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Are there explicit relations between hypergeometric functions over finite fields of different size? In particular, if we consider the function 2F1(A,B,C|x) over Fp (i.e., with characters A,B and C defined over Fp), and the same hypergeometric function but over Fp^2 (i.e., with characters defined over Fp^2) or any other field extension of Fp, can we relate them in any explicit way?

Thank you very much!

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Assuming by the hypergeometric function, you mean the trace of Frobenius the hypergeometric sheaves, there is a nice relation.

First I will mention the right way to move a character to a higher degree finite field. If $A$ is a character of $\mathbb F_q^\times$, then the corresponding character of $\mathbb F_{q^n}^\times$ is $A$ composed with the norm from $\mathbb F_{q^n}$ to $\mathbb F_q$. This is correct because both characters correspond to the same sheaf.

Thus a hypergeometric function $2F1(A,B,C|x)$ over $\mathbb F_{q^2}$ corresponds to a hypergeometric function $2F1(A',B',C'|x)$ over $\mathbb F_{q^2}$ because they come from the same sheaf, if $A$ corresponds to $A'$, $B$ corresponds to $B'$, and $C$ corresponds to $C'$.

But what does this say about the hypergeometric function? Well the function is just the trace of Frobenius. Since $Frob_{q^n}=\left(Frob_q\right) ^n$, we can use an identity about the traces of powers of matrices. Since the sheaf has rank $2$, the matrices are two-dimensional, and we have the identities:

\[ \operatorname{tr}(M^2) = \operatorname{tr}(M)^2 - 2 \det(M)\]

\[ \operatorname{tr}(M^3) = \operatorname{tr}(M)^3 - 3 \det(M) \operatorname{tr}(M)\]

and so on, using the Chebychev polynomials. Here $ \operatorname{tr}(M)$ corresponds to the hypergeometric function of a point over $\mathbb F_q$, $ \operatorname{tr}(M^2)$ to the corresponding function of the same point over $\mathbb F_{q^2}$, and $ \operatorname{tr}(M^3)$ to the corresponding function of the same point over $\mathbb F_{q^3}$. It remains to give a nice formula for the determinant.

I'm pretty sure there is one, although I don't know exactly what it is. It should be something like $p^3 A\left(\frac{x}{x-1}\right)B\left(\frac{x}{x-1}\right)C(x-1)$. I'm not sure precisely what notation you are using. Without an exact formula, you still get a relation among $\mathbb F_p$, $\mathbb F_{p^2}$, and $\mathbb F_{p^3}$, or between any three finite fields, using the symmetric functions.

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Will, I did some editing (and I think some more is needed). Please have a look at what I've done to see that I haven't introduced any mistakes. –  Gerry Myerson Jun 14 '13 at 3:20
    
Thank you very much for your response! I really appreciate you taking the time to answer my question. Now, let me ask you something else. What exactly do you mean when you say that the function is the trace of Frobenius? I know there are explicit relations between traces of Frobenius endomorphisms of certain families of elliptic curves and special values of 2F1-hypergeometric functions over Fq, but I am not sure if that can be done for any 2F1. Or maybe this is not what you were referring to. Thanks. –  Val Jul 2 '13 at 19:29
    
I mean that the way the hypergeometric function has trace corresponding to some l-adic sheaf.This sheaf may or may not be the sheaf of Tate modules of some family of elliptic curves. Part of the power of sheaf theory is that it allows us to describe things which cannot be expressed so naturally geometrically! Essentially, the formula for the hypergeometric function as a sum immediately suggests a construction of a sheaf via cohomology. The construction of generalized hypergeometric sheaves is described, for instance, in "G2 and hypergeometric sheaves" by Nick Katz –  Will Sawin Jul 2 '13 at 19:52
    
but I don't necessarily expect this will be tremendously helpful! –  Will Sawin Jul 2 '13 at 19:55
    
Thanks for the reference. I will check it. –  Val Jul 3 '13 at 15:08

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