The Wiles unit

Abstract

In this chapter I want to introduce a conjecture (Conjecture 6.3.4) whose cryptic form would be: “The Wiles Unit is a determinant” The “Wiles Unit” unit is the p-adic unit-valued function on Galois representations given (approximately) by the ratio of the p-adic L-function to the Iwasawa polynomial. The values of this function are p-adic units by the main result of [166]. In §6.1 I give a functorial treatment of the material of [64], which results in Iwasawa polynomials even in the non-abelian case. In §6.2 we shall recall the definition of the p-adic L-function. In §6.3 we examine what determinantal functions are and how to detect them when the Galois group is an elementary abelian p-group. Then, having set up the background material, we state the conjecture and accompany it with some preliminary evidence in its favour.