Units and norm residue symbol (Q2717613)

Let \(p\geq 5\) be a prime. Let \(K=\mathbb Q_p(\zeta_p)\), where \(\zeta_p\) is a \(p\)th primitive root of unity. Let \(U_K\) and \({\mathfrak p}_K\) be the group of units and the maximal ideal of the integral closure of \(\mathbb Z_p\) in \(K\), respectively. Let \(U_K^{(n)}=1+{\mathfrak p}_K^n\) and \(V=\mathbb Q(\zeta_p)\cap U_K\). For any \(B\subseteq V\), let \(B^{\text{Kum}}=B\cap\mu_{p-1}U_K^{(p)}\), where \(\mu_{p-1}\) is the group of \((p-1)\)th roots of unity. Then \({\mathcal V}=V/V ^{\text{Kum}} \) is a \(\mathbb F_p[G]\)-module, where \(\mathbb F_p=\mathbb Z/p\mathbb Z\) and \(G\) is the Galois group of \(\mathbb Q(\zeta _p)/\mathbb Q\). For any \(\mathbb F_p[G]\)-module \(A\) and any \(1\leq i\leq p-1\), let \(A(i)\) mean the \(i\)th component of \(A\), i.e. \(A(i)=\{a\in A:\forall \sigma \in G, \sigma (a)=\omega ^i(\sigma)a\}\), where \(\omega\) is the Teichmüller character. For any \(\mathbb F_p[G]\)-submodule \(B\) of \({\mathcal V}\), let \(B^\perp\) be the orthogonal of \(B\) with respect to the norm residue symbol. NEWLINENEWLINENEWLINEUsing Lubin-Tate theory, the author proves that for any \(1\leq i\leq p-1\) and for any \(\mathbb F_p[G]\)-submodule \(B\) of \({\mathcal V}\), we have NEWLINE\[NEWLINE\dim_{F_p} B^\perp (i)+\dim_{F_p} B(p-i)=1.NEWLINE\]NEWLINE As a corollary he obtain the following result for the group of units \(E_F\) and the group of cyclotomic units \( \text{Cyc}_F \) of a real subfield \(F\neq\mathbb Q\) of \(\mathbb Q(\zeta _p)\): NEWLINE\[NEWLINE\dim_{F_p}\frac{( \text{Cyc}_F/\text{Cyc}_F^{\text{Kum}})^\perp} {(E_F/E_F^{\text{Kum}})^\perp} (i)= \dim_{F_p} \frac{E_F}{\text{Cyc}_FE_F^{\text{Kum}} }(p-i).NEWLINE\]NEWLINE Another result of the paper concerns the Terjanian conjecture. The author shows that this conjecture is equivalent to the statement that the Kummer system of congruences \(B_{2j}M_{p-2j}\equiv 0\pmod p\), \(1\leq j\leq (p-3)/2\), has only trivial solutions. Here \(B_{2j}\) is the Bernoulli number and \(M_{p-2j}\) is the Mirimanoff polynomial. Let \(i(p)\) be the index of irregularity of \(p\). He proves that if Terjanian's conjecture is false then \(2^{p-1}\equiv 1\pmod {p^2}\), \(B_{p-3}\equiv 0\pmod p\), and \(i(p)\geq\sqrt p-2\). NEWLINENEWLINENEWLINEFinally, using a class number congruence obtained by Metsänkylä, the author shows that the condition \(E_F^{\text{Kum}} =(E_F)^p\) in Kummer's lemma is equivalent to \(\frac{R_p(F)}{\sqrt{d(F)}}\not\equiv 0\pmod p\), where \(d(F)\) and \(R_p(F)\) are the discriminant and the \(p\)-adic regulator of \(F\), respectively.