Introduction to chapter 19 of ``The arithmetic of polynomials''
In 1935 \textit{L. Carlitz} wrote a beautiful paper on the arithmetic of function fields over a finite field. This paper, ``On certain functions connected with polynomials in a Galois field'' [Duke Math. J. 1, 137-168 (1935; Zbl 0012.04904)] contained the first instance of what is called today the ``Carlitz module''. The Carlitz module is the simplest example of a Drinfeld module and in many respects, perhaps the most important such object. Carlitz's method of construction was very concrete and yielded a number of elegant formulae. In 1973 \textit{V. G. Drinfeld} wrote his seminal paper on Drinfeld modules and established their existence and many basic properties using moduli-theoretic techniques [Math. USSR, Sb. 23 (1974), 561-592 (1976); translation from Mat. Sb., Nov. Ser. 94(136), 594-627 (1974; Zbl 0321.14014)]. Now the Carlitz module is associated to a rank one lattice, whereas Drinfeld's theory works for lattices of all ranks. So it is natural to wonder whether Carlitz had realized that his theory could be generalized to higher ranks. It turns out, in fact, that he had indeed known of this possibility and even presented parts of the general theory in some unpublished notes entitled ``The arithmetic of polynomials''. In the paper being reviewed, Carlitz's student David Hayes gives a very enjoyable personal introduction to chapter 19 of these notes where such ``multiply-periodic functions'' are described.
scientific article; zbMATH DE number 759424