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I want to examine nontrivial examples of what we call Iwasawa class formula, $c(n)=\mu p^n + \lambda n + \nu$, where $\lambda, \mu \in \mathbf N$ and $\nu \in \mathbf Z$ are parameters depending only on $K$ (number field), and $c(n)$ is order of $p$-Sylow subgroup of ideal class group of $K$.

Order of $37$-Sylow subgroup of ideal class group of $K_{37} = \Bbb Q(\mu_{37^{n}})$ is known to be $37^n$.

I heard this fact follows from Iwasawa theory. But how? And why $37$ is special (Maybe because it is an irregular prime? Does this become difficult with another (regular) prime?)

Self contained proof or reference (pdf, website...etc) is also welcomed. Thank you in advance.

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  • $\begingroup$ This link mentions that the $37$-group of $C(\mathbb{Q}(\zeta_{37^n}))$ is the cyclic group of order $37^n$ and that the $691$-group of $C(\mathbb{Q}(\zeta_{691^n}))$ is the product of two cyclic groups of order $691^n$. $\endgroup$ Nov 10, 2023 at 22:32

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Much more than Iwasawa's original theorem is known by now. First of all, the $p$-primary part of the class group stays trivial in the field $K_n=\mathbb{Q}(\mu_{p^{n+1}})$ unless it is already non-trivial for $K_0=\mathbb{Q}(\mu_p)$. See Proposition 13.22 in Washington's "Introduction to cyclotomic fields". Therefore the only primes of interest are the irregular primes and the first one is $p=37$. Ribet's results showing the converse to Herbrand's theorem lets you even determine the structure of the class group of $K_0$ with its action by the Galois group of $K_0/\mathbb{Q}$ in terms of Bernoulli numbers. It is known that $\mu=0$. Furthermore, the main conjecture is known, which means that the constant $\lambda$, and in fact most things about the limit class group as a $\Lambda$-module, can be determined from a $p$-adic zeta-function that can be calculated either using Stickelberger elements (so Bernoulli numbers) or modular forms. The constant $\nu$ and the constant $n_0$ such that Iwasawa's theorem is valid for $n\geq n_0$ can often be determined, too.

A lot of these things are in Washington's book. Lang's "Cyclotomic Field" is another standard reference.

I know too little about explicit calculations of these invariants. But I would expect that it becomes harder as the irregularity of the prime increases.

For the special case of $p=37$, I once did the explicit calculation for a lecture series in Heidelberg. The Bernoulli number $B_{32}$ is divisible by $37$. Accordingly, we expect a non-trivial $\omega^5$ part in the $p$-primary part of the class group of $\mathbb{Q}(\mu_{37})$ where $\omega$ is the Teichmüller character. Indeed the approximation to the $37$-adic $L$-function for this character is \begin{multline*} 14\cdot 37 + 33\cdot 37^2 + 13\cdot 37^3 + \mathbf{O}(37^4) + \bigl(16 + 6\cdot 37 + 32\cdot 37^2 + \mathbf{O}(37^3)\bigr)\cdot T \\ + \bigl(29 + 9\cdot 37 + 13\cdot 37^2 + \mathbf{O}(37^3)\bigr)\cdot T^2 + \mathbf{O}(T^3). \end{multline*} This is not a unit as $-B_{1,\omega^{-5}}$ is divisible by $37$. From the fact that the second coefficient is a unit, we conclude that the $p$-adic $L$-function is a unit times a linear factor. Hence the limit class group $X$ is a free $\mathbb{Z}_{37}$-module of rank $1$ and hence the $p$-primary part of the class group of $K_n$ is of order $p^{n+1}$ for all $n\geq 0$, i.e., $c(n)=n$ in your notations. The fact which underlies the proof of Ribet's theorem is that the Eisenstein series \begin{align*} G &= -\frac{B_{32}}{2\cdot 32} + \sum_{n\geq 1} \sum_{d\mid n} d^{31} q^n \\ &={\scriptstyle \frac{7709321041217}{32640} + q + 2147483649\,q^2 + 617673396283948\,q^3 + 4611686020574871553\,q^4 %+ 4656612873077392578126\,q^5 +\cdots} \end{align*} of weight $32$ is congruent modulo one of the primes above $37$ in $\mathbb{Q}(\mu_{12})$ to the cuspform $$ f = q + \zeta_{12} \,q^2 + \bigl(-\zeta_{12}^3 +\zeta_{12}^2 -\zeta_{12}\bigr)\,q^3 -\zeta_{12}^2\, q^4 + \bigl(2\,\zeta_{12}^3 +\zeta_{12}^2-2\,\zeta_{12}^2 -2\bigr)\, q^5 + \cdots $$ of weight $2$ for the group $\Gamma_1(37)$ and character $\omega^{30}$.

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  • $\begingroup$ A follow-up to this: how did you calculate the approximation for the Kubota-Leopoldt p-adic L function? Is there a reference that details how to compute these approximations? $\endgroup$ Apr 22, 2022 at 20:09
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    $\begingroup$ @chaad Using Stickelberger elements, so Bernoulli numbers, as in 1.5 of maths.nottingham.ac.uk/plp/pmzcw/download/heidelberg.pdf. Lang's book "Cyclotomic fields" is a good reference. $\endgroup$ Apr 23, 2022 at 11:11
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    $\begingroup$ Only one puzzle. How do you deduce that $X$ is a free $\mathbb{Z}_{37}$-module, i.e. why $X$ can not have finite torsion part? Do you implicitly use the fact that X has no nontrivial $\Lambda$-finite submodule? $\endgroup$
    – J.Li
    Apr 30, 2022 at 11:06

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