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3 Transcendental computation of the Class number

We first need to introduce the Dedekind zeta function for a number field K, and its Euler product expansion

$\displaystyle \zeta_{K}^{}$(s) = $\displaystyle \sum_{I}^{}$$\displaystyle {\frac{1}{N(I)^s}}$ = $\displaystyle \prod_{Q}^{}$$\displaystyle {\frac{1}{(1-{\frac{1}{N(Q)^s}})}}$

where the sum runs over all ideals I of R and the product runs over all prime ideals Q of R. The two expressions give us two ways of computing $ \lim_{s\to 1}^{}$(s - 1)$ \zeta_{K}^{}$(s). The left-hand side is expressed in terms of ``arithmetic'' invariants and the right-hand side in terms of invariants for the Galois group. The resulting identity gives a way for computing the Class number h of K. The left-hand limit can be computed to be

$\displaystyle \lim_{s\to 1}^{}$(s - 1)$\displaystyle \sum_{I}^{}$$\displaystyle {\frac{1}{N(I)^s}}$ = $\displaystyle \lim_{r\to\infty}^{}$$\displaystyle {\frac{\char93 \{I\mid N(I)\leq r\}}{r}}$

The set {I | N(I) $ \leq$ r} can be split according to ideal classes. We try to compute for each ideal class C,

z(C) = $\displaystyle \lim_{r\to\infty}^{}$$\displaystyle {\frac{\char93 \{I\in C\mid N(I)\leq r\}}{r}}$.

Fixing an ideal I0 $ \in$ C, this latter set is bijective to the set {aR $ \subset$ I0-1 | N(a) $ \leq$ r . N(I0)-1}. (Here N(a) denotes the modulus of the norm of a.) We have a natural embedding K $ \hookrightarrow$ K $ \otimes_{{\Bbb Q}}^{}$ $ \Bbb$R. The image of J = I0-1 is a lattice in K $ \otimes_{{\Bbb Q}}^{}$ $ \Bbb$R. Let $ \Lambda$ denote the image of J - {0} in the quotient $ \cal {S}$ = (K $ \otimes_{{\Bbb Q}}^{}$ $ \Bbb$R)*/U where U is the image of the group of units in R under the above embedding. There is a natural homomorphism N : $ \cal {S}$$ \to$$ \Bbb$R* which restricts to the modulus of the norm on the image of K. We obtain a natural bijection between {aR $ \subset$ I0-1 | N(a) $ \leq$ r} and {l $ \in$ $ \Lambda$ | N(l ) $ \leq$ r}. Let $ \Lambda_{r}^{}$ denote the image of (1/r)J - {0} in $ \cal {S}$, then we have a natural bijection between {l $ \in$ $ \Lambda$ | N(l ) $ \leq$ rd} and {l $ \in$ $ \Lambda_{r}^{}$ | N(l ) $ \leq$ 1}, where d denotes the degree of K oover $ \Bbb$Q. Let $ \cal {S}$$\scriptstyle \leq$ 1 denote locus of l $ \in$ $ \cal {S}$ such that N(l ) $ \leq$ 1. Let $ \mu$ denote the Haar measure on K $ \otimes_{{\Bbb Q}}^{}$ $ \Bbb$R. This is invariant under the action of U and thus gives a measure also denoted by $ \mu$ on $ \cal {S}$. Since J is a lattice in K $ \otimes_{{\Bbb Q}}^{}$ $ \Bbb$R we have

$\displaystyle \lim_{r\to\infty}^{}$$\displaystyle {\frac{\char93 \{l\in\Lambda_r\mid N(l)\leq 1\} }{r^d}}$ = $\displaystyle {\frac{\mu({\cal S}_{\leq 1})}{\mu(K\otimes _{{\Bbb Q}}{\Bbb R}/J)}}$

Moreover, the denominator can be re-written

$\displaystyle \mu$(K $\displaystyle \otimes_{{\Bbb Q}}^{}$ $\displaystyle \Bbb$R/J) = N(J)$\displaystyle \mu$(K $\displaystyle \otimes_{{\Bbb Q}}^{}$ $\displaystyle \Bbb$R/R).

