Serge Lang

Computation of a Unit Index

This chapter is due to Kersey. We determine the index of the group of Klein units in the group of all units in case the conductor is the power of a prime ideal p n . For the composite case, we refer to [Ke 2]. The treatment differs from, say, Robert’s [Ro 1] in several ways. First we deal with the somewhat larger group of Klein units rather than Robert units; and second, we introduce intermediate groups other than those of Robert, more in line with the Sinnott computation of the index in the composite case for cyclo-tomic fields.

Modular Units on Tate Curves

The generic units (which are algebraic functions of j) can be specialized whenever j is specialized, say into a number field. Three cases arise: when j is not integral, when j is integral without complex multiplication, and when j is integral with complex multiplication. The first case will be discussed in this chapter. The third case is in some sense the oldest and will be discussed later. The middle case is the one about which the least is known. A recent result of Harris [Har] gives an asymptotic estimate for the rank of the specialized units.

Siegel-Robert Units in Arbitrary Class Fields

In this chapter, we shall define units in arbitrary class fields by means of values of modular functions, following again Siegel, Ramachandra, and Robert. We follow the latter especially, with the modifications which arise from having a more systematic theory of the modular forms used to construct units. It has been apparent from the beginning that the 12th power was used in the definition of the Siegel functions in order to catch the delta function and not one of its roots. In the present chapter, we keep this 12th power (and the additional N-th power) which give rise to relatively simple explicit formulas. In the next chapter, we shall develop a more general systematic approach, resulting in a refinement of the group of modular units in class fields of K by using the Klein forms.

More Units in the Modular Function Field

In this chapter we give some brief preliminaries in the modular function field, for application to the complex multiplication case afterward.

The Siegel Units Are Generators

In this chapter we essentially prove that the Siegel units generate all units, and we also get a precise description of all the units of a given level.

Klein Units in Arbitrary Class Fields

In this chapter we define in a natural way a large group of units which may be called the modular units in class fields of K. This construction is done with the Klein forms exclusively, but because of the distribution relations, the group which we obtain also contains the unramified units constructed previously with the delta function.