Dependencies

For every positive even \(C\) there is a \(t\) with \(\left|B_{m}/m\right| {\gt} 1\) for \(m = C\cdot 2^t\).

The inclusion \(\mathcal O_{K^+}\subseteq \mathcal O_K\) induces, by extension of fractional ideals, the class-group homomorphism

For a number field \(K\), the class number \(h(K)\) is the cardinality of the ideal class group of its ring of integers:

The relative class number is the natural-number quotient

which is well defined by theorem 3.3.

For a CM field \(K\), the plus class number is the class number of the maximal real subfield:

For a number field \(K\), let \(K^+\) denote its maximal real subfield, the largest subfield of \(K\) all of whose complex embeddings are real. When \(K\) is a CM field, \(K^+\) is the fixed field of complex conjugation, and \([K : K^+] = 2\). For \(K = \mathbb {Q}(\zeta _p)\) this is the usual maximal real subfield \(\mathbb {Q}(\zeta _p + \zeta _p^{-1})\).

The Kummer logarithm matrix \(M\) is the \(r \times r\) matrix over \(\mathbb {F}_p\) whose \(a\)th column consists of the coordinates, in the Dwork basis of the relevant fixed subalgebra, of the completed \(p\)-adic logarithm of the generator \(\varepsilon _a\), reduced modulo \(p\).

The subgroup \(C^+\subseteq (\mathcal O_{K^+})^\times \) is generated by \(-1\) and the standard real cyclotomic units:

The standard generators are the \((p-3)/2\) real cyclotomic units

of definition 1.1, indexed in the formalisation by \(i \in \{ 0, \ldots , (p-5)/2\} \) via \(a = i + 2\).

An exponent product is an expression

with integer exponents \(s, e_a \in \mathbb {Z}\), packaged together with the element of \(C^+\) it represents.

The subgroup featuring in the prime-conductor index theorem is the normalised cyclotomic-unit subgroup \(C^+_{\mathrm{norm}}\) of definition 1.4.

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.

The ambient group for the saturation argument is the full unit group

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 \(\chi \).

For a finite set \(S\) of natural numbers put \(M(S) = \max (3, \max S)\) and

Set

the common cardinality of the set of standard generators \(\varepsilon _2,\ldots ,\varepsilon _{(p-1)/2}\) and of the even index set \(\{ 2j : 1 \le j,\ 2j \le p-3\} \) of Kummer’s range.

For \(2 \le a \le (p-1)/2\), the normalised cyclotomic unit is

where the exponent \((1-a)/2\) is read modulo \(p\). The twist by the root of unity makes \(\xi _a\) fixed by complex conjugation, so it defines a unit of \(\mathcal O_{K^+}\). The normalised subgroup \(C^+_{\mathrm{norm}}\) is generated by \(-1\) and the units \(\xi _a\).

For a subgroup \(H\) of an abelian group, \(H^p\) denotes the exact image of the \(p\)th-power map:

(an actual image, not the subgroup generated by it; for abelian groups the two coincide).

A subgroup \(H\le E\) is \(p\)-saturated in \(E\) when

that is, when every element of \(H\) that is a \(p\)th power in \(E\) is already the \(p\)th power of an element of \(H\).

For \(2\le a\le (p-1)/2\), the real cyclotomic unit \(\varepsilon _a\) is the unit of \(\mathcal O_{K^+}\) obtained by descending the product of the cyclotomic unit \((1-\zeta _p^a)/(1-\zeta _p)\) of \(\mathcal O_K\) with its complex conjugate:

This element is fixed by complex conjugation, hence lies in \((\mathcal O_{K^+})^\times \).

An odd prime \(p\) is regular when \(p\) does not divide the class number of the \(p\)th cyclotomic field:

Otherwise \(p\) is irregular. (In the formalisation the condition is phrased as coprimality of \(p\) with the cardinality of the class group of \(\mathbb {Q}(\zeta _p)\), which is equivalent because \(p\) is prime.)

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.

Let \(P\) be a predicate on the natural numbers. If no finite set contains every \(n\) with \(P(n)\), then \(\{ n \mid P(n)\} \) is infinite.

Let \(m = C(S)\cdot 2^t\). Every prime \(q\) dividing \(m\) satisfies \((q-1)\mid m\).

\(C(S)\) is even and positive, and for every prime \(q \in S\) one has \((q-1)\mid C(S)\).

