1 Characters and Bernoulli factors
A Dirichlet character modulo \(N\) with values in a commutative ring \(R\) is a multiplicative character of \((\mathbb {Z}/N\mathbb {Z})^\times \) with values in \(R^\times \), extended by zero to all of \(\mathbb {Z}/N\mathbb {Z}\). A character \(\chi \) modulo \(N {\gt} 1\) is even if \(\chi (-1) = 1\) and odd if \(\chi (-1) = -1\). In this chapter the modulus is the odd prime \(p\), in which case every nontrivial character is primitive and exactly \((p-1)/2\) of the characters are odd.
For a Dirichlet character \(\chi \) modulo \(N\) and \(n \ge 0\), the generalised Bernoulli number is
where \(B_n(X)\) is the \(n\)th Bernoulli polynomial. For nontrivial \(\chi \) the case \(n = 1\) simplifies to
because \(\sum _a \chi (a) = 0\) kills the constant term of \(B_1(X) = X - \tfrac 12\). The class-number formula only uses the values \(B_{1,\chi ^{-1}}\) for odd \(\chi \).
The Teichmüller character of the prime \(p\) is the unique character
such that \(\omega (a) \equiv a \pmod p\) and \(\omega (a)^{p-1} = 1\) for every \(a\); its values are the \((p-1)\)st roots of unity in \(\mathbb {Z}_p\). Composing with the inclusion \(\mathbb {Z}_p \hookrightarrow \mathbb {Q}_p\) gives the \(\mathbb {Q}_p\)-valued Dirichlet character used below. The characters \(\omega ^j\) for \(0 \le j \le p-2\) exhaust the characters modulo \(p\), and \(\omega ^j\) is odd exactly when \(j\) is odd.
For \(0{\lt}n{\lt}p-1\), the classical Bernoulli number \(B_n\) is \(p\)-adically integral: \(B_n \in \mathbb {Z}_p\).
By the von Staudt–Clausen theorem, the denominator of \(B_n\) (for \(n\) even, the only nontrivial case) is the product of the primes \(q\) with \((q-1)\mid n\). If \(p\) occurred, then \(p - 1 \le n\), contradicting \(n {\lt} p-1\). Hence the denominator is prime to \(p\) and \(B_n \in \mathbb {Z}_p\).
For \(0{\lt}n{\lt}p-1\), the prime \(p\) does not divide the denominator of \(B_n\).
This is the rational form of theorem 1.4: a rational number lies in \(\mathbb {Z}_p\) exactly when its reduced denominator is prime to \(p\).
For \(0{\lt}n{\lt}p-1\), the quotient \(B_n/n\) lies in \(\mathbb {Z}_p\), and it is a \(p\)-adic unit if and only if \(p\) does not divide the numerator of \(B_n\).
The denominator of \(B_n\) is prime to \(p\) by theorem 1.5, and \(n {\lt} p-1 {\lt} p\) is also prime to \(p\), so \(B_n/n \in \mathbb {Z}_p\). In \(\mathbb {Z}_p\) the non-units are exactly the elements of \(p\mathbb {Z}_p\); since the denominator and \(n\) are \(p\)-adic units, \(B_n/n \in p\mathbb {Z}_p\) holds precisely when \(p\) divides the numerator of \(B_n\).
For the odd exponents \(j\) occurring in the relative class-number formula (odd \(j\) with \(j + 1\) divisible by neither \(p\) nor \(p-1\)),
This is the standard congruence linking generalised Bernoulli values of Teichmüller powers to classical Bernoulli numbers. Writing \(B_{1,\omega ^j} = \tfrac 1p \sum _a \omega ^j(a)\, a\) and using \(\omega (a) \equiv a^{p^r}\) to high \(p\)-adic accuracy, the sum is compared with the power sum \(\sum _a a^{\, j+1}\), which Faulhaber’s formula relates to \(B_{j+1}\); the hypothesis that \(j+1\) is divisible by neither \(p\) nor \(p-1\) makes the comparison factor a \(p\)-adic unit and keeps the error terms in \(p\mathbb {Z}_p\).