Iwasawa theory for motives.

*(English)*Zbl 0727.11043
L-functions and arithmetic, Proc. Symp., Durham/UK 1989, Lond. Math. Soc. Lect. Note Ser. 153, 211-233 (1991).

[For the entire collection see Zbl 0718.00005.]

The author formulates a “main conjecture” for motives which should give a link between the zeros of the p-adic L-function and the structure of the Selmer group. At present, the theory seems to work only under the hypothesis that the compatible system \(V=\{V_{\ell}\}\) of \(\ell\)-adic representations of \(G_ Q\) supplied by the homology of the motive M is ordinary at p.

Let \(L_ V(s)\) be the complex L-function attached to V and assume that the value \(L_ V(1)\) is critical in the sense of Deligne. In general, the existence of a p-adic L-function attached to such V and satisfying the usual interpolation properties is only conjectural. These properties are described at the beginning of the paper. As for the ordinary character of V at the prime p, the proposed definition is the following: There should be a filtration \(F^ iV_ p\) (where \(i\in {\mathbb{Z}})\) of subspaces of \(V_ p\), invariant under \(G_{{\mathbb{Q}}_ p}\), such that:

i) \(F^{i+1}V_ p\subset F^ iV_ P\), \(F^ iV_ p=V_ p\) for \(i\ll 0\), \(F^ iV_ p=0\) for \(i\gg 0.\)

ii) The inertia group \(I_{{\mathbb{Q}}_ p}\) acts on \(gr^ i(V_ p)=F^ iV_ p/F^{i+1}V_ p\) by the i-th power of the p-cyclotomic character.

Let \(T_ p\) be a \(G_{{\mathbb{Q}}}\)-invariant lattice in \(V_ p\) and let \(A=V_ p/T_ p\). The author defines a Selmer group \(S_ A({\mathbb{Q}}_{\infty})\) and conjectures that its Pontryagin dual \(X=S_ A({\mathbb{Q}}_{\infty})\) is a torsion module over the Iwasawa algebra \(\Lambda\), so that its characteristic ideal \((\lambda_ A)\) should be nonzero. The “main conjecture” predicts a relationship between the generators of this ideal and the zeros and the poles of the p-adic L- functions \(L_ p(\phi,V)\), for some characters \(\phi \in Hom_{cont}(\Gamma,{\mathbb{C}}_ p^{\times})\). Here \(\Gamma\) denotes, as usual, the Galois group of the cyclotomic \({\mathbb{Z}}_ p\)-extension of \({\mathbb{Q}}.\)

The definition of the Selmer group attached to V proposed by the author depends on the choice of the lattice \(T_ p\). This fact motivates the appearance of some extra rational factors in the formulation of the “main conjecture”. The truth of the conjecture, as formulated in the paper, is independent of the choice of the lattice \(T_ p\) and is compatible with exact sequences. For \(M={\mathbb{Q}}(-n)\), it is a reformulation of the classical “Main Conjecture” of the Iwasawa theory, proved by Mazur and Wiles in 1984 and by Rubin in 1989.

If E is an elliptic curve defined over \({\mathbb{Q}}\) with good, ordinary reduction at p, Greenberg’s conjecture for \(V=\{V_{\ell}(E)\}\) is equivalent to a conjecture formulated by Mazur in 1972 and it is known to be true if E has complex multiplication.

Further interesting examples which show the viability of the conjecture are also discussed.

The author formulates a “main conjecture” for motives which should give a link between the zeros of the p-adic L-function and the structure of the Selmer group. At present, the theory seems to work only under the hypothesis that the compatible system \(V=\{V_{\ell}\}\) of \(\ell\)-adic representations of \(G_ Q\) supplied by the homology of the motive M is ordinary at p.

Let \(L_ V(s)\) be the complex L-function attached to V and assume that the value \(L_ V(1)\) is critical in the sense of Deligne. In general, the existence of a p-adic L-function attached to such V and satisfying the usual interpolation properties is only conjectural. These properties are described at the beginning of the paper. As for the ordinary character of V at the prime p, the proposed definition is the following: There should be a filtration \(F^ iV_ p\) (where \(i\in {\mathbb{Z}})\) of subspaces of \(V_ p\), invariant under \(G_{{\mathbb{Q}}_ p}\), such that:

i) \(F^{i+1}V_ p\subset F^ iV_ P\), \(F^ iV_ p=V_ p\) for \(i\ll 0\), \(F^ iV_ p=0\) for \(i\gg 0.\)

ii) The inertia group \(I_{{\mathbb{Q}}_ p}\) acts on \(gr^ i(V_ p)=F^ iV_ p/F^{i+1}V_ p\) by the i-th power of the p-cyclotomic character.

Let \(T_ p\) be a \(G_{{\mathbb{Q}}}\)-invariant lattice in \(V_ p\) and let \(A=V_ p/T_ p\). The author defines a Selmer group \(S_ A({\mathbb{Q}}_{\infty})\) and conjectures that its Pontryagin dual \(X=S_ A({\mathbb{Q}}_{\infty})\) is a torsion module over the Iwasawa algebra \(\Lambda\), so that its characteristic ideal \((\lambda_ A)\) should be nonzero. The “main conjecture” predicts a relationship between the generators of this ideal and the zeros and the poles of the p-adic L- functions \(L_ p(\phi,V)\), for some characters \(\phi \in Hom_{cont}(\Gamma,{\mathbb{C}}_ p^{\times})\). Here \(\Gamma\) denotes, as usual, the Galois group of the cyclotomic \({\mathbb{Z}}_ p\)-extension of \({\mathbb{Q}}.\)

The definition of the Selmer group attached to V proposed by the author depends on the choice of the lattice \(T_ p\). This fact motivates the appearance of some extra rational factors in the formulation of the “main conjecture”. The truth of the conjecture, as formulated in the paper, is independent of the choice of the lattice \(T_ p\) and is compatible with exact sequences. For \(M={\mathbb{Q}}(-n)\), it is a reformulation of the classical “Main Conjecture” of the Iwasawa theory, proved by Mazur and Wiles in 1984 and by Rubin in 1989.

If E is an elliptic curve defined over \({\mathbb{Q}}\) with good, ordinary reduction at p, Greenberg’s conjecture for \(V=\{V_{\ell}(E)\}\) is equivalent to a conjecture formulated by Mazur in 1972 and it is known to be true if E has complex multiplication.

Further interesting examples which show the viability of the conjecture are also discussed.

Reviewer: P.Bayer (Barcelona)

##### MSC:

11R23 | Iwasawa theory |

11G09 | Drinfel’d modules; higher-dimensional motives, etc. |

14F30 | \(p\)-adic cohomology, crystalline cohomology |

11S40 | Zeta functions and \(L\)-functions |