下面是一个使用ASP.NET Core Identity的AuthController示例，包括登录和注册操作：

using Microsoft.AspNetCore.Identity;
using Microsoft.AspNetCore.Mvc;
using System.Threading.Tasks;

namespace YourNamespace.Controllers
{
    [ApiController]
    [Route("[controller]")]
    public class AuthController : ControllerBase
    {
        private readonly UserManager _userManager;
        private readonly SignInManager _signInManager;

        public AuthController(UserManager userManager, SignInManager signInManager)
        {
            _userManager = userManager;
            _signInManager = signInManager;
        }

        [HttpPost("register")]
        public async Task Register(RegisterViewModel model)
        {
            if (!ModelState.IsValid)
            {
                return BadRequest(ModelState);
            }

            var user = new IdentityUser { UserName = model.Email, Email = model.Email };
            var result = await _userManager.CreateAsync(user, model.Password);

            if (result.Succeeded)
            {
                await _signInManager.SignInAsync(user, isPersistent: false);
                return Ok();
            }

            return BadRequest(result.Errors);
        }

        [HttpPost("login")]
        public async Task Login(LoginViewModel model)
        {
            if (!ModelState.IsValid)
            {
                return BadRequest(ModelState);
            }

            var result = await _signInManager.PasswordSignInAsync(model.Email, model.Password, model.RememberMe, lockoutOnFailure: false);

            if (result.Succeeded)
            {
                return Ok();
            }

            return BadRequest("Invalid login attempt");
        }
    }

    public class RegisterViewModel
    {
        public string Email { get; set; }
        public string Password { get; set; }
    }

    public class LoginViewModel
    {
        public string Email { get; set; }
        public string Password { get; set; }
        public bool RememberMe { get; set; }
    }
}

在上面的示例中，我们使用UserManager来管理用户，SignInManager用于用户的登录和登出操作。Register和Login方法分别处理注册和登录请求。