Kummer theory for abelian varieties over local fields.

*(English)*Zbl 0858.11032In this work the authors introduce a new class of infinite algebraic extensions of \(\mathbb{Q}_p\), which they call deeply ramified extensions, and then deduce several remarkable theorems on the arithmetic of abelian varieties over deeply ramified extensions. The approach is partly motivated by the paper of J. Tate on \(p\)-divisible groups [in: Proc. Conf. Local Fields, NUFFIC Summer School Driebergen 1966, 158-183 (1967; Zbl 0157.27601)].

An algebraic extension \(K\) over \(\mathbb{Q}_p\) is called deeply ramified if it has infinite conductor, that is, the set of breaks of \(K/ \mathbb{Q}_p\) with respect to the upper numbering is unbounded. The authors provide several equivalent reformulations, including the following: \(H^1(K,M) = 0\), where \(M\) is the maximal ideal of the ring of integers of the algebraic closure \(\mathbb{Q}^a_p\) of \(\mathbb{Q}_p\). According to Sen’s theorem [S. Sen, Invent. Math. 17, 44-50 (1972; Zbl 0242.12012)] an infinitely ramified \(p\)-adic Lie extension of a finite extension of \(\mathbb{Q}_p\) is deeply ramified. The authors express their belief that the class of deeply ramified extensions is essentially larger than that of \(p\)-adic Lie extensions, namely, that there exists a deeply ramified extension \(L\) of any finite extension \(K\) of \(\mathbb{Q}_p\) with the property that no subfield \(L'\) of \(L\) is an infinitely ramified \(p\)-adic Lie extension of a finite extension of \(\mathbb{Q}_p\). In section 3 it is shown that if \(F\) is a commutative formal group law in several variables defined over the ring of integers of a finite extension \(K\) of \(\mathbb{Q}_p\), then for a deeply ramified extension \(L\) of \(K\) the groups \(H^i(L,F(M))\) are trivial for \(i\geq 1\).

Let \(A\) be an abelian variety which is defined over a finite extension \(K\) of \(\mathbb{Q}_p\). Let \(L\) be an algebraic extension of \(K\). The preceding results are applied to the study of the local Kummer homomorphism \(\kappa_L:A(L) \otimes \mathbb{Q}_p/ \mathbb{Z}_p \to H^1(L,A [p^\infty])\) where \(A[p^\infty]\) is the \(p\)-primary subgroup of \(A(\mathbb{Q}^a_p)\). The main result is that in the case where \(L\) is a deeply ramified extension, the image of \(\kappa_L\) coincides with the image of \(\lambda_L:H^1(L,C) \to H^1(K,A [p^\infty])\). Here \(C\) is the image of the \(G_K\)-invariant \(\mathbb{Q}_p\)-subspace of the Tate module \(T_p(A)\) tensored with \(\mathbb{Q}_p\), of minimal dimension such that some subgroup of the inertia group of \(G_K\) acts trivially on the quotient space (if \(A\) has semistable reduction, then \(C\) is the \(p\)-primary subgroup of the \(M\)-points of the formal group attached to the Neron model for \(A\) over the ring of integers of \(K)\). If \(A\) has good, ordinary reduction over \(K\), then the latter result holds for any infinitely wildly ramified extension \(L\) of \(K\).

In the last section the authors provide a very general explicit description of the group of the universal norm \(N_{L/K}(A^t)\) of the dual abelian variety of \(A\), which includes all special cases studied earlier by Mazur, Vvedenskij, Hazewinkel, and Schneider. The main theorem states that provided \(im(\kappa_L) = im (\lambda_L)\) there is a canonical dual pairing between the maximal pro-\(p\)-subgroup of \(N_{L/K} (A^t)\) and the quotient group \(im (\pi_K)/(im (\pi_K) \cap\ker (\text{res}_{L/K}))\) with values in \(\mathbb{Q}_p/ \mathbb{Z}_p\). Here \(\pi_K : H^1(K,A[p^\infty]) \to H^1(K,A [p^\infty]/C)\) and \(\text{res}_{L/K} : H^1 (K,A [p^\infty]/C) \to H^1 (L,A[p^\infty]/C)\). Since a lot is known about the homomorphisms \(\pi_K\) and \(\text{res}_{L/K}\), one can describe the group of universal norms in many cases.