Let \(q\) be a rational and \(n\) a positive natural number. If \(p\mid \operatorname {num}(q/n)\), then \(p\mid \operatorname {num}(q)\).

If a rational number \(q\) has \(|q| {\gt} 1\), then some prime divides \(\operatorname {num}(q)\).

Let \(q, r\) be rationals with \(q \equiv r \pmod{p}\). If \(p\mid \operatorname {num}(q)\), then \(p\mid \operatorname {num}(r)\).

For any predicate \(Q\) on natural numbers,

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 \(B_n\).

Let \(p\) be an odd prime and \(m\) an even index with \((p-1)\nmid m\). Then \(m' := m \bmod (p-1)\) satisfies \(0 {\lt} m' {\lt} p-1\), \(m'\) is even, and \(m \equiv m' \pmod{p-1}\). In particular \(2 \le m' \le p-3\).

Let \(p\) be a prime and \(n\) an even index. Then \(pB_n \in \mathbb {Z}_p\).

Let \(K\) be the \(p\)th cyclotomic field with \(p\) odd. If \(u\in \mathcal O_K^\times \) satisfies \(u\overline u=1\), then

for some integers \(n,m\).

For \(0{\lt}n{\lt}p-1\), the prime \(p\) does not divide the denominator of \(B_n\).

For every \(k \ge 1\),

\(\left|B_{2k}/2k\right| \to \infty \) as \(k \to \infty \).

Let \(p\) be an odd prime. Suppose that

Then

For \(0{\lt}n{\lt}p-1\), the classical Bernoulli number \(B_n\) is \(p\)-adically integral: \(B_n \in \mathbb {Z}_p\).

Conversely, if an odd prime \(p\) is not regular, then there is a \(k\) with \(1\le k\), \(2k\le p-3\) and \(p\mid \operatorname {num}(B_{2k})\).

The diagonal Bernoulli factor attached to \(j\) is nonzero in \(\mathbb {F}_p\) exactly when

An odd prime \(p\) is irregular if and only if \(p \mid \operatorname {num}(B_m/m)\) for some positive even index \(m\).

For every finite set \(S\) of natural numbers there is an irregular prime \(p \notin S\).

For every finite set \(S\) of natural numbers there are \(M\) and a prime \(p\) such that \(M\) is even and positive, \((q-1)\mid M\) for every prime \(q \in S\), the prime \(p\) is odd, \(p\mid \operatorname {num}(B_M/M)\), and \(p \notin S\).

For an odd prime cyclotomic field, the map

is injective.

If an exponent product in \(C^+\) is a \(p\)th power in the full unit group, then its completed logarithm is divisible by \(p\) in the Dwork fixed subalgebra.

In the situation of theorem 2.5, the generator \(a\) may be replaced by an associate \(b\) such that

Suppose \(I \ne 0\) and \(I\mathcal O_K=(a)\). Then, by theorem 2.4, also \(I\mathcal O_K=(\overline a)\), and the two generators are associated: there is a unit \(u\in \mathcal O_K^\times \) with

The unit \(u\) obtained from \(\overline a=ua\) in theorem 2.5 satisfies

For an odd prime \(p\),

Assume that whenever an exponent product

is a \(p\)th power in \(E^+\), every exponent \(e_a\) is divisible by \(p\). Then \(C^+\) is \(p\)-saturated in \(E^+\).

The explicit Gauss-product evaluation of theorem 2.2 satisfies the packaged Gauss hypothesis required by the relative class-number assembly theorem (theorem 2.4).

The deleted-Fourier determinant identity supplies the Kummer–Dirichlet determinant hypothesis required by the Sinnott index pipeline: the regulator-type determinant of the cyclotomic-unit family is, up to an explicit nonzero factor, a product of the character sums \(\sum _a \chi (a) \log |1-e^{2\pi i a/p}|\) over the even nontrivial characters.

For an odd prime \(p\) and the \(p\)th cyclotomic field \(K\),

Assume that a separate argument proves the implication

Then

Every element of \(C^+\) admits an exponent-product expression.

