On torsion line bundles and torsion points on Abelian varieties
In this paper, the author studies torsion line bundles on Abelian schemes and their restriction to the subschemes of torsion points. This is motivated by problems on the Galois module structure of rings of integers obtained by adjoining values of Abelian functions at torsion points. By definition, an Abelian scheme \(A@> f >>S\) of relative dimension \(g\) is a group scheme that is proper and smooth with geometrically connected fibers of dimension \(g\). Denote by \(0:S\rightarrow A\) the identity section. For any line bundle \({\mathcal L}\) on \(A\), a rigidification of \({\mathcal L}\) is an isomorphism \({\mathcal O}_S @>\sim>> 0^*{\mathcal L}\). For each nonzero integer \(m\), we can consider the (closed) subgroup scheme \(A_m\rightarrow S\) of \(m\)-torsion points of \(A\). The group scheme \(A_m\) is finite and locally free of rank \(m^{2g}\) over \(S\). The first aim of this paper is to show the following theorems. Theorem A. Assume \(g=1\) and \(\gcd(6,m)=1\), and let \({\mathcal L}\) be a torsion line bundle on \(A\) with a rigidification. Then the restriction of \({\mathcal L}\) to the subscheme of \(m\)-torsion points \(A_m\) is a trivial line bundle. Theorem B. Assume \(g=1\), and denote by \({\mathcal P}\) the Poincaré line bundle on \(A\times_S A\). Assume that one of the two nonzero integers \(n\) and \(m\) is coprime to 6. Then the restriction of \({\mathcal P}\) to the subscheme \(A_n\times_S A_m\) is a trivial line bundle. The proofs actually give trivializations for these restrictions. When \(g\geq 2\), the situation is dramatically different. He shows the following. Theorem C. Assume that \(g\geq 2\). For any two distinct prime numbers \(r\), \(l\), there is a smooth affine curve \(D\) over a finite field of characteristic \(r\), an Abelian scheme \(A\rightarrow D\) of relative dimension \(g\), and an \(l\)-torsion line bundle \({\mathcal L}\) on \(A\) with a rigidification, such that the restriction of \({\mathcal L}\) to \(A_l\) is not trivial. The restriction of (torsion) line bundles on the subschemes of torsion points of elliptic curves has been studied extensively in a different formulation as a problem on the Galois module structure of the rings of integers in certain fields that are obtained by adjoining values of elliptic functions. Indeed, suppose that the base \(S\) is \(\text{Spec}(R)\), where \(R\) is the ring of integers of a number field. Then, as it was first observed in [\textit{A. Agboola}, J. Théor. Nombres Bordx. 6, 273-280 (1994; Zbl 0833.11055)], the statement of Theorem A with \(S\) as above, is equivalent to the vanishing of the so-called class-invariant homomorphism on torsion points of order coprime to 6. The class-invariant homomorphism was first introduced by W. Waterhouse. In the context of Abelian varieties, it was investigated by M. J. Taylor, who first considered elliptic curves with complex multiplication. The vanishing of the class-invariant homomorphism on torsion points of elliptic curves with complex multiplication implies the existence of Galois generators for certain rings of integers of Abelian extensions of imaginary quadratic fields. More precisely, it implies that these rings of integers are free over certain Hopf algebras associated to group schemes of torsion points. In [\textit{A. Srivastav} and \textit{M. J. Taylor}, Invent. Math. 99, 165-184 (1990; Zbl 0705.14031)], it was proved that the class invariant homomorphisms vanishes on torsion points of order coprime to 6 for elliptic curves with complex multiplication and everywhere good reduction. This may be regarded as an integral version of Kronecker's Jugendtraum. In [\textit{A. Agboola}, Invent. Math. 123, 105-122 (1996; Zbl 0864.11055)] the same statement was proven for all elliptic curves over number fields with everywhere good reduction. This corresponds to a proof of the theorem in the case \(S= \text{Spec}(R)\), with \(R\) the ring of integers of a number field. The main tool in Srivastav-Taylor's article was the use of special elliptic functions. These were shown to have the required integrality properties by the use of the \(q\)-expansion principle for modular functions. In [Agboola (1996; loc. cit.)] similar elliptic functions were used, but the Neron intersection pairing replaced the role of the \(q\)-expansion principle in proving the required integrality. The author's approach towards the proof of Theorem A is a lot simpler and independent of cosntructions involving special functions. It essentially uses only basic facts regarding families of elliptic curves and applies to any base scheme \(S\). On the author hand, the situation regarding Abelian varieties of dimension greater than 1 has been less clear. Theorem C provides the first examples in which the principal homogeneous spaces obtained by dividing torsion points on Abelian varieties (with everywhere good reduction) do not have trivial Galois module structure.
Galois module structure of rings of integers
scientific article; zbMATH DE number 1425081
scientific article