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

Agboola, A.

Invent. Math. 123, No. 1, 105-122 (1996).

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

Keywords:

\(p^ i\)-torsion points; modified integral closure; Galois module structure; elliptic curve

Citations:

Zbl 0705.14031

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