Theorem F ([15, Theorem 1, §5.1]).
\begin{align}
K_{4m-1}(\mathcal{O}_F)\cong\left\{\begin{matrix}
\Z/2w_{2m}(F)\Z\oplus(\Z/2\Z)^{d-1} \amp 4m-1\equiv3\pmod{8}\\\Z/w_{2m}(F)\Z \amp 4m-1\equiv7\pmod{8}
\end{matrix}\right.,\tag{4.21}
\end{align}
Section 4.3 Applications
A famous conjecture of Kummer and Vandiver states that for a prime \(p\) the class number of \(\Q(\mu_p)^+\text{,}\) the maximal totally real subfield of \(\Q(\mu_p)\text{,}\) is not divisible by \(p\text{.}\) The conjecture was first made by Ernst Kummer on 28 December 1849 and 24 April 1853 in letters to Leopold Kronecker, reprinted in [27, pages 84, 93, 123–124], and independently rediscovered around 1920 by Philipp Furtwängler and Harry Vandiver [46, p. 576]. This question has a direct relation with Milnor \(K\)-theory. Note that a for number field \(F\text{,}\) \(K_0(\mathcal{O}_F)\cong\Cl{F}\oplus\Z\text{,}\) where \(\Cl{F}\) denotes the class group of \(F\text{.}\) Therefore, Kummer – Vandiver is equivalent to \(p\)-indivisibility of \(\#K_0(\mathcal{O}_F)_{\rm tors}\) for \(F= Q(\mu_p)^+\text{.}\) This motivates us to study the torsion of higher \(K\)-groups \(K_n(\mathcal{O}_F)\) of number fields.
Borel proved [4] that for \(n\gt 0\text{,}\) \(K_{2n}(\mathcal{O}_F)\) is finite and hence \(K_{2n}(\mathcal{O}_F)_{\rm tors}=K_{2n}(\mathcal{O}_F)\text{.}\) The simplest case after \(K_0\) is \(K_2\text{.}\) Studying even \(K\)–groups in the spirit of Iwasawa’s work is not new. Coates [9] proved that the \(p\)–primary part of \(K_2\) grows like the \(p\)–primary part of the class group in \(\Z_p\)–tower. Later, it was generalized [25] to higher even \(K\)–groups of totally imaginary fields.
Subsection 4.3.1 Birch-Tate and Lichtenbaum’s Conjecture
We focus on the relation between special values of Dedekind Zeta functions and the torsion in even \(K\)–groups of number fields. The first significant conjecture of this theme was posed by Bryan Birch and John Tate, which states that
\begin{gather}
\#K_2(\mathcal{O}_K)=w_2(K)\abs{\zeta_K(-1)},\tag{4.19}
\end{gather}
where \(w_2(K)\) is the largest natural number \(N\) such that the Galois group of the cyclotomic extension over \(K\) obtained by adjoining the \(N\)-th roots of unity to \(K\text{,}\) is an elementary abelian \(2\)-group. Mazur and Wiles’s work [30] in the Iwasawa Main Conjecture confirmed Birch-Tate Conjecture up to \(2\)-torsion for abelian number fields. Later, Wiles resolved [49] Birch-Tate, up to \(2\)-torsion, for any totally real field.
We write \(a\sim b\) for two complex numbers \(a,b\) if there exists an integer \(n\) such that \(a=2^nb\text{.}\) Wiles proved [49] Lichtenbaum’s conjecture, a vast generalization of Birch-Tate, which states that if \(F\) is a totally real field, then
\begin{gather}
\abs{\zeta_F(1-2m)}\sim\frac{\#K_{4m-2}(\mathcal{O}_F)}{\#K_{4m-1}(\mathcal{O}_F)}.\tag{4.20}
\end{gather}
No doubt this is relevant because we know the structure of odd \(K\)-groups quite well but we know very little about the even \(K\)-groups. But equation (4.20) hints at the fact that questions about \(p\)-torsion in certain even \(K\)-groups can be turned into questions about \(p\)-adic valuations of special values of Dedekind zeta functions. That’s where our congruence comes into play!
