There is no such thing as a blue emerald. First off, by definition an emerald color ranges from a slight yellowish green to a bluish green. While the tone and saturation of the color are factors in this, the primary color must be green.
Other colors of beryl (the mineral species of emeralds) do exist in a wide variety of colors, including blue. These other varieties include Heliodor, Gohsenite, Morganite, Bixbite, Maxixe, and Aquamarine. Aquamarines and Maxixe are the blue varieties of beryl, though there are complications with their color stability
Aquamarines have issues with their color fading with exposure to sunlight, though radiation will restore the color. The dark blue variety of beryl is called Maxixe, and fades even faster in light than aquamarines do.
Most aquamarines in the world come from Brazil (an international emerald supplier). Unlike their green cousins, aquamarines can grow to immense sizes, some equivalent to your forearm, with perfect clarity. The clarity and size also makes them extremely popular with cutters to make highly unusual “fancy” or “fantasy” cuts.
A famous example of this is the Dom Pedro Aquamarine, the largest aquamarine ever faceted as well as the most expensive one (its value sits around 5.5 million USD). The original, uncut gem was around 100lbs, though that does not necessarily mean the whole piece was gem quality. The amount of gem material usable for faceting may have only been 15-20lbs, which is still impressive for top-quality aquamarine.
When dealing with large gems, there are usually multiple gems faceted from the same rough, uncut stone. There are probably a number of smaller aquamarines that were faceted and sold after polishing the main piece, similarly to the Cullinan Diamond.
Just like aquamarines, the dark blue Maxixe comes primarily from Brazil. It was originally thought to only come from the Maxixe mine, (hence the name) though it has been found elsewhere in limited quantities.
This one will only ever be used as a collector gem, and never widely-used in the gem trade. Any exposure to sunlight will turn this super-rare collector gem into a piece of aquamarine, and not even a particularly nice one. Mining it can be very problematic since the miners do not necessarily know that this gem will fade in the sun. It also means Maxixe will never be found on the surface.
Normal indoor light causes fading to Maxixe beryl too, though it fades much more slowly like this than in daylight. This means it can’t even be viewed in a collector’s case with special lighting.
The only way to completely prevent fading in this type of beryl is to put it where light can’t reach, which limits its appeal to collectors.