Transcendence of the Carlitz-Goss gamma function at rational arguments
Let \({\mathbf A} = \mathbb{F}_q [T]\) where \(q= p^{m_0}\) is a prime power and set \({\mathbf k} = \mathbb{F}_q (T)\) and \({\mathbf K} = \mathbb{F}_q ((1/T))\). One checks easily that \({\mathbf A}\) is contained in \({\mathbf K}\) discretely (the topology on \({\mathbf K}\) is given by measuring the order of zero of a function at \(\infty)\), and that \({\mathbf K} / {\mathbf A}\) is compact. Of course this is similar to \(\mathbb{Z} \subset \mathbb{R}\). It is by now well known that one can construct a theory of ``objects with \({\mathbf A}\)-multiplication'' in analogy with the theory of ``objects with \(\mathbb{Z}\)-multiplication'' (i.e., \(\mathbb{G}_m\), elliptic curves, abelian varieties, etc.); these objects with \({\mathbf A}\)-multiplication are named Drinfeld modules after V. G. Drinfeld who initiated their general theory in 1973 (we note that more general objects with \({\mathbf A}\)-multiplication, called \({\mathbf A}\)-modules, were constructed by G. Anderson in the 1980's). However, back in the 1930's L. Carlitz had actually worked out the details of a particularly simple Drinfeld module called the Carlitz module. The Carlitz module, like all Drinfeld modules defined over extensions of \({\mathbf k}\), has both an algebraic and analytic side to it; in particular there is an entire analytic function called the Carlitz exponential which uniformizes the Carlitz module just as \(e^z\) uniformizes \(\mathbb{G}_m\) classically. The Carlitz exponential, \(e_C(z)\), may be expressed as \(e_C(z) = \sum^\infty_{i=0} (z^{q^i}/D_i)\), where \(D_0 = 1\) and for \(i>0\), \(D_i = (T^{q^i} -T) (T^{q^i} -T^q) \cdots (T^{q^i} - T^{q^{i -1}})\). With a little work, one can also see that \(D_i\) is the product of all monic polynomials of degree \(i\); thus, the polynomials \(D_i\) look very much like they are a ``factorial.'' However, as Carlitz realized, the elements \(D_i\) alone cannot be a good factorial for \({\mathbf A}\) as \(D_i\) is the coefficient of \(z^{q^i}\) in \(e_C(z)\) and not \(z^i\). Carlitz then fixed this by using the \(D_i\) as basic building blocks for a good factorial: let \(j\) be any nonnegative integer which we write \(q\)-adically as \(j= \sum^\nu_{t = 0} c_tq^t\), where \(0\leq c_t < q\) for all \(t\). We then set \(\Pi(j): = \prod^\nu_{t=0} D_t^{c_t}\); one can now show e.g. that \(\Pi (j)\) has the correct factorization at the finite primes, etc. One can also interpolate the \(\Pi (j)\) into continuous gamma functions in the following simple fashion: set \(\langle D_t \rangle: = T^{- \deg D_t} D_t\) -- upon writing this out in terms of \(1/T\) it is obvious that \(D_t \to 1\) in \({\mathbf K}\) as \(t\to \infty\). Thus if \(y= \sum^\infty_{t=0} c_tq^t\), \(0\leq c_t <q\) all \(t\), is any \(p\)-adic integer we can set \[ \Pi_\infty (y): = \prod^\infty_{t=0} \langle D_t \rangle^{c_t}, \] (the author calls this ``\(y!\)'') and this is now a continuous function from \(\mathbb{Z}_p\) to \({\mathbf K}\). Many important results about such functions have been established by Thakur starting with his Harvard thesis in the 1980's. In the paper, ``Transcendence of gamma values for \(\mathbb{F}_q [T]\)'' [Ann. Math. 144, 181-188 (1996; Zbl 0861.11043)]. \textit{D. Thakur} began a very original and powerful study of the transcendency properties of these functions using automata theory. He established that many values of this function at proper fractions inside \(\mathbb{Z}_p\) are transcendental over \(\mathbb{F}_q(T)\). More precisely, Thakur can handle all denominators (with certain restrictions on the numerators) where previous results using classical techniques were stuck at analogs of denominators 4 and 6. In the paper being reviewed, Thakur's result is extended by the author to show \(\Pi_\infty (y)\) is transcendental at all \(y\in (\mathbb{Q} \cap \mathbb{Z}_p) \backslash \mathbb{N}\) (where \(0\in \mathbb{N})\). Finally, in ``Transcendence and the Carlitz-Goss Gamma function'' [J. Number Theory (to appear)], \textit{M. Mendès France} and \textit{J. Yao} establish the remarkable result that \(\Pi(y)\) is transcendental over \(\mathbb{F}_q(T)\) for all \(y\in \mathbb{Z}_p \backslash \mathbb{N}\). Thakur has also pointed out that the author's results establish the transcendence of all those monomials in Gamma values at fractions which general theory implies should be transcendental. These monomials come from analogs in characteristic \(p\) of Deligne's classical results on Gamma values.
scientific article; zbMATH DE number 955262