The Galois module structure of certain arithmetic principal homogeneous spaces
This paper describes a new connection between Galois module theory and values of \(p\)-adic \(L\)-functions. The author demonstrates such a connection both for unramified Kummer extensions of order \(p\) of the cyclotomic field \(\mathbb{Q}(\zeta_ p)\), and for principal homogeneous spaces arising from an elliptic curve with complex multiplication defined over an imaginary quadratic field \(K\). For the cyclotomic case, let \(N=\mathbb{Q}(\zeta_ p)\), \(p\) an odd prime number, let \(G\) be cyclic of order \(p\), and let \({\mathfrak B}=\text{Map}(G,{\mathcal O}_ N)\). The group \(\text{PH}({\mathfrak B})\) of principal homogeneous spaces for \({\mathfrak B}\) is the same as the group of isomorphism classes of Galois extensions of \({\mathcal O}_ N\) with group \(G\). The map \(\psi: \text{PH}({\mathfrak B})\to\text{Cl}({\mathfrak A})\) defined by viewing a principal homogeneous space as a (locally free) module over \({\mathfrak A}={\mathcal O}_ N G\) is a \(\Delta\)-homomorphism with the natural action of \(\Delta=\text{Gal}(N/\mathbb{Q})\) on \(\text{PH}({\mathfrak B})\) and \(\text{Cl}({\mathfrak A})\). So \(\text{ker}(\psi)\) is a \(\mathbb{Z}_ p\Delta\)- module, hence decomposes into eigenspaces \(\text{ker}(\psi)_ i\) corresponding to the \(i\)-th power of the character \(\kappa: \Delta\to\mathbb{Z}^ \times_ p\) given by \(\zeta^ \delta=\xi^{\kappa(\delta)}\) for \(\delta\) in \(\Delta\). The author shows that \(\text{ker}(\psi)_ i=(0)\) if \(i=0\) or \(i\) is odd, while if \(i>0\) is even then \(\text{ker}(\psi)_ i=(0)\) iff \(L_ p(1,\kappa^ i)h_{F,i}^{-1}\not\equiv 0\pmod p\). Here \(L_ p(1,\kappa^ i)\) is the \(p\)-adic \(L\)-function associated to \(\kappa^ i\), and \(h_{F,i}\) is the cardinality of the \(i\)-part of the \(p\)-Sylow subgroup of the class group of \(F\). In an explicitly analogous way, the author obtains a similar result for principal homogeneous spaces arising from an elliptic curve \(E\) defined over \(K\) as above with complex multiplication by \({\mathcal O}_ K\). Let \({\mathfrak p}\) be a prime ideal of \({\mathcal O}_ K\) not dividing the conductor \({\mathfrak f}\) of \(E\), let \(p{\mathcal O}_ K={\mathfrak p}{\mathfrak p}^*\), \({\mathfrak p}=\pi{\mathcal O}_ K\), \({\mathfrak p}^*=\pi^*{\mathcal O}_ K\), let \(E_ \pi\) denote the group of \(\pi\)-division points of \(E\), let \(F=K(E_ \pi)\), \(N=F(\zeta_ p)\), and analogously define \(E_{\pi^*}\), \(F_ *\), \(N_ *\). Let \(G\) be an isomorphic copy of \(E_{\pi^*}\). Let \(\Delta=\text{Gal}(N/F_ *)\), let \(\kappa\) be the \(\mathbb{Z}^ \times_ p\)-valued character which gives the action of \(\text{Gal}(N/K)\) on \(E_ \pi\). Let \({\mathfrak B}\) be the unique Hopf order in \(B=\text{Map}(G,N)\) with \(\text{Spec}({\mathfrak B})=E_{\pi^*}/{\mathcal O}_ N\), and let \({\mathfrak A}=\Hom_{{\mathcal O}_ N}({\mathfrak B},{\mathcal O}_ N)\). Again, the map \(\psi: \text{PH}({\mathfrak A})\to\text{Cl}({\mathfrak A})\) is a \(\Delta\)-module homomorphism, and \(\text{ker}(\psi)\) decomposes into eigenspaces under the action of \(\Delta\) via powers of the character \(\kappa\). Then for \(i\not\equiv 0\pmod{p-1}\), \(\text{ker}(\psi)_ i=(0)\) iff \(L_{{\mathfrak p},{\mathfrak f}}(\kappa^{-i})h_{F,i}^{-1}\not\equiv 0\pmod p\). Here \(h_{F,i}\) is a class number defined as before, and \(L_{{\mathfrak p},{\mathfrak f}}\) is the \({\mathfrak p}\)-adic \(L\)-function defined on the Größencharaktere of the Galois group over \(K\) of the union of all ray class fields of \(K\) whose conductor is \({\mathfrak f}\) times a power of \(p\).
elliptic curve with complex multiplication
scientific article; zbMATH DE number 130471