Creating a specific mysql query

I have to create a function where for each customer i will find the year and month where he had the most rents. As for now I've got this done:

Select CUSTOMER_ID, NAME, SURNAME, count(distinct RENT_ID_PK) as total 
from t_rent
inner join t_customer on t_customer.CUSTOMER_ID_PK=t_rent.CUSTOMER_ID 
group by CUSTOMER_ID 
order by total desc;

To be honest got no idea how to move forward. Any tips?

@edit

CREATE TABLE `database`.`t_rent` (
  `RENT_ID_PK` INT(10) NOT NULL AUTO_INCREMENT,
  `CAR_ID` INT(10) NOT NULL,
  `CUSTOMER_ID` INT(10) NOT NULL,
  `STARTING_DATE` DATE NOT NULL,
  `FINISH_DATE` DATE NOT NULL,
  `COMPANY_ID_START` INT NOT NULL,
  `COMPANY_ID_FINISH` INT NOT NULL,
  `RENT_COST` INT UNSIGNED NOT NULL,
  PRIMARY KEY (`RENT_ID_PK`));

CREATE TABLE `database`.`t_customer` (
  `CUSTOMER_ID_PK` INT(10) NOT NULL AUTO_INCREMENT,
  `NAME` VARCHAR(45) NOT NULL,
  `SURNAME` VARCHAR(45) NOT NULL,
  `E_MAIL` VARCHAR(45) NOT NULL,
  `PHONE_NO` VARCHAR(45) NOT NULL,
  `PERSONAL_NO` VARCHAR(45) NOT NULL,
  `CREDIT_CARD_NO` VARCHAR(45) NOT NULL,
  `ADRESS` INT(10) NOT NULL,
  PRIMARY KEY (`CUSTOMER_ID_PK`));