Subsection 4.3.2 \(p\)–torsion in even \(K\)–groups and other results
The following Theorem describes the structure of odd \(K\)-groups:
where the quantity \(w_{2n}(F)\) will be defined later. Equations (4.20) and (4.21) together imply that
\begin{gather}
\#K_{4m-2}(\mathcal{O}_F)\sim w_{2n}(F)\abs{\zeta_F(1-2m)}.\tag{4.22}
\end{gather}
For a prime \(\ell\text{,}\) we define
\begin{gather*}
w_{i}^{(\ell)}(F)\coloneqq\max\{\ell^\nu:\nu\in\Z_{\geq0},\Gal(F(\mu_{\ell^\nu})/F)\text{ has exponent dividing $i$}\}.
\end{gather*}
The exponent of a finite group \(G\) is the least common multiple of orders of the elements of \(G\text{.}\) For a positive integer \(i\text{,}\) the quantity \(w_{i}(F)\) is defined as the product \(\prod_{\ell}w_i^{(\ell)}(F)\) over all rational primes. For any local or global field \(F\text{,}\) \(w_i(F)\) is finite.
We are only interested in the \(p\)-torsion in even \(K\)-groups for a fixed odd prime \(p\text{.}\) So, we compute \(w_{2m}^{(p)}(F_\alpha)\text{:}\)
Lemma 4.11.
Let \(p\) be an odd prime and \(m\) be a positive integer. Then
\begin{gather}
w_{2m}^{(p)}(F_\alpha)=1\tag{4.23}
\end{gather}
if \(p-1\nmid2m\text{.}\)
Proof.
Consider the following diagram
Hasse diagram of the field extensions relating \(\Q\text{,}\) \(\Q(\mu_p)\text{,}\) \(F_\alpha\text{,}\) \(F_\alpha(\mu_p)\) and \(\Q(\mu_{p^{\alpha+1}})\text{,}\) with the edges from \(\Q\) labelled \(p-1\) and \(p^\alpha\text{.}\)
Using [2, Corollary 15.3.8] one can check that \(p-1\mid\#\Gal(F_\alpha(\mu_{p^\nu})/F_\alpha)\) whenever \(\nu\geq1\text{.}\) Therefore, by the definition of \(w_{2m}(F)\text{,}\) the largest power of \(p\) that divides \(w_{2m}(F_\alpha)\) is 1 whenever \(p-1\nmid 2m\text{.}\)
Corollary 4.12.
Let \(p\geq5\) be a prime, \(F_\alpha\) as in Theorem 4.1, and \(m\geq1\) be an integer not divisible by \((p-1)/2\text{.}\) If \(p\) does not divide \(B_{2m}/2m\text{,}\) then \(K_{4m-2}(\mathcal{O}_{F_\alpha})[p^\infty]=0\) for all \(\alpha\geq1\text{.}\)
Proof.
Note that Theorem 4.1 implies that
\begin{gather}
\zeta(1-2m)=\zeta_{F_0}(1-2m)\equiv\zeta_{F_\alpha}(1-2m)\pmod{p}\tag{4.24}
\end{gather}
for all \(\alpha\in\Z_{\geq1}\) and even positive integer \(2m\) not divisible by \(p-1\text{.}\) Therefore, if \(p\) does not divide \(B_{2m}/2m\text{,}\) then \(p\nmid\zeta(1-2m)=-B_{2m}/2m\text{.}\) As a result, \(p\nmid\zeta_{F_\alpha}(1-2m)\) for all \(\alpha\geq0\text{.}\) By Lemma 4.11, \(p\nmid\#K_{4m-1}(F_\alpha)\zeta_{F_\alpha}(1-2m)\text{.}\) By Lichtenbaum’s Conjecture (Wiles’s Theorem [15, Chapter 5, Theorem 77, p. 170]), \(p\nmid\#K_{4m-2}(F_\alpha)\) as \(p\) is odd. Consequently, \(K_{4m-2}(F_\alpha)[p^\infty]=0\text{.}\)
Corollary 4.13.
