Class-invariants: examples of higher-dimensional nonvanishing. (Invariants de classes: exemples de non-annulation en dimension supérieure.) (French. English summary) Zbl 1137.11042
Let \(G\) be a finite flat group scheme over \(S = \text{Spec}(R)\) where \(R\) is the ring of integers in a number field \(K\). We furthermore assume that \(G\) is the kernel of an isogeny \({\mathcal A} \rightarrow {\mathcal B}\) between the Néron models \({\mathcal A}, {\mathcal B}\) of abelian varieties \(A, B\) over \(K\). It was conjectured by M. J. Taylor [Ill. J. Math. 32, 428–452 (1988; Zbl 0631.14033)] and later proved by A. Srivastav and M. J. Taylor [Invent. Math. 99, 165–184 (1990; Zbl 0705.14031)] and A. Agboola [Invent. Math. 123, 105–122 (1996; Zbl 0864.11055)] that the associated class-invariant homomorphism \(\psi: {\mathcal B}(S) \rightarrow \text{Pic}(G^D)\) vanishes on torsion points in the case \(A,B\) are elliptic curves and the order of \(G\) is coprime to \(6\).
In the paper under review, the author constructs examples where \(\psi\) does not vanish on torsion points. In these examples \(G=\mu_p\), the abelian variety \(A\) is the Weil restriction of an elliptic curve \(E_{K'}\) over a finite extension \(K'\) of \(K\) where the elements of \(\text{Pic}(R)[p]\) capitulate, \(E_{K'}\) has good reduction over \(p\) and \(E_{K'}(K')\) is a finite group containing an element of order \(p\). A crucial step in the author’s construction is a theorem giving a rather general situation when the coboundary map \({\mathcal B}(S) \rightarrow H^1(S, G)\) is surjective.
In the paper under review, the author constructs examples where \(\psi\) does not vanish on torsion points. In these examples \(G=\mu_p\), the abelian variety \(A\) is the Weil restriction of an elliptic curve \(E_{K'}\) over a finite extension \(K'\) of \(K\) where the elements of \(\text{Pic}(R)[p]\) capitulate, \(E_{K'}\) has good reduction over \(p\) and \(E_{K'}(K')\) is a finite group containing an element of order \(p\). A crucial step in the author’s construction is a theorem giving a rather general situation when the coboundary map \({\mathcal B}(S) \rightarrow H^1(S, G)\) is surjective.
Reviewer: Bernhard Köck (Southampton)
MSC:
| 11G10 | Abelian varieties of dimension \(> 1\) |
| 14K15 | Arithmetic ground fields for abelian varieties |
| 11R33 | Integral representations related to algebraic numbers; Galois module structure of rings of integers |
Keywords:
class-invariant homomorphism; Galois module structure; abelian variety; Néron model; Weil restriction; semi-stable reduction; isogeny; monodromy pairing; Weil biextensionSoftware:
ecdataReferences:
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