3 The Kummer logarithm matrix
The Kummer logarithm matrix has rank \((p-3)/2\), matching the number of standard real cyclotomic-unit generators.
The concrete Kummer logarithm matrix is the matrix over \(\mathbb F_p\) whose columns are the completed \(p\)-adic logarithms of the real cyclotomic-unit generators, expressed in the Dwork basis.
The concrete Kummer logarithm matrix factors as a diagonal matrix of Bernoulli factors times a Teichmuller Vandermonde matrix.
The proof computes each logarithm coefficient. The row-dependent part is the Bernoulli factor, while the column-dependent part is a power of a Teichmuller node. This separates the matrix into a diagonal factor and a Vandermonde factor.
The Teichmuller Vandermonde determinant occurring above is nonzero.
The Teichmuller nodes are distinct and no node is equal to \(1\). The Vandermonde determinant is therefore a product of nonzero differences.
The Kummer logarithm determinant is nonzero exactly when every Bernoulli diagonal factor is nonzero.
Use the diagonal-times-Vandermonde factorisation. The Vandermonde determinant is nonzero by theorem 3.4, so the determinant is nonzero exactly when every diagonal entry is nonzero.
The diagonal Bernoulli factor attached to \(j\) is nonzero modulo \(p\) exactly when
The factor is the reduction modulo \(p\) of a \(p\)-integral rational expression whose unit part is the numerator of \(B_{2j}\). Denominator terms are units in the permitted range.
The determinant is nonzero precisely when every Bernoulli numerator in Kummer’s range is nonzero modulo \(p\):
Theorem 3.5 reduces the determinant condition to the diagonal Bernoulli factors. Then theorem 3.6 rewrites each factor as the corresponding Bernoulli numerator. The finite matrix index is identified with the classical range \(1\le j\), \(2j\le p-3\).
If a standard 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.
The completed logarithm is additive on the product expression and sends a \(p\)th power to \(p\) times the logarithm of the root. Expanding the product therefore expresses the weighted sum of Kummer logarithm columns as a multiple of \(p\).
The completed-log divisibility relation gives
over \(\mathbb F_p\), where \(e\) is the vector of exponents modulo \(p\).
Take coordinates in the Dwork basis. Divisibility by \(p\) makes every coordinate vanish after reduction modulo \(p\), and those coordinates are exactly the entries of the matrix-vector product \(Me\).
If \(\det M\ne 0\) and \(Me=0\), then every coordinate of \(e\) is zero modulo \(p\).
Over the field \(\mathbb F_p\), a square matrix with nonzero determinant has trivial right kernel.
If the Kummer logarithm determinant is nonzero, then every exponent product in \(C^+\) which is a \(p\)th power in the full unit group has all standard exponents zero modulo \(p\).
Apply theorem 3.8 to the \(p\)th-power relation. Convert the logarithmic divisibility statement to \(Me=0\) using theorem 3.9. Since the determinant is nonzero, theorem 3.10 forces \(e=0\).
If the Kummer logarithm determinant is nonzero, then \(C^+\) is \(p\)-saturated in the full plus-side unit group.
Theorem 3.11 gives exactly the exponent-vanishing hypothesis required by theorem 2.6.