NON-COMMUTATIVE IWASAWA THEORY

R. SUJATHA

1. Motivation

A typical problem motivating non-commutative Iwasawa theory is the following from the arithmetic of elliptic curves:

Let E be an elliptic curve over ℚ, p an odd prime number and F_{∞} a Galois extension of ℚ whose Galois group is a p-adic Lie group of dimension ≥ 1. How does the rank of the Mordell-Weil group E(L) vary as L ranges over all finite extensions of ℚ contained in F_{∞}?

The classical commutative example of this situation is when F_{∞} is obtained by adjoining all p-power roots of unity to ℚ, i.e. F_{∞} = ℚ(μ_{p^{∞}}). In this case, a recent deep theorem of Kato asserts that the rank of E(F_{∞}) is finite. Perhaps the simplest non-commutative example is the field F_{∞} = ⋃_{n≥0} ℚ(μ_{p^{∞}}, a^{1/p^{n}}), a ∈ ℚ^{×}, a ≠ ±1. In this case the Galois group G(F_{∞}/ℚ(μ_{p})) ≅ ℤ_{p}×ℤ_{p}, a semidirect product of two copies of ℤ_{p}.

Theorem 1. (Hachimori-Venjakob) Let F_{m} = ℚ(μ_{p^{m+1}}, a^{1/p^{m}}), m ≥ 0. Then

rk_{ℤ}(E(F_{m})) ≤ c p^{m} ,

where c is a constant independent of m.

2. Iwasawa theory

Let G be a profinite p-adic Lie group. The Iwasawa algebra of G, denoted by Λ(G), is the completed group algebra

Λ(G) = lim_{U⊂G} ℤ_{p}[G/U] ,

where U varies over the open normal subgroups of G. This Iwasawa algebra appears to be a very interesting example of a non-commutative algebra which is of great importance in number theory.

A torsion Λ(G)-module M is said to be pseudonull if Ext_{Λ(G)}^{1}(M, Λ(G)) = 0. Every M has a maximal pseudonull submodule which we denote by M^{∘}.

Theorem 2. (Coates, Schneider, Sujatha) Let G be a compact p-valued Lie group in the sense of Lazard, and let M be any torsion, finitely generated left Λ(G)- module. Then there exist non-zero reflexive left-ideals L_{1}, …, L_{m} of Λ(G) and an exact sequence

0 → ⨁_{i=1}^{m} Λ(G)/L_{i} → M/M^{∘} → D → 0

where D is a pseudonull Λ(G)-module.

This structure theorem is a generalization of an analogous theorem when G = ℤ_{p}^{d}.

Numerical examples: Let E/ℚ be the curve X_{1}(11) of conductor 11, let p = 5 and F = ℚ(E_{5^{∞}}). The Galois group G = G(F_{∞}/ℚ) is of dimension 4. Let F_{n} = ℚ(E_{5^{n+1}}), then Coates-Howson show, using techniques from Iwasawa theory, that rk_{ℤ}E(F_{n}) ≤ 4(5^{3n} − 5^{2n−1}) ∀n ≥ 1.

3. A conjecture

Definition. The fine Selmer group R(E/F_{∞}) of E/F_{∞}, where F_{∞} is a p-adic Lie extension, is defined to be

ker (H^{1}(F_{∞}, E_{p^{∞}}) → ∏_{ν} H^{1}(F_{∞,ν}, E_{p^{∞}}))

where the map is the natural restriction map.

Let Y (E/F_{∞}) denote the compact Pontryagin dual of R(E/F_{∞}). This module has a natural structure as a finitely generated Λ(G)-module. It is conjectured to always be a torsion Λ(G)-module when F_{∞} ⊃ ℚ(μ_{p^{∞}}). We make the following stronger conjecture:

Conjecture (Coates–Sujatha). For any p-adic Lie extension F_{∞}/ℚ of dimension > 1 and containing ℚ(μ_{p^{∞}}), Y (E/F_{∞}) is always pseudonull.

Numerical example: Let E/ℚ be the elliptic curve y^{2} + xy = x^{3} − 3x − 3, of conductor N = 150. Let p = 5 and F_{∞} = ℚ(E_{p^{∞}}). Again G = G(F_{∞}/ℚ) is a p-adic Lie group of dimension 4.

Theorem 3. (Coates, Sujatha) In the above example, either Y (E/F_{∞}) is pseudonull as a Λ(G)-module or E(F_{∞}) has no point of infinite order.

Remark. We strongly suspect that the latter case in the theorem does not occur.

References

[1] Coates, J., Howson, S.; Euler characteristics and elliptic curves II, Jour. Math. Soc. Japan, 53, 2001, 175-235.

[2] Coates, J., Schneider, P., Sujatha, R., Modules over Iwasawa algebras, Journal of Inst. Math. Jussieu, 2, 2003, 73-108.

[3] Hachimori, Y., Venjakob, O., Completely faithful Selmer groups over Kummer extensions, Doc. Mathematika, To appear.