×

Torsion points on elliptic curves and Galois module structure. (English) Zbl 0864.11055

Let \(E\) be an elliptic curve defined over a number field \(F\) with everywhere good reduction. For a prime \(p>3\) let \({\mathfrak B}_i(F)\) denote the \({\mathcal O}_F\)-Hopf algebra which belongs to the \({\mathcal O}_F\)-group scheme of \(p^i\)-torsion points on \(E\). Let furthermore \({\mathfrak A}_i(F)\) be its Cartier dual. For any point \(Q\in E(F)\) let \(G_Q(i)= \{Q'\in E (\overline F) \mid p^iQ'=Q\}\) and the Kummer algebra \(F_Q(i)= \text{Map} (G_Q(i), \overline F)^{\text{Gal} (\overline F/F)}\). One can now consider the \({\mathfrak A}_i(F)\)-module \({\mathfrak C}_Q(i)\) which is the “modified” integral closure of \({\mathcal O}_F\) in \(F_Q(i)\).
The author proves that \({\mathfrak C}_Q(i)\) is a globally free \({\mathfrak A}_i(F)\)-module if \(Q\) is a torsion point in \(E(F)\). This generalizes a result of A. Srivastav and M. J. Taylor [Invent. Math. 99, 165-184 (1990; Zbl 0705.14031)] who proved the same result under the additional assumption that \(E\) has complex multiplication. The proof depends upon the use of intersection theory on arithmetic surfaces.
Reviewer: H.-G.Rück (Essen)

MSC:

11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11G05 Elliptic curves over global fields
14G25 Global ground fields in algebraic geometry
14H25 Arithmetic ground fields for curves

Citations:

Zbl 0705.14031
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] [A1] A. Agboola: A geometric interpretation of the class invariant homomorphism. J. Théorie Nombres de Bordeaux6 (1994) 273-280
[2] [A2] A. Agboola: A note on elliptic curves and Galois module structure in global function fields. Amer. J. Math. (to appear)
[3] [BT] N. Byott, M.J. Taylor: Hopf orders and Galois module structure. In: Group rings and classgroups, K.W. Roggenkamp, M.J. Taylor (eds), Birkhauser, Basel Boston 1992, pp. 53-210. · Zbl 0811.11068
[4] [CNS] Ph. Cassou-Noguès, A. Srivastav: On Taylor’s conjecture for Kummer orders. Seminaire de Théorie des Nombres de Bordeaux2 (1990) 349-363
[5] [CNT] Ph. Cassou-Noguès, M.J. Taylor: Elliptic functions and rings of integers. Birkhauser, Basel Boston 1987 · Zbl 0621.12012
[6] [G] B. Gross: Local heights on curves. In: Arithmetic geometry, G. Cornell, S. Silverman (eds) Springer, Berlin Heidelberg Newyork. 1986, pp. 327-339
[7] [L] S. Lang: Introduction to Arakelov theory. Springer, Berlin Heidelberg New York 1988 · Zbl 0667.14001
[8] [S] J.-P. Serre: Lectures on the Mordell-Weil theorem. Vieweg, Wiesbaden 1989
[9] [Sh] S. Shatz: Group schemes, formal groups andp-divisible groups. In: Arithmetic geometry, G. Cornell, S. Silverman (eds) Springer, Berlin Heidelberg New York 1986, pp. 29-78
[10] [Si] J. Silverman: The arithmetic of elliptic curves. Springer, Berlin Heidelberg New York 1986 · Zbl 0585.14026
[11] [ST] A. Srivastav, M.J. Taylor: Elliptic curves with complex multiplication and Galois module structure. Invent. Math.99 (1990) 165-184 · Zbl 0705.14031
[12] [T] M.J. Taylor: Mordell-Weil groups and the Galois module structure of rings of integers. Ill. J. Math.32 (1988) 428-452 · Zbl 0631.14033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.