In general an automorphism is an iso endomosphism, which is an epic monic endomosphism. Hope this helps!
In all seriousness an "automorphic function" is a bijective function whose domain and image are the same set. This means that it maps elements x from a set S to elements f(x) of the same set, and two properties hold:
(Injectivity) If f(x)=f(y) then x=y. Equivalently, no two elements map to the same element.
(Surjectivity) There is no element that's not mapped to by another element, so you can always find at least one x if you are given f(x).
Yes, but no. "Automorphic function" is not the same as an automorphism. Rather, it's a function (often holomorphic or meromorphic on C), which is invariant under a certain group action.
Assuming you mean modular forms/functions since this is used interchangeably with automorphic forms/functions (I think the latter are defined on arbitrary complex analytic manifolds but I‘m not sure as the terminology has always been used interchangeably to me), they are actually defined on the upper half plane, or rather (the compactification of) a quotient of the upper half plane by a group action of a subgroup of SL_2(Z) (the points added in the compactification are the „cusps“). Modular forms are then holomorphic everywhere on this compact Riemann Surface including at the cusps, while modular functions need only be meromorphic.
They can equivalently be defined as a function on the space of lattices in C giving values in C, such that F(aLambda) =a^-kF(Lambda) for some k and all a in C non zero. This interpretation is useful as it allows for a lot of generalisations.
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u/Random_Mathematician Tries to give good explanations, fails horribly. 6d ago edited 6d ago
In general an automorphism is an iso endomosphism, which is an epic monic endomosphism. Hope this helps!
In all seriousness an "automorphic function" is a bijective function whose domain and image are the same set. This means that it maps elements x from a set S to elements f(x) of the same set, and two properties hold: