Chapter 19 of ``The arithmetic of polynomials'' (Q1892899)

In 1935 \textit{L. Carlitz} wrote the paper ``On certain functions connected with polynomials in a Galois field'' [Duke Math. J. 1, 137-168 (1935; Zbl 0012.04904)]. This paper discussed a certain ``one-periodic function'' \(\psi(x)\), now called the ``Carlitz exponential'', which is a wonderful \(\mathbb{F}_ q [T]\)-analog of the classical exponential function with many ``exponential-like'' properties. This is the first, and perhaps most important instance of the ``exponential function of a Drinfeld module''. Carlitz's method of construction was very concrete and yielded a number of highly useful formulae. In 1973 \textit{V. G. Drinfeld} wrote his seminal paper on Drinfeld modules and established their existence, as well as many basic properties, with more modern, deformation-theoretic, techniques [Math. USSR, Sb. 23 (1974), 561-592 (1976); translation from Mat. Sb., Nov. Ser. 94(136), 594-627 (1974; Zbl 0321.14014)]. Drinfeld developed \(d\)-periodic exponentials for all \(d\geq 1\). It is natural to wonder if Carlitz had realized that his theory could be generalized to multiply-periodic exponential functions. In fact this is indeed the case and Carlitz presented parts of the multiple-period theory in some unpublished notes entitled ``The arithmetic of polynomials''. This theory was for the polynomial ring \(\mathbb{F}_ q [T]\), but in practice an arbitrary Drinfeld exponential may always be viewed as an exponential for \(\mathbb{F}_ q [T]\) equipped with ``complex multiplication''. The paper being reviewed is chapter 19 of Carlitz's ``Arithmetic'' which concerns such exponentials. As Dinesh Thakur has pointed out, while the results are certainly superseded by Drinfeld's, publication of this chapter is very much warranted on historical grounds.