The leaderboards are determined by 75% from the horse itselves and 25% from the descendents that are still "alive" in the game.
Freshman stallions (age 3yo) need to have 10 foals bred from them to make the leaderboard, 1 of those needs to be from a live cover.
Returning stallions (over age 3yo) just need to have bred a minimum of 1 live cover foal each game year to keep themselves on the leaderboard.
Mares need to have at least 3 foals, no matter if they are still in the game or not to get on the boards.
Stallions LB is deleted every gameyear´s reroll and built up new in the next few days after the studs got used.
Mares LB never gets deleted.
All horses at the ago 18+ vanish from the boards as they dont have live covers available any more.
Single discipline LBs only look at the traits, that are important for this discipline. So even e.g. an EAA stallion can be high on the Dressage LB ;)
When a horse is created, it gets "points" - but they never show up for us players. They are the basics to determine the rating of a horse. So if a horse has more of these "points" by itselves, it is higher on the boards than one with less "points".
But these "points" and the LB position are NOT a marker, if the horse is a good producer or not! To determine a good producer you will have to take a close look at all foals of a horse and the foals´ pedigrees.
Even great producers do not need to show up on the LBs, if their own "points" are to low as, the top LBs are very competitive.
Low ratings in a pedigree ( e.g. a visible S rating ) do not have any influence on the possibilies of a horse as a good-foal-producer, too!