WikiPedia has an article about E-sharp minor scale (that is a scale relative to G-sharp major) there is no article of E-sharp major, instead it redirects to F major that is a scale using flats instead of sharps. I want to know if the E-sharp major scale is fake? PS. I am asking specifficaly this scale.
Doesn't look fake to me. But sort of cumbersome to read. E major has 4 sharps, so it's not surprising E sharp major has 4 double sharps and the other notes sharped.
In general, going to the enharmonically equivalent key implies a jump by net 12 sharps/flats (2 for each whole step in the scale, 1 for each half step). F major has -1 sharps (well, 1 flat), so E sharp major has a net of 11, namely 3 single and 4 double ones.
It's not fake, it's theoretical.
There is a theoretical difference between E sharp major and F major because they are spelled differently and reside in different places on the circle (spiral?) of fifths. In practical terms, they are identical because they are enharmonically equivalent, thanks to equal temperament.
It exists but probably isn't included as an article because, whereas a scale like Db minor is a flat note with no "simpler" spelling*, E# is enharmonic to F which is a white key and arguably simpler / more common to read. Db minor also has double accidentals.
Basically, E# refers to the same pitch as F, and there's not much reason to use E# as the root of your scale in the first place, so there's no article for E# minor.
As a sidenote, E# and F don't necessarily refer to the same pitch. For example, in the tuning system 19TET, E# is enharmonic to Fb instead, which then lies halfway between E and F.
*typically Db minor is replaced with C# minor because C# minor, being the relative minor of E major, has a much simpler key signature without any double accidentals. But if a piece is in Db major and borrows chords / notes from the parallel minor, then Db minor would theoretically be used.
As other answers point out, E# minor exists in theory, but is in used vanishingly rarely (if ever) in practice. But this provokes the followup question: if E# minor so rare as to be “purely theoretical”, why is G# major (with the same key signature) less rare in practice?
In isolation, G# major is just as impractical: for the primary key of a work, one would almost always write it as Ab major instead. But G# major is closely related to several minor keys which have fewer sharps in their key signature — most closely, C# minor and G# minor. These keys do appear often in practice (since their enharmonic equivalents, Db minor and Ab minor, have more flats than these have sharps). So if a piece is primarily in C# minor and modulates to its dominant major, it’s often more readable to write the section in G# major than to switch between sharp spellings and flat spellings within the piece.
Generally, major keys have closely related minor keys with more flats/fewer sharps, while minor keys have closely related major keys with more sharps/fewer flats, so extremely sharp key spellings occur more often as major keys, while extremely flat key spellings occur more often as minor keys. On the sharp end, E# minor and A# minor are rarer than G# major and C# major respectively, as mentioned in the question. On the flat end, Fb major and Cb major are rarer than Db minor and Ab minor, because (for instance) these minors can appear secondarily in a piece with primary key Ab major.