Introduction

It’s fine to work on any problem, so long as it generates interesting mathematics along the way – even if you don’t solve it at the end of the day.
―Andrew Wiles

This Chapter aims to present a complete proof of two main results of this thesis. Recall (as defined in Chapter 2) that \(\zeta_K(s)\) denotes the Dedekind zeta function attached to a number field \(K\text{.}\)

Theorem 4.1.

Let \(\alpha\) be a nonnegative integer, \(p\geq3\) be a prime, and \(F_\alpha\) be the unique subfield of \(\Q(\mu_{p^{\alpha+1}})\) which is cyclic of degree \(p^\alpha\) over \(\Q\) (recall the definition on page 30). Then \(\zeta_{F_\alpha}(1-n)\in\Z_p\) and

\begin{gather} \zeta_{F_\alpha}(1-n)\equiv\zeta_{F_{\alpha+1}}(1-n)\pmod{p^{\alpha+1}\Z_p}\tag{4.1} \end{gather}

for all positive even integers \(n\) not divisible by \(p-1\text{.}\)

The proof uses Iwasawa’s construction of \(p\)–adic \(L\)–functions as described in Chapter 3 (for a detailed account, we refer the reader to [24] and [47, Chapter 7]) and Local Class Field Theory. The special case when \(n=2\text{,}\) of Theorem 4.1, first appeared in [7, Proposition 2.5], which we obtain as a simple Corollary of Theorem 4.1:

Corollary 4.2.

Let \(p\geq5\) be an odd prime and \(F_\alpha\) be as in Theorem 4.1. Then \(\zeta_{F_\alpha}(-1)\) is a \(p\)–adic integer and

\begin{gather*} \zeta_{F_\alpha}(-1)\equiv\zeta_{F_{\alpha+1}}(-1)\pmod{p} \end{gather*}

for all nonnegative integers \(\alpha\text{.}\)

Proof.

Take \(n=2\) in Theorem 4.1 and observe that \(p-1\) does not divide \(2\) for any prime \(p\geq5\text{.}\)

An alternate proof, utilizing Coates’s suggestion to use \(p\)–adic \(L\)–functions, was sketched in [6, Proposition 2]. We show that Coates’s idea extends from \(n=2\) to all even integers not divisible by \(p-1\) and an application of Local Class Field Theory improves the congruence from \(\pmod{p}\) to \(\pmod{p^{\alpha+1}}\text{.}\)

Our second main result is a congruence of Dirichlet \(L\)–functions. Let \(p\) be an odd prime, and \(\chi\) be a Dirichlet character \(\chi:\Gal(\Q(\mu_{p^{m+1}})/\Q)\longrightarrow\C^\times\text{.}\) Recall that there is a canonical isomorphism \(\Gal(\Q(\mu_{p^{\alpha+1}})/\Q)\cong(\Z/p^{\alpha+1}\Z)^\times\) and hence \(\chi\) is a Dirichlet character modulo \(p^{\alpha+1}\text{.}\) Fixing an embedding of \(\overline{\Q}\) into \(\overline{\Q}_p\text{,}\) we may regard the values \(\chi(a),a\in(\Z/p^{\alpha+1}\Z)^\times\) as elements of \(\overline{\Z}_p\text{,}\) in particular, of the local ring \(\mathscr{O}_\chi=\Z_p[\chi(1),\chi(2),\ldots]\) (recall the definition on page 26). Let \(\mf{p}\) be the unique maximal ideal of \(\mathscr{O}_\chi\text{.}\) For two such characters \(\chi_1,\chi_2\text{,}\) we say that \(\chi_1\equiv\chi_2\pmod{\mf{p}}\) if \(\chi_1(a)\equiv\chi_2(a)\pmod{\mf{p}}\) for all \(a\in(\Z/p^{\alpha+1}\Z)^\times\text{.}\)

Theorem 4.3.

Let \(p\) be an odd prime, \(n\) be either 1 or an even positive integer not divisible by \(p-1\text{,}\) \(\chi_i\neq1,i=1,2\) be two characters of the group \(\Gal(\Q(\mu_{p^{m+1}})/\Q)\) for some \(m\geq1\) such that \(\chi_i\) does not contain the character \(\omega^{-n}\) modulo \(p\text{,}\) where \(\omega\) is the Teichmüller character modulo \({p}\text{,}\) as defined on page 30. If \(\chi_1\equiv\chi_2\pmod{\mf{p}}\text{,}\) then

\begin{align} \left\{ \begin{matrix} (1-\chi_1(p))L(0,\chi_1)\amp\equiv\amp(1-\chi_2(p))L(0,\chi_2)\amp\pmod{\mf{p}}\amp n=1\\L(1-n,\chi_1)\amp\equiv\amp L(1-n,\chi_2)\amp\pmod{\mf{p}}\amp n\gt1 \end{matrix}\right.\tag{4.2} \end{align}

We will also provide some new applications of our main results. In particular, elementary arguments show that Theorem 4.1 implies the following results:

Corollary 4.4. see 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 nonnegative integers \(\alpha\text{.}\)

Corollary 4.5. see Corollary 4.13.

Let \(p\) be an odd prime and \(n\) be an even positive integer not divisible by \(p-1\text{.}\) Then \(B_{n,\chi}\) is \(p\)-indivisible for all Dirichlet character \(\chi\) of \(\Gal(F_\alpha/\Q)\text{,}\) for some \(\alpha\in\Z_{\geq1}\text{,}\) if and only if \(p\nmid B_n/n\text{.}\)

For a group \(G\text{,}\) with certain finiteness conditions [41, No. 1.8], we can attach to it a rational number, denoted by \(\chi(G)\text{,}\) called the (homological) Euler Characteristic (see [5, §4] for the precise definition). There are beautiful results [18, 5] that realize special values of zeta functions as Euler Characteristics of arithmetic groups. In particular, we will use one such result when \(G\) is the symplectic group. Let \(\Sp_{2n}(\mathcal{O}_{K})\) denote the symplectic group (see [16, Chapter 7] for the definition) with coefficients in the ring \(\mathcal{O}_K\) of integers of a number field \(K\text{.}\)

Corollary 4.6. see 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 \(\Sp_{2n}(\mathcal{O}_{F_\alpha})\text{.}\)

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