Computing in Component Groups of Elliptic Curves
Let K be a p-adic local field and E an elliptic curve defined over K. The component group of E is the group E(K)/E 0(K), where E 0(K) denotes the subgroup of points of good reduction; this is known to be finite, cyclic if E has multiplicative reduction, a
- PDF / 390,543 Bytes
- 7 Pages / 430 x 660 pts Page_size
- 85 Downloads / 222 Views
Abstract. Let K be a p-adic local field and E an elliptic curve defined over K. The component group of E is the group E(K)/E 0 (K), where E 0 (K) denotes the subgroup of points of good reduction; this is known to be finite, cyclic if E has multiplicative reduction, and of order at most 4 if E has additive reduction. We show how to compute explicitly an isomorphism E(K)/E 0 (K) ∼ = Z/N Z or E(K)/E 0 (K) ∼ = Z/2Z×Z/2Z.
1
Introduction
Let K be a p-adic local field (that is, a finite extension of Qp for some prime p), with ring of integers R, uniformizer π, residue field k = R/(π) and valuation function v. Let E be an elliptic curve defined over K. The component group of E is the finite abelian group Φ = E(K)/E 0 (K), where E 0 (K) denotes the subgroup of points of good reduction. When E has split multiplicative reduction, we have Φ ∼ = Z/N Z, where N = v(Δ) and Δ is the discriminant of a minimal model for E. In all other cases, Φ has order at most 4, so is isomorphic to Z/nZ with n ∈ {1, 2, 3, 4} or to Z/2Z × Z/2Z. The order of Φ is called the Tamagawa number of E/K, usually denoted c or cp . In this note we will show how to make the isomorphism κ : E(K)/E 0 (K) → A explicit, where A is the one of the above standard abelian groups. The most interesting case is that of split multiplicative reduction. Here the map κ is almost determined by a formula for the (local) height in [3]. Specifically, if the minimal Weierstrass equation for E has coefficients a1 , a2 , a3 , a4 , a6 as usual, for a point P = (x, y) ∈ E(K) \ E 0 (K) we have κ(P ) = ±n (mod N ), where n = min{v(2y + a1 x + a3 ), N/2}, and 0 < n ≤ N/2. In computing heights, of course, one need not distinguish between P and −P , but for our purposes this is essential. We show how to determine the appropriate sign in a consistent way to give an isomorphism κ : E(K)/E 0 (K) ∼ = Z/N Z. Note that for an individual point this is not a well-defined question since negation gives an automorphism of Z/N Z; but when comparing the values of κ at two or more points it is important. We first establish the formula for Tate curves, and then see how to apply it to a general minimal Weierstrass model. We also make some remarks about the other reduction types, which are much simpler to deal with, and also the real case. A.J. van der Poorten and A. Stein (Eds.): ANTS-VIII 2008, LNCS 5011, pp. 118–124, 2008. c Springer-Verlag Berlin Heidelberg 2008
Computing in Component Groups of Elliptic Curves
119
One application for this, which was our motivation, occurs in the determination of the full Mordell-Weil group E(K), where E is an elliptic curve defined over a number field K. Given a subgroup B of E(K) of full rank, generated by r independent points Pi for 1 ≤ i ≤ r, one method for extending this to a Z-basis for E(K) (modulo torsion) requires determining the index in B of B ∩ p≤∞ E 0 (Qp ). [For p = ∞, we denote as usual R = Q∞ , and then E 0 (Qp ) denotes the connected component of the identity in E(R).] The component group maps κ for each prime p may be used for this, and are accordingly
Data Loading...