In particular, we see that the limit z(C) is independent of the class C. Let (K $ \otimes_{{\Bbb Q}}^{}$ $ \Bbb$R)*1 denote the kernel of the norm map. This is a group and thus we have a Haar measure $ \nu$ on it. One shows that

$\displaystyle \mu$($\displaystyle \cal {S}$$\scriptstyle \leq$ 1) = $\displaystyle \nu$((K $\displaystyle \otimes_{{\Bbb Q}}^{}$ $\displaystyle \Bbb$R)*1/U)

Combining the above calculations one obtains

$\displaystyle \lim_{s\to 1}^{}$(s - 1) . $\displaystyle \zeta_{K}^{}$(s) = h . $\displaystyle {\frac{\nu((K\otimes _{{\Bbb Q}}{{\Bbb R}})^{*}_1 / U)}{\mu(K\otimes _{{\Bbb Q}}{\Bbb R}/R)}}$

This often called the ``Class number formula'' for K. Note that the denominator can be computed in closed form in terms of the discriminant D of the field K and the number of pairs of conjugate complex embeddings r2 of K.

$\displaystyle \mu$(K $\displaystyle \otimes_{{\Bbb Q}}^{}$ $\displaystyle \Bbb$R/R) = $\displaystyle {\frac{1}{2^{r_2}}}$ . $\displaystyle \sqrt{\vert D\vert}$

However, the numerator is in general more complicated since it involves computing the group of units of K. To expand the right-hand term we restrict our attention to abelian extensions K of $ \Bbb$Q. The product term on the left can be first grouped according to rational primes

$\displaystyle \prod_{Q}^{}$$\displaystyle {\frac{1}{(1-{\frac{1}{N(Q)^s}})}}$ = $\displaystyle \prod_{q}^{}$$\displaystyle \prod_{Q\mid q}^{}$$\displaystyle {\frac{1}{(1-{\frac{1}{N(Q)^s}})}}$

Now for each rational prime q which is unramified in K we have

$\displaystyle \prod_{Q\mid q}^{}$$\displaystyle {\frac{1}{(1-{\frac{1}{N(Q)^s}})}}$ = $\displaystyle \prod_{\chi}^{}$$\displaystyle {\frac{1}{(1-{\frac{\chi(q)}{q^s}})}}$

where $ \chi$ runs over all characters of the Galois group and $ \chi$(q) = $ \chi$(Frobq) is the value of $ \chi$ on a Frobenius element associated with q. We define the Dirichlet L-series and their Euler product formulas as follows

L(s,$\displaystyle \chi$) = $\displaystyle \sum_{n}^{}$$\displaystyle {\frac{\chi(n)}{n^s}}$ = $\displaystyle \prod_{p}^{}$$\displaystyle {\frac{1}{(1-{\frac{\chi(p)}{p^s}})}}$

where we set $ \chi$(p) = 0 when $ \chi$ is ramified at p. We also define the additional factor

F(s) = $\displaystyle \prod_{p \text{ ramified}}^{}$$\displaystyle {\frac{1}{(1-{\frac{1}{p^{f_p}}})^{g_p}}}$

where the product runs over all ramified primes and fp denotes the residue field extension over p and gp the number of distinct primes in K lying over p. The product expansion of $ \zeta_{K}^{}$(s) becomes

$\displaystyle \zeta_{K}^{}$(s) = F(s) . $\displaystyle \prod_{\chi}^{}$L(s,$\displaystyle \chi$).

Thus the computation of the limit can be reduced to the corresponding computation for the Dirichlet L-series. For the case of the unit character we get by comparison with the zeta function

$\displaystyle \lim_{s\to 1}^{}$(s - 1)F(s)L(s, 1) = 1.