Let \(p\) be an odd prime and \(n\) be an even positive integer such that \(n\) is not divisible by \(p-1\text{.}\) Then \(B_{n,\chi}/n\) is \(p\)-indivisible for all Dirichlet character \(\chi\) in \(\widehat{\Gamma_\alpha}\text{,}\) for some \(\alpha\in\Z_{\geq1}\text{,}\) if and only if \(B_n/n\) is.
Proof.
Since \(p-1\nmid n\text{,}\) \(B_n/n\) is a \(p\)-adic integer and hence \(v_p(B_n/n)\geq0\text{.}\) Theorem 4.1 implies that
\begin{gather}
\zeta(1-n)=\zeta_{F_0}(1-n)\equiv\zeta_{F_\alpha}(1-n)\pmod{p}\tag{4.25}
\end{gather}
and hence, using the Theorem 2.16,
\begin{gather}
v_p(\zeta(1-n))=0\,\Longleftrightarrow\,\sum_{\chi\in\widehat{\Z/p^\alpha\Z}}v_p(L(1-n,\chi))=0\tag{4.26}
\end{gather}
Since \(F_n(T)\in\Z_p\llbracket T\rrbracket\text{,}\) \(L(1-n,\chi)=F_n(\zeta_\chi-1)\in\mathscr{O}_\chi\text{.}\) Therefor, \(v_p(L(1-n,\chi))\geq0\text{;}\) hence \(v_p(L(1-n,\chi))=0\) for all Dirichlet characters \(\chi\) of \(p\)–power order. Moreover, \(L(1-n,\chi)=-B_{n,\chi}/n\text{.}\) Thus, \(p\nmid B_n/n\) if and only if \(p\nmid B_{n,\chi}/n\) for all \(\chi\in\widehat{\Gamma_\alpha}\text{.}\)
Corollary 4.14.
Let \(\Sp_{2n}(\mathcal{O}_{F})\) be the symplectic group of order \(2n\) with coefficients in \(\mathcal{O}_F\text{,}\) for a number field \(F\text{.}\) Let \(p\gt\max(3,n+1)\) be any prime and \(F_\alpha\) be as defined above. Then
\begin{gather*}
\chi\bparen{\Sp_{2n}(\mathcal{O}_{F_\alpha})}\equiv\chi\bparen{\Sp_{2n}(\mathcal{O}_{F_{\alpha+1}})}\pmod{p},
\end{gather*}
where \(\chi\bparen{\Sp_{2n}(\mathcal{O}_{F_\alpha})}\) is the homological Euler characteristic of the arithmetic group \(\Sp_{2n}(\mathcal{O}_{F_\alpha})\text{.}\)
Proof.
A formula of Harder [18, §2.2] connects special values of the Dedekind zeta function and the homological Euler characteristic of arithmetic groups: for a totally real field \(F\) of degree \(d\text{,}\)
\begin{gather}
\chi\bparen{\Sp_{2n}(\mathcal{O}_{F})}=\frac{1}{2^{n(d-n)}}\prod_{i=1}^n\zeta_F(1-2i).\tag{4.27}
\end{gather}
For \(p\gt\max(3,n+1)\text{,}\) \(p-1\) does not divide \(i\) for all \(1\leq i\leq n\text{.}\) Therefore, Theorem 4.1 implies
\begin{gather*}
\prod_{i=1}^n\zeta_{F_\alpha}(1-2i)\equiv\prod_{i=1}^n\zeta_{F_{\alpha+1}}(1-2i)\pmod{p}.
\end{gather*}
Equation (4.27) finishes the proof.
