A six exponentials theorem in finite characteristic (Q1063639)

It has been known for a number of years that the theory of Drinfeld modules over a global field \(k\) of finite characteristic offers many analogies to classical arithmetic. One example is the ``exponential function'', \(e_ L(z)\), of a lattice \(L\). For instance, if the lattice is of rank one and \(e_ L(z)\) has ``rational'' multiplications, then the division values of \(e_ L(z)\) generate abelian extensions, etc. A natural conjecture is that when \(e_ L(z)\) has rational (or algebraic) multiplications, then \(L\) should be transcendental. For instance, one would expect analogs for \(e_ L(z)\) of all the transcendence results on the classical exponential function and the analytic functions associated to elliptic curves. In the case of Drinfeld modules, pioneering work of L. I. Wade in 1946 established that the period of the ``Carlitz module'' (i.e., the simplest rational Drinfeld module in the case \(k\) is the field of rational functions) is transcendental. In an elegant recent series of papers the author has built on this and has established the results mentioned above. In the paper being reviewed he establishes a ``six-exponentials'' theorem for \(k\) which asserts that under suitable hypotheses on a rank one lattice \(L\), at least one of a certain collection of six values of \(e_ L(z)\) is transcendental. More generally, if \(L\) has rank \(d\), then the result may be generalized to a ``\(4d+2\)'' theorem for \(e_ L(z)\).