Class invariants of Mordell-Weil groups.

*(English)*Zbl 0799.11049Let \(E/K\) be an elliptic curve with complex multiplication by the ring of integers \({\mathcal O}_ K\) of an imaginary quadratic field \(K\) with class number 1. Let \(F/K\) be a finite abelian extension over which \(E/F\) acquires everywhere good reduction. Choose a prime number \(p>3\), not dividing \([F:K]\), such that:

(a) \(p\) is non-anomalous and splits in \({\mathcal O}_ K\) with \(p=(\pi) (\pi^*)\);

(b) \((\pi)\) and \((\pi^*)\) are primes of good reduction of \(E/K\).

The authors introduce \({\mathcal O}_ F\)-orders \({\mathcal A}_ n\) (within suitable Hopf algebras) such that \(\text{Spec}({\mathcal A}_ n)= E_{(\pi^*)^ n}\) as \({\mathcal O}_ F\)-group schemes. Then they construct a natural morphism \[ \psi: E_{1,\pi^*} \otimes_{{\mathcal O}_ K} {\mathcal O}_{K,\pi}\to \varprojlim C\ell({\mathcal A}_ n), \] where \(E_{1,\pi^*}\) is the kernel of the reduction map at all primes of \(F\) dividing \(\pi^*\) and \({\mathcal O}_{K,\pi}\) denotes semi-localization at the prime \((\pi)\). Their main results concerning the map \(\psi\) are:

1) \(\text{Im } \psi\) is isomorphic to a submodule of finite index in \(H^ 0(F,\operatorname{Hom} (T_{\pi^*}, \varprojlim C\ell_ n))\), where \(C\ell_ n\) is the class-group of the field \(F(E_{\pi^{*n}})\) and \(T_{\pi^*}\) is the Tate module;

2) If \((\pi^*)\) is completely split in \(F\), and \(K(E_{\pi^ \infty})\) and \(F\) are linearly disjoint over \(K\), then \(\psi\) is injective on \(E_{1,\pi^*}\).

Another map \(\phi\), closely related to \(\psi\), can be constructed starting from the prime \((\pi)\). Modulo standard conjectures, its kernel yields distinguished one dimensional subspaces of the completed Mordell- Weil group.

It is the authors’ belief that these results suggest a new and as yet not understood relationship between class invariants and \(p\)-adic height pairing on abelian varieties.

(a) \(p\) is non-anomalous and splits in \({\mathcal O}_ K\) with \(p=(\pi) (\pi^*)\);

(b) \((\pi)\) and \((\pi^*)\) are primes of good reduction of \(E/K\).

The authors introduce \({\mathcal O}_ F\)-orders \({\mathcal A}_ n\) (within suitable Hopf algebras) such that \(\text{Spec}({\mathcal A}_ n)= E_{(\pi^*)^ n}\) as \({\mathcal O}_ F\)-group schemes. Then they construct a natural morphism \[ \psi: E_{1,\pi^*} \otimes_{{\mathcal O}_ K} {\mathcal O}_{K,\pi}\to \varprojlim C\ell({\mathcal A}_ n), \] where \(E_{1,\pi^*}\) is the kernel of the reduction map at all primes of \(F\) dividing \(\pi^*\) and \({\mathcal O}_{K,\pi}\) denotes semi-localization at the prime \((\pi)\). Their main results concerning the map \(\psi\) are:

1) \(\text{Im } \psi\) is isomorphic to a submodule of finite index in \(H^ 0(F,\operatorname{Hom} (T_{\pi^*}, \varprojlim C\ell_ n))\), where \(C\ell_ n\) is the class-group of the field \(F(E_{\pi^{*n}})\) and \(T_{\pi^*}\) is the Tate module;

2) If \((\pi^*)\) is completely split in \(F\), and \(K(E_{\pi^ \infty})\) and \(F\) are linearly disjoint over \(K\), then \(\psi\) is injective on \(E_{1,\pi^*}\).

Another map \(\phi\), closely related to \(\psi\), can be constructed starting from the prime \((\pi)\). Modulo standard conjectures, its kernel yields distinguished one dimensional subspaces of the completed Mordell- Weil group.

It is the authors’ belief that these results suggest a new and as yet not understood relationship between class invariants and \(p\)-adic height pairing on abelian varieties.

Reviewer: Thong Nguyen Quang Do (BesanĂ§on)

##### MSC:

11R33 | Integral representations related to algebraic numbers; Galois module structure of rings of integers |

11G05 | Elliptic curves over global fields |