Hypergeometric functions for function fields
Let \(\{a,b,c\}\) be complex constants. Then the famous Gauss hypergeometric equation is given by \[ z(1 - z) {d^2y \over dz^2} + \bigl( c - (a + b + 1) z \bigr) {dy \over dz} - aby = 0. \] One defines the Pochhammer symbol \((a)_n\) by \((a)_0 : = 1\) and for \(n > 1\), \((a)_n : = a(a + 1) (a + 2) (a + 3) \cdots (a + n - 1)\). A series solution to the Gauss differential equation is given by \(F(a,b;c;z) : = \sum^\infty_{n = 0} {(a)_n (b)_n \over n! (c)_n} z^n\). (So one has the identity \((1 - z)^s = F(- s,b;b;z).)\) The hypergometric functions \(F(a,b;c;z)\) have many applications to mathematics and physics. As an example, there is the beautiful application by Dwork of the above equation to \(p\)-adic unit roots of characteristic polynomials of Frobenius [see \textit{N. M. Katz}, Sémin. Bourbaki 1971/72, Exp. No. 409, 167-200 (1973; Zbl 0259.14007)]. Now let \({\mathbf A} = \mathbb{F}_q [T]\), \(q = p^{m_0}\), \({\mathbf k} = \mathbb{F}_q (T)\) and \({\mathbf K} = \mathbb{F}_q ((1/T))\). Let \(D_0 : = 1\) and for \(i > 0\) \(D_i : = (T^{q^i} - T) D^q_{i - 1}\). These \(\{D_i \}\) can be fit into the Carlitz exponential \(e_C (x) = \sum^\infty_{i = 0} z^{q^i}/D_i\). This power series is entire and, since we are in characteristic \(p\), we find that \(e_C (x)\) is also \(\mathbb{F}_q\)-linear --- the zeros \(\Lambda\) of \(e_C (x)\) are thus an \(\mathbb{F}_q\)-vector space; one can see that there is an element \(\xi \neq 0\) with \(\Lambda = {\mathbf A} \xi\). Moreover, \(e_C (x)\) satisfies the well-known Carlitz functional equation \(e_C (Tx) = Te_C (x) + e^q_C (x)\). Thus \(e_C (T^2x) = T^2 e_C (x) + (T^q + T) e^q_C (x) + e_C^{q^2} (x)\) and so on. Just as the classical functional equation \(e^{nz} = (e^z)^n\) leads to the usual action of \(\mathbb{Z}\) on \(\mathbb{C}^*\), the Carlitz functional equation leads to a new action of \({\mathbf A}\) on the additive group called the Carlitz module. In classical theory, the cyclotomic extensions of \(\mathbb{Q}\) are given by the division values \(\{e^{{\alpha \over \beta} 2 \pi i}\}\), \((\{\alpha, \beta\} \subset \mathbb{Z},\;\beta \neq 0)\) of \(e^z\). In a similar way the division values \(\{e_C ({\alpha \over \beta} \xi)\}\), \((\{\alpha, \beta \} \subset {\mathbf A},\;\beta \neq 0)\) generate good abelian extensions of \({\mathbf k}\). But for \({\mathbf k}\) the situation is actually more complicated, and interesting, because one also can generate abelian extensions simply by adjoining nontrivial roots of unity; of course this is the same as generating constant field extensions of \({\mathbf k}\). The constant field extensions are everywhere unramified; however, the division values of the Carlitz module may generate ramified extensions. In this fashion one can prove that the division fields of the Carlitz exponential generate geometric extensions (i.e., these extensions contain no nontrivial constant field extensions). Thus notions related to the division values of the Carlitz exponential are labeled ``geometric'' whereas notions related to constant field extensions are labeled ``arithmetic''. The Carlitz exponential also leads to the construction of \(\Gamma\)-functions. As there are two types of cyclotomic extensions, there are also two types of \(\Gamma\)-functions: an arithmetic one (with a characteristic 0 domain) related to arithmetic extensions and a geometric one (with a characteristic \(p\) domain) and related to geometric cyclotomic extensions. Into this theory the author introduces arithmetic and geometric hypergeometric functions via analogs of Pochhammer symbols. Analogs of many classical phenomena are given (such as a version of the Gauss differential equations, Bessel functions, Jacobi polynomials, etc.). Moreover a very interesting application is given to the exponential function of the \(n\)-th tensor power of the Carlitz module. Still one wonders what other applications exist for all the marvelous and surprising structure described in this paper.
scientific article; zbMATH DE number 759428