An algebraic extension \(K\) over \(\mathbb{Q}_p\) is called deeply ramified if it has infinite conductor, that is, the set of breaks of \(K/ \mathbb{Q}_p\) with respect to the upper numbering is unbounded. The authors provide several equivalent reformulations, including the following: \(H^1(K,M) = 0\), where \(M\) is the maximal ideal of the ring of integers of the algebraic closure \(\mathbb{Q}^a_p\) of \(\mathbb{Q}_p\). According to Sen’s theorem [S. Sen, Invent. Math. 17, 44-50 (1972; Zbl 0242.12012)] an infinitely ramified \(p\)-adic Lie extension of a finite extension of \(\mathbb{Q}_p\) is deeply ramified. The authors express their belief that the class of deeply ramified extensions is essentially larger than that of \(p\)-adic Lie extensions, namely, that there exists a deeply ramified extension \(L\) of any finite extension \(K\) of \(\mathbb{Q}_p\) with the property that no subfield \(L'\) of \(L\) is an infinitely ramified \(p\)-adic Lie extension of a finite extension of \(\mathbb{Q}_p\). In section 3 it is shown that if \(F\) is a commutative formal group law in several variables defined over the ring of integers of a finite extension \(K\) of \(\mathbb{Q}_p\), then for a deeply ramified extension \(L\) of \(K\) the groups \(H^i(L,F(M))\) are trivial for \(i\geq 1\).

Let \(A\) be an abelian variety which is defined over a finite extension \(K\) of \(\mathbb{Q}_p\). Let \(L\) be an algebraic extension of \(K\). The preceding results are applied to the study of the local Kummer homomorphism \(\kappa_L:A(L) \otimes \mathbb{Q}_p/ \mathbb{Z}_p \to H^1(L,A [p^\infty])\) where \(A[p^\infty]\) is the \(p\)-primary subgroup of \(A(\mathbb{Q}^a_p)\). The main result is that in the case where \(L\) is a deeply ramified extension, the image of \(\kappa_L\) coincides with the image of \(\lambda_L:H^1(L,C) \to H^1(K,A [p^\infty])\). Here \(C\) is the image of the \(G_K\)-invariant \(\mathbb{Q}_p\)-subspace of the Tate module \(T_p(A)\) tensored with \(\mathbb{Q}_p\), of minimal dimension such that some subgroup of the inertia group of \(G_K\) acts trivially on the quotient space (if \(A\) has semistable reduction, then \(C\) is the \(p\)-primary subgroup of the \(M\)-points of the formal group attached to the Neron model for \(A\) over the ring of integers of \(K)\). If \(A\) has good, ordinary reduction over \(K\), then the latter result holds for any infinitely wildly ramified extension \(L\) of \(K\).

In the last section the authors provide a very general explicit description of the group of the universal norm \(N_{L/K}(A^t)\) of the dual abelian variety of \(A\), which includes all special cases studied earlier by Mazur, Vvedenskij, Hazewinkel, and Schneider. The main theorem states that provided \(im(\kappa_L) = im (\lambda_L)\) there is a canonical dual pairing between the maximal pro-\(p\)-subgroup of \(N_{L/K} (A^t)\) and the quotient group \(im (\pi_K)/(im (\pi_K) \cap\ker (\text{res}_{L/K}))\) with values in \(\mathbb{Q}_p/ \mathbb{Z}_p\). Here \(\pi_K : H^1(K,A[p^\infty]) \to H^1(K,A [p^\infty]/C)\) and \(\text{res}_{L/K} : H^1 (K,A [p^\infty]/C) \to H^1 (L,A[p^\infty]/C)\). Since a lot is known about the homomorphisms \(\pi_K\) and \(\text{res}_{L/K}\), one can describe the group of universal norms in many cases.

Reviewer: I.Fesenko (Nottingham)

##### MSC:

11G25 | Varieties over finite and local fields |

14K15 | Arithmetic ground fields for abelian varieties |

11S15 | Ramification and extension theory |