Simple Steps to Training the Rider's Core

Most sources agree that core strength is important for riders of all disciplines. In fact whether you practice a sport or not core strength is a significant key to athletic performance good posture and joint and spine health.
A strong core is especially important because riders ride mostly with the torso. Whether you realize it or not your limbs are completely secondary aids to your seat weight and torso orientation.
For example observe a paraequestrian such as Canadian Olympic gold medalist Lauren Barwick who is paralyzed from the waist down. In addition to competing as a paraequestrian Barwick rides in open FEI levels. She and similar riders show that balancing on your horse and giving correct and effective aids depends less on use of limbs as torso position and usage. 

The impact of your limbs on your horses way of going is completely secondary to that of your seat and torso. If confused between a leg aid and a seat aid your horse will follow your seat and weight. This explains why he may seem to drift in one direction even though you are giving him leg aids to the contrary. In clinics and private training I constantly see riders who think they are giving aids in one direction but their torso position is giving a contrary aid which the horse obeys instead. For example they may be trotting on the left rein but the riders shoulders are pointed away from the circle so the circle looks like an octagon. This happens because instead of bending inward the riders body position is telling the horse to orient his body away from the circle.
It can be helpful to have a friend or your coach check your alignment and weight shifts. In addition to lateral weight shifts side to side you can tip forward which shuts down the horses movement and puts him on the forehand lean back which shuts down the horses movement and causes his back to hollow or rotate asymmetrically e g. one shoulder back more than another which interrupts the horses ability to move with straightness. Tipping forward with a weak core is not the same as deliberately putting your body in twopoint position. Even though the seat is in the air and the back is tipped forward a correct jumping position must include a strong core to help the rider maintain hip and heel alignment and grounding through the seat. Otherwise the rider would be falling onto his horses neck or involuntary shifts of bodyweight might cause the horse to bear onto the forehand.
Your bodyweight distribution and motion affects your horses movement with the tiniest shifts. Muscles in your core are like a sheath of muscles wrapping your torso and hips. They layer like plywood some of them orienting horizontally some vertically and some responsible for angled or rotational force. Most of them have a higher percentage of slowtwitch muscle fibre which means they are built for endurance not fast movement because they have to stabilize your body while your limbs do other tasks that generate load to your spine area. 
Your core muscles control your hip mobility and stability sitting on your horse postural stamina and independence of shoulders and hips. They are responsible for your control over seat and leg placement protecting your spine so that it is free to absorb your horses motion without injury and returning you to straight symmetrical posture after a dynamic interruption such as posting trot a change in gait a spook or a fence.
Your core muscles are the bridge connecting your upper and lower body and providing an anchor for your