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The context is the definition of Hecke Größencharakter:


This is supposed to generalize the Dirichlet $L$-series for number fields. Dirichlet characters are characters of the multiplicative groups of $\mathbb Z/p\mathbb Z$. An appropriate generalization would be instead to consider characters of the multiplicative group of $\mathcal O_K/\cal P$ where $\mathcal P$ is a prime ideal in the ring of integers of a number field $K$.

But Hecke Größencharakter goes to more trouble than this. It brings in ideles and such for a more complicated generalization. Why is this necessary?

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3 Answers 3

It's natural to think that the correct analogue of the groups $({\mathbf Z}/m{\mathbf Z})^\times$ in a number field $K$ should be the groups $({\mathcal O}_K/{\mathfrak a})^\times$, and for some purposes that is true. But for other purposes it is not. Hecke's motivation for creating "his" characters was to produce $L$-functions of them as Euler products over prime ideals in $K$ that generalize Dirichlet $L$-functions. If you start off with a character $\chi$ on a group $({\mathcal O}_K/{\mathfrak a})^\times$, how do you make it into a function of ideals? You want some kind of series like $$ \sum_{\mathfrak a} \frac{\chi({\mathfrak a})}{{\rm N}(\mathfrak a)^s} $$ running over (nonzero) integral ideals ${\mathfrak a}$ of ${\mathcal O}_K$, and if $K$ has class number greater than 1 it's hard to imagine how to get a function of ideals out of a function on $({\mathcal O}_K/{\mathfrak a})^\times$. Even if $K$ has class number 1 you'd have pretty serious problems making such a transition if there are units of infinite order, which there are except for ${\mathbf Q}$ and imaginary quadratic fields.

The key to understanding how Hecke generalized Dirichlet characters is to reinterpret the group $({\mathbf Z}/m{\mathbf Z})^\times$ as the multiplicative quotient group of fractional ideals $I_{(m)\infty}/P_{(m)\infty}$, not as a group of unit cosets of a ring. That leads to generalized ideal class groups $I_{\mathfrak m}/P_{\mathfrak m}$ for a generalized modulus $\mathfrak m$ in any number field, and it is characters of generalized ideal class groups that fit in more smoothly with Hecke's definition of his characters. The characters of generalized ideal class groups all have finite order, but Hecke's definition allowed not just these but also infinite-order characters that are not closely related to any finite order characters. Generalized ideal class groups are the original way in which class field theory was developed, and you're not going to find anyone telling you that the formalism of class field theory is easy to grasp the first time through it.

Hecke's original definition of his characters did not make any use of ideles, which in fact weren't created until later (by Chevalley). His paper introducing his characters came in two parts in Mathematische Zeitschrift (vol. 1 in 1918 and vol. 6 in 1920) and he gives explicit examples of his characters for real and imaginary quadratic fields. The classical definition is discussed on the Wikipedia page you link to in your question, although the definition there is (at the moment) all largely in words and is kind of opaque. I think you would find the classical formulas defining Hecke characters in general pretty frightening. You can find them in, for instance, Narkiewicz's book on algebraic number theory. Hecke's original papers do not offer much in the way of gentle motivation for his definition. These developments, at the time, were not at all obvious. In the 1940s, Matchett showed in her thesis how to interpret Hecke's characters more conceptually as the characters of the idele class group, and that is often how they are viewed today because it is a cleaner and more conceptual definition.

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One reason for the greater generality is that Hecke characters do more than describe Abelian extensions of number fields (essentially Dirichlet characters describe Abelian extenions of $\mathbb Q$). For instance the L-function of an elliptic curve with complex multiplication is the L-function of a Hecke character of infinite order.

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Apart from specific applications, for all number fields larger than $Q$, Hecke L-functions are simply "there", resulting from the full harmonic analysis on the idele class group. Most of these do assume non-algebraic values, so are not "motivic" (not "type $A_0$" in Weil's sense). But for the harmonic analysis of characters, and then for harmonic analysis of automorphic forms generally, Hecke characters are inescapable. If they are omitted, Plancherel theorems fail, pointwise representation theorems fail, etc.

An even simpler application than to elliptic curves with CM is to distribution of Gaussian primes in angular sectors. All the Hecke L-functions $\sum (\frac{\alpha}{|\alpha|})^k \frac{1}{|\alpha|}^{2s}$ are necessary/natural to treat this question. In effect, the index $k$ is a Fourier series index in the circle coordinate, doing a separation of variables writing $C^\times$ as circle $\times(0,+\infty)$.

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