So the right-hand limit gives

$\displaystyle \lim_{s\to 1}^{}$(s - 1)$\displaystyle \zeta_{K}^{}$(s) = $\displaystyle \prod_{\chi\neq 1}^{}$L(1,$\displaystyle \chi$).

There is a positive integer m such that $ \chi$ is determined on classes modulo m and $ \chi$ is zero on all primes p dividing it; m is called the conductor of $ \chi$. We rewrite the L-function associated with $ \chi$ as follows

L(s,$\displaystyle \chi$) = $\displaystyle \sum_{x\in ({\Bbb Z}/{m{\Bbb Z}})^{*}}^{}$$\displaystyle \left(\vphantom{
\chi(x) \cdot \sum_{n\cong x \pmod m} {\frac{1}{n^s}}
}\right.$$\displaystyle \chi$(x) . $\displaystyle \sum_{n\cong x \pmod m}^{}$$\displaystyle {\frac{1}{n^s}}$$\displaystyle \left.\vphantom{
\chi(x) \cdot \sum_{n\cong x \pmod m} {\frac{1}{n^s}}

The latter sum can be rewritten using the identity

$\displaystyle \sum_{i=0}^{m-1}$$\displaystyle \omega^{xi}_{}$ = \begin{displaymath}\begin{cases}
0, \text{ if } x \not\cong 0 {\pmod m} \\
m, \text{ if } x \cong 0 {\pmod m}

where $ \omega$ is a primitive m-th root of unity. The second sum then becomes

$\displaystyle \sum_{n\cong x \pmod m}^{}$$\displaystyle {\frac{1}{n^s}}$ = $\displaystyle {\frac{1}{m}}$$\displaystyle \sum_{n=1}^{\infty}$$\displaystyle {\frac{1}{n^s}}$$\displaystyle \sum_{i=0}^{m-1}$$\displaystyle \omega^{(x-n)i}_{}$

Thus we obtain

L(s,$\displaystyle \chi$) = $\displaystyle {\frac{1}{m}}$$\displaystyle \sum_{i=0}^{m-1}$$\displaystyle \left(\vphantom{
\sum_{x\in ({\Bbb Z}/{m{\Bbb Z}})^{*}} \chi(x) \omega^{ix}
}\right.$$\displaystyle \sum_{x\in ({\Bbb Z}/{m{\Bbb Z}})^{*}}^{}$$\displaystyle \chi$(x)$\displaystyle \omega^{ix}_{}$$\displaystyle \left.\vphantom{
\sum_{x\in ({\Bbb Z}/{m{\Bbb Z}})^{*}} \chi(x) \omega^{ix}
}\right)$ . $\displaystyle \sum_{n=1}^{\infty}$$\displaystyle {\frac{\omega^{-in}}{n^s}}$

The expression

$\displaystyle \tau_{i}^{}$($\displaystyle \chi$) = $\displaystyle \sum_{x\in ({\Bbb Z}/{m{\Bbb Z}})^{*}}^{}$$\displaystyle \chi$(x)$\displaystyle \omega^{ix}_{}$

is called the Gaussian sum associated with the integer i and the character $ \chi$. If $ \chi$ is not the unit character then $ \tau_{0}^{}$($ \chi$) = 0. Moreover, if i $ \neq$ 0 then we have the identity

$\displaystyle \sum_{n=1}^{\infty}$$\displaystyle {\frac{\omega^{-in}}{n}}$ = - log(1 - $\displaystyle \omega^{-i}_{}$)

Hence, we obtain the formula when $ \chi$ is not the unit character

L(1,$\displaystyle \chi$) = - $\displaystyle {\frac{1}{m}}$$\displaystyle \sum_{i=1}^{m-1}$$\displaystyle \tau_{i}^{}$($\displaystyle \chi$) . log(1 - $\displaystyle \omega^{-i}_{}$)

next up previous
Next: 4 Divisiblity of the Up: Kummer's proof of Fermat's Previous: Case 2: p| XYZ
Kapil Hari Paranjape 2002-11-22