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Let $K = \mathbb Q(\mu_m)$ and $\zeta_K$ it's Dedekind zeta function. We know from the class number formula that, around $0$: $$\zeta_K(s) \sim s^{r_1+r_2-1}h(K)R(K)/w(K) $$ where $h,R,w$ stand for the size of the class group, the regulator and the size of the subgroup of roots of unity.

On the other hand, we have the decomposition: $$\zeta_K(s) = \prod_\chi L(s,\chi)$$ where the product is over suitable Dirichlet characters and moreover, $L(0,\chi) = -B_{1,\chi}$ - the generalized Bernoulli numbers.

On the other hand, $R(K)$ should not be algebraic and there should be a corresponding transcendental contribution from $L(0,\chi)$ and the explicit formula for $B_{1\chi}$ shows that this comes only from those $\chi$ such that $L(s,\chi) = 0. $

Question 1: In the case that $L(s,\chi) = 0$, is it possible to say what the first non zero Taylor coefficient is?

Question 2: Can we decompose (even if it is only conjecturally) $h(K),R(K),w(K)$ into factors corresponding to each Dirichlet character appearing in the decomposition of $\zeta_K$ . What about the order of vanishing of $L(0,\chi)$?

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    $\begingroup$ All of your questions have comprehensive and beautiful answers: providing them is exactly what the subject of Iwasawa theory was developed to do. You should perhaps read either Washington's Cyclotomic Fields or Coates and Sujatha's Cyclotomic Fields and Zeta Values. $\endgroup$ – David Loeffler Feb 15 '18 at 15:09
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    $\begingroup$ I am a little familiar with Washington's book, could you suggest which chapters or theorems I should look at? $\endgroup$ – Asvin Feb 15 '18 at 15:14
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All of your questions have comprehensive and beautiful answers: providing them is exactly what the subject of Iwasawa theory was developed to do. You should perhaps read either Washington's Cyclotomic Fields or Coates and Sujatha's Cyclotomic Fields and Zeta Values.

The order of vanishing of $L(s, \chi)$ at $s = 0$, and its leading term there, is described quite precisely by the results of Chapter 4 of Washington, particularly Corollary 4.4 and Theorem 4.9. (These are stated in terms of values at s = 1, but the functional equation given just after Theorem 4.5 relates $L(s, \chi) $and $L(1-s, \bar\chi)$.) From these results it follows that:

  • if $\chi(-1) = -1$, or $\chi$ is trivial, then $L(\chi, 0)$ is a non-zero algebraic number.
  • if $\chi(-1) = +1$ and $\chi$ is non-trivial, then $L(\chi, 0) = 0$ and $L'(\chi, 0)$ is a non-zero $\overline{\mathbf{Q}}$-linear combination of logarithms of algebraic numbers (more specifically, of cyclotomic units).

As for decomposing $h(K)$ etc into pieces corresponding to Dirichlet characters, you might want to look at Chapter 10 -- in particular Corollary 10.15 and the remark following it, which explain such a decomposition for the p-part of the class group when $K = \mathbf{Q}(\zeta_p)$.

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