There are two-way prepositions that can be either accusative or dative. If the preposition in the sentence answers the question "where to?" (meaning the object in question is at work at reaching its destination) it is accusative. If the sentence answers the question "where?" (meaning the object in question is already at that place) it is dative. In this case, the "best wishes" are directed "to" your mother, so as it implies motion, it's accusative.
I suspect it's because 'an' translates to 'to' rather than 'on' in this situation; however, I can't find the answer elsewhere. Edit: Nope, I'm wrong. It is something to do with motion.
You're kinda right though, 'to' would only be used for movement, and 'on' only for location. So it's fair to say that when 'an' translates to 'to', it takes accusative, and for 'on' it takes dative. (Although if it translates as 'onto', it's movement, so it's accusative). This is only for two-way prepositions though, 'zu' can mean 'to' as well, but it isn't two-way, and it always takes the dative.
"at" is only for location, but "on" is sometimes used for destination of motion as well (he puts the book on the table = he puts the book onto the table).
But I'm pretty sure when it comes to physical locations and directions "on" translates to auf with dative and accusative respectively.
In general, yes. But for example, a picture hangs "on the wall" in English but "an der Wand" in German (and you would hang it on the wall but "an die Wand").
Ah very true. Because the painting isn't actually atop the wall, but attached to it.
I'm not sure what auf has to do with wishes.
Are you thinking of toasts? (Auf deine Mutter! = "To your mother!")
But that's not the same as conveying wishes to someone.
Yeah, the way I understand it the 'greetings' are being sent, hence it is considered a motion. :P
Viele Grüße = many greetings, regards. Best wishes = gute Wünsche, alles Gute
if i would want to tell "to" when i have to use an, zu, für, um, bei, or mit.
"Best wishes ...." is good, but kind of formal. Simply "Greetings to .." is much better, but probably not accepted without the adjective.
Literally, yes. But a slightly more natural translation would be "to your mother".
Could anybody explain why after 'an', that is a two-ways preposition, and there is no movement involved, there is the Accusative instead of Dative case?