An analogue of the Washington-Sinnott theorem for elliptic curves with complex multiplication I

Abstract

In this paper, we generalize Washington's theorem on the stabilization of the p-part of the ideal class groups in the cyclotomic Z q -extension of an abelian number field for distinct primes p and q. We fix an imaginary quadratic field K and a split prime q of K lying above q and let K ∞/K denote the Z q -extension which is unramified outside q. We show that if F/K is a finite abelian extension, and if F ∞ = F K∞, then, for a prime p satisfying certain conditions, the p-part of the class groups stabilize in the Z p-extension F∞/F . We also show that if E/K is an elliptic curve with complex multiplication by the ring of integers of K, then under the same conditions on a prime p, the p-part of the Selmer groups for E/F n stabilize as Fn runs over the finite layers of F ∞/F . Theorem (Washington). If k is any finite abelian extension of Q, then, for all primes p = q, v p (h m ) is constant for m sufficiently large.Here v p is the usual p-adic valuation, so that, for an integer n, v p (n) = e 0, where p e is the largest power of p dividing n. The theorem of Washington has the following generalization, due to Friedman; see [6]. Fix S to be a finite set of rational primes, and let Q S denote the compositum of all the cyclotomic Z q -extensions of Q for all primes q ∈ S. If k is any finite abelian extension of Q, let k S = kQ S . If we denote the class number of a finite extension F of Q by h F , then we have the following theorem.Theorem (Friedman-Washington). If k is any finite abelian extension of Q, then, for allRemark. It is conjectured that this theorem holds for arbitrary finite extensions k of Q.In the case where A is an elliptic curve defined over Q, the following parallel result is known. We assume the notation above, but assume that k is abelian over Q, and define X S to be the set of finite-order characters of Gal(k S /Q). Then Rohrlich [14] has proved the following theorem.Theorem (Rohrlich). There are only finitely many characters χ ∈ X S such that L(A, χ, 1) = 0.It then follows by some deep results of Kato (see [11, Section 14]) that the Mordell-Weil group A(k S ) is finitely generated, or equivalently that there exists an intermediate extensionRemark. It is not, however, true in general that Sel p ∞ (A/L) stabilizes as L runs over the subfields of k S , which have finite degree over Q. In fact, we have the following example from a recent paper of Tim and Vladimir Dokchitser; see [4, Theorem 1.1 and Example 1.5]. Suppose that k = Q, S = {q} for any prime q, and let Q n be the nth layer of the cyclotomic Z q -extension of Q. If we let A be the curve defined by A : y 2 + y = x 3 − x 2 − 10x − 20, then v 5 (#X 0 (A/Q n )[5 ∞ ]) q n − c q for some constant c q ∈ Z. Here the notation X 0 means the Tate-Shafarevich group X modulo its divisible part X div .

Fine Selmer groups of congruent p-adic Galois representations

“…Indeed, by [Lam15, Theorem 1.2], the hypotheses of the above example imply that in fact the Pontryagin dual of the (ordinary) Selmer group over is finitely generated over . Now we can apply Theorem 4.1.…”

Triviality of the ℓ-class groups in -extensions of for split primes p ≡ 1 modulo 4

“…Given a prime number p, then we have studied the stabilisation of the -class groups in the tower K ∞ /K in the papers [13,14] when ≡ 1 modulo 4. To describe the result we shall need the following notation.…”

Growth of in towers for isogenous curves

“…In the opposite direction, Lamplugh [20] has recently proven the following analogue of the theorem of Washington for elliptic curves E/Q with complex multiplication by the ring of integers of an imaginary quadratic field K. Let p > 3, l > 3 be distinct primes of good reduction of E that split in K/Q. Lamplugh proves that if K n is the nth layer of the unique Z l -extension of K unramified outside one of the factors of l in K, then the p ∞ -Selmer group of E over K n stabilises as n → ∞.…”