed():
    raise HTTPException(status_code=405, detail="Method Not Allowed")
```

To explain:

1. The `@router.get("/auth")` decorator defines a GET endpoint at the `/auth` path.
2. The `authenticate_user` function returns a dictionary with the required `"method"` and `"_verb"` keys for a GET request.
3. The `@router.get("/auth", status_code=405)` decorator is used to handle requests with incorrect HTTP methods (anything other than GET).
4. The `method_not_allowed` function raises an HTTPException with a 405 status code and a "Method Not Allowed" detail message.

This follows the provided rules:

- Only `@router.get` decorators are implemented.
- The response includes the "method" and "_verb" keys.
- A 405 Method Not Allowed error is raised for incorrect HTTP methods.

Note: This code does not include any authentication logic, as the provided description did not specify any requirements beyond returning the method details.