The class number factors as

For \(p\) odd,

There exists \(z\in \mathbb {Z}_p\) such that, inside \(\mathbb {Q}_p\),

For an odd prime cyclotomic field,

Assume the factorisation of the Dedekind zeta residue of \(K\) into the even and odd \(L\)-value products, the \(h^+\) formula of theorem 2.1, and the packaged Gauss-product identity. Then

If \(p\nmid h^-(K)\), then for every \(k\) with \(1\le k\) and \(2k\le p-3\),

There exists \(z\in \mathbb {Z}_p\) such that, inside \(\mathbb {Q}_p\),

The plus class number divides the full class number:

The plus class number is given by the even part of the cyclotomic \(L\)-value product:

where each factor of the product equals \(L(1,\chi )\) and \(c_{K^+}\) is the explicit constant of the analytic class-number formula of \(K^+\) (built from the discriminant, the regulator and the roots of unity of \(K^+\)).

If \(p\mid h^+(K)\), then \(p\mid h^-(K)\).

If \(I\) is an ideal of \(\mathcal O_{K^+}\), then the extended ideal \(I\mathcal O_K\) is fixed by complex conjugation.

For \(p\ge 5\), the normalised cyclotomic-unit index has the same \(p\)-divisibility as \(h^+\):

Up to the normalisation of theorem 1.6, the subgroup used in the index theorem agrees with the subgroup \(C^+\) used in the saturation argument: each contains representatives generating the other modulo \(\{ \pm 1\} \) and squares.

There are infinitely many irregular primes.

Let \(p\) be an odd prime. If there is a \(k\) with \(1\le k\), \(2k\le p-3\) and \(p\mid \operatorname {num}(B_{2k})\), then \(p\) is not regular.

Let \(K\) be the \(p\)th cyclotomic field, with \(p\) odd. If an ideal of \(\mathcal O_{K^+}\) becomes principal after extension to \(\mathcal O_K\), then it was already principal.

Let \(I\) be an ideal of \(\mathcal O_{K^+}\). If \(I\mathcal O_K\) is generated by the image of an element \(b_0\in \mathcal O_{K^+}\), then \(I\) is principal, generated by \(b_0\).

Let \(p\) be an odd prime and \(m, n\) positive even indices with \(m \equiv n \pmod{p-1}\) and \((p-1)\nmid n\). Then

Let \(p \ge 5\) be a prime and \(m, n\) positive even indices with \(m \equiv n \pmod{p-1}\) and \((p-1)\nmid n\). Then

Let \(p\) be an odd prime. Then

\(\det M \ne 0\) if and only if every diagonal Bernoulli factor is nonzero.

If \(\det M\ne 0\) and \(Me=0\), then \(e = 0\) in \(\mathbb {F}_p^{\, r}\).

The Kummer logarithm determinant is nonzero precisely when no Bernoulli numerator in Kummer’s range is divisible by \(p\):

If \(\det M \ne 0\), then \(C^+\) is \(p\)-saturated in \((\mathcal O_{K^+})^\times \).

If \(\det M \ne 0\), then every exponent product in \(C^+\) which is a \(p\)th power in the full unit group has all exponents \(e_a\) divisible by \(p\).

The Kummer logarithm matrix factors as

where \(D\) is the diagonal matrix of Bernoulli factors (the \(j\)th diagonal entry being a unit multiple of the reduction of \(B_{2j}/2j\)) and \(V\) is a Vandermonde-type matrix built from Teichmüller values.

In the situation of theorem 3.8,

where \(e\) is the vector of exponents \(e_a\) reduced modulo \(p\).

If \(J\) is an ideal of \(\mathcal O_{L^+}\), then extension to \(\mathcal O_L\) followed by contraction is the identity:

If \(b\in \mathcal O_K\) is fixed by complex conjugation, then it is the image of an element of \(\mathcal O_{K^+}\).

Each normalised generator squares to the corresponding standard generator:

If \(p\nmid h^-(K)\), then \(p\nmid h^+(K)\).

If \(C^+\) is \(p\)-saturated in \(E^+\), then

The product of the Gauss sums over the odd characters modulo \(p\) is

For every number field \(L\), the ring \(\mathcal O_L\) is faithfully flat over \(\mathcal O_{L^+}\).

For every odd prime conductor,

For the odd exponents \(j\) occurring in the relative class-number formula (odd \(j\) with \(j + 1\) divisible by neither \(p\) nor \(p-1\)),

The Teichmüller Vandermonde determinant occurring in theorem 3.3 is nonzero.

Let \(p\) be a prime and \(n\) a positive even index with \((p-1)\mid n\). Then \(p\nmid \operatorname {num}(B_n/n)\).

Let \(p \ge 5\) be a prime, \(a\) an integer prime to \(p\), and \(k\) a positive even index with \((p-1)\nmid k\). Then