Federico's webpage

A theory of ``special functions" and ``periods" emerged in the framework of function fields of positive characteristic after the early works of Carlitz in the 1930s and later, in the hands of Anderson, Goss, Hayes, Thakur and others. In this theory, the role of the ring of relative numbers, prominent in the classical arithmetic theory, is played by the ring A=k[T] with k a finite field; in other words, the ring of polynomials in an indeterminate T with coefficients in a finite field k. Instead of the real line, one then looks at the ``Carlitz line", that is, the local field completion of the fraction field of A with respect to the infinite place. One of the most important features available here is the possibility to see all our modules as k-vector spaces. My interest in these topics began around 2005. The first transcendental special function ever considered in this framework is the Carlitz exponential function. Later, the theory was developed along lines parallel to the classical theory of ``special functions" and ``periods," including, for instance, zeta- and L-functions, modular forms, Galois representations, etc. In most developments, the analogy real line/Carlitz line is assumed to be a basic point of view. All I did since then, is to try to overcome this analogy which I find superficial, looking for more significative and deeper structures.