So today, I’m going to write about something I’ve learnt that I think is really cool. Unsurprisingly, it’s related to the Class Group; I will get back to my set of posts on Baker’s Theorem and the Class Number Problem, think of this as a nice excursion. I might go quite fast over the basics of the Class Group, I refer to previous posts (and Wikipedia) to get the necessary background. Throughout, is a number field with ring of integers . I refer heavily to Neukirch’s “Algebraic Number Theory” in this post, a current favourite text of mine.
There are two major things that I think should be in a first course on Algebraic Number Theory; the Class Group (specifically, the finiteness of the Class Group), and Dirichlet’s Unit Theorem, both of which I state below for reference.
Theorem 1 Let be an algebraic number field. The Class Group of is finite.
Theorem 2: Dirichlet’s Unit Theorem Let be an algebraic number field with ring of integers . Let be the number of pairs of complex embeddings of , and the number of real embeddings. The group of units is the direct product of the finite cyclic group (the group of roots of unity contained in ) and a free abelian group of rank .
My aim for this post is to connect these two seemingly unrelated results, ending with a statement equivalent to both these theorems. I assume familiarity with a classic proof of both, especially of Dirichlet’s Unit Theorem. To get there however, we’ll have to generalise our ideas of fractional ideals.
As a reminder, a fractional ideal is a finitely generated -submodule of . I’ve discussed these in previous posts, so won’t spend too much time on them here. When we consider the set of places on (that is, the absolute values on up to equivalence), in a way these fractional ideals only take into account the finite places (after finishing my set of posts on Baker’s Theorem, I’ll write about places in some depth). Finite places correspond to prime ideals, hence we can see a link. We can extend this notion of fractional ideals then, by taking into consideration the infinite places/primes.
Define a replete ideal of to be an element of the group , where is the group of fractional ideals of and denotes the multiplicative group of positive real integers.
I will unify this idea with the standard notation and put, for any infinite prime and any real number , . That is, I follow the convention where we treat as an infinite prime number.
Further, given a set of real numbers with (that is, is an infinite prime), I choose to let denote the vector rather than the product of the quantities. This allows every replete ideal to admit a unique product representation . We note that when is a finite prime, and is a real number when infinite.
Important takeaways from here so we don’t get bogged down in theory; all fractional ideals are replete ideals and we can “factorise” replete ideals into a product over finite places times a product over infinite places. To we associate the replete principal ideal , where is the principal fractional ideal.
From here, think back to the definition of the class group, we can see what we’re going to do. The replete principal ideals form a normal subgroup of the replete ideals , so we define the replete ideal class group (or replete Picard group) to be .
These replete ideals behave essentially as we would want them to, we can define the absolute norm of a replete ideal which is multiplicative, we have some nice results if we consider extensions… I’m going to briefly discuss norms as we’ll shortly need them. Let be the residue class degree of . We set if is finite (where lies above ), and if is infinite. This norm is multiplicative, so we can find the norm of any replete ideal . The extension stuff is less relevant here, so I’ll leave it for another day.
We’re now going to consider divisors. I’ll define these straight away for replete divisors, but the analogous definition holds for fractional ideals as one would expect.
A replete (or Arakelov) divisor of is a formal sum , where is an integer for the finite primes, a real number for the infinite primes and 0 for almost all . These divisors form a group, which we’ll denote by . I’ll leave it as a comment that this is a locally compact topological group, if you want details comment.
We not consider the map , given by . We call elements of this form replete principal divisors.
We’re now in a position to start to see how this relates to Dirichlet’s Unit Theorem. Consider the map , given by , where for infinite primes, , with being the closure of with respect to .
This is where I’m going to need familiarity with the “normal “, “standard” proof of Dirichlet’s Unit Theorem. The composite of the mapping above with the mapping is equal, up to sign, to the logarithm map of Minkowski theory, . This maps the unit group onto a complete lattice in the trace-zero space .
From here, we could prove that the kernel of is the group of roots of unity of , and that the image is a discrete subgroup of , but I’ll leave that for another day (it’s all about the exact sequence , where the second to last arrow is ).
We now define to be the replete divisor (or Arakelov) class group of . We note we can do the exact same thing with fractional ideals, coming up with the divisor (or Chow) class group , where we use the exact same construction, but with fractional rather than replete ideals. We’re almost at the point we wanted to get to, hang in there!
As is discrete in , it is closed, so is a locally compact Hausdorff topological group with respect to the quotient topology. We consider a degree map from onto the reals, induced by the continuous homomorphism . We define the by sending the replete divisor to the real number . By the product formula for places, we find that for any replete principal divisor . This shows that is a well-defined continuous homomorphism.
We are now at the point where we can finish and connect all the things we’ve been talking about. The kernel of the degree map is made up from the unit group and the ideal class group , as can be seen from the following proposition.
Proposition Let denote the complete lattice of units in trace-zero space . There is an exact sequence
The proof of this isn’t too hard, and isn’t too instructive, so I omit it. From this point, the finiteness of the class number and Dirichlet’s Unit Theorem merge into (and are equivalent to) the fact that is compact.
Theorem The group is compact.
To see this we consider the exact sequence from the previous proposition. As is a complete lattice in , is a compact torus, of dimension , with defined as above (this is where we get Dirichlet’s Unit Theorem from!). We obtain as the union of finitely many (indeed, many) compact cosets of , giving us that is itself compact, and that is finite.