When will the value of automorphic function $f(x)$ satisfy an algebraic equation? Or what is the value of $x$ such that the value of automorphic function $f(x)$ is algebraic?
If the question is too broad, may anyone familiar with such a topics give example or class of examples?
Any reference and answer is welcome.
Again, I think there is too much that could be said in response to this question to fit into this format, but I can give some indications.
Again, thinking that exponential functions are a sort of automorphic function, certainly special values of them (at "division points", hence, roots of unity) generate many abelian extensions of $\mathbb{Q}$. Kronecker (and Weber) showed that these are all the abelian extensions of $\mathbb{Q}$.
Similarly, for complex quadratic fields, many people in the 19th century knew that division-point coordinates of corresponding elliptic curves (with "complex multiplication" by the ring of integers) in Weierstrass form generated abelian extensions. Similarly, special values of automorphic forms, in effect giving corresponding points on moduli spaces, generate abelian extensions. Kronecker's "youth dream" was (to prove) that these (maybe with suitable roots of unity) are all abelian extensions.
For real quadratic fields, some of Hecke's early work gave similar (partial) results for values of Hilbert-Blumenthal modular forms.
Shimura-Taniyama developed analogous ideas for abelian varieties, in part, describing possible endomorphism algebras analogous to the simpler situation of elliptic curves. Throughout the 1960s and into the 1970s, Shimura looked at the automorphic-form version of the problem of generating classfields.
Going in a somewhat different direction, values of automorphic forms at points can be construed as a special case of "periods" of automorphic forms, along subgroups or cosets thereof. A general pattern one seeks is that for $f$ an automorphic form on $G$, and for $H\subset G$, and $\phi$ an automorphic form on $H$, the integral of the restriction of $f$ against $\phi$ can be an Euler product multiplied by another (smaller) period, and that (up to various normalizations), modulo the periods, the outcome is algebraic, with explicit Galois properties. Shimura's work in the early 1970's on rationality properties of the Rankin-Selberg L-functions is another prototype, quite different from point-wise evaluation. Many people investigated ranges of possibilities for this: Piatetski-Shapiro, Rallis, Shimura, Michael Harris, myself, and ever-more subsequently.
The p-adic interpolation of various of these algebraic numbers has a long history, but/and seems to have "taken off" in the last 10-20 years, as a refinement of a mere assertion of algebraicity, much as with Iwasawa-Leopoldt years ago.
