On the parity of ranks of Selmer groups. III.

*(English)*Zbl 1201.11067
Doc. Math. 12, 243-274 (2007); erratum 14, 191-194 (2009).

In this article, the author studies the invariance of the parity conjecture for Selmer groups under deformation in \(p\)-adic families of representations.

In fact, let \(F\) and \(L\) be number fields inside some fixed algebraic closure of the rationals, and let \(M\) be a motive over \(F\) with coefficients in \(L\). The \(L\)-function of \(M\), if it is well defined, is a Dirichlet series \(\sum a_n n^{-s}\). For each embedding \(\iota\) of the algebraic closure of the rationals into the complex numbers, the \(L\)-function \[ L(\iota M, s) = \sum_{n \geq 1} \iota(a_n) n^{-s} \] converges absolutely for all \(s\) with \(\text{Re } s \gg 0\). It is expected to admit a meromorphic continuation to the whole complex plane and to satisfy a functional equation.

Let \(p\) be a prime number, \({\mathfrak p}\) a prime in \(L\) above \(p\), and \(F_S\) the maximal extension of \(F\) unramified outside a suitable finite set \(S\) of primes of \(F\). The \({\mathfrak p}\)-adic realization \(M_{\mathfrak p}\) of \(M\) is a finite-dimensional \(L_{\mathfrak p}\)-vector space, on which the Galois group \(G_{F,S}\) acts continuously.

By conjectures due to Bloch and Kato, generalized by Fontaine and Perrin-Riou, the order of vanishing of \(L(\iota M,s)\) at \(s = 0\) is equal to the difference of the \(L_{\mathfrak p}\)-dimensions \(h_f^1 (F,M_{\mathfrak p}^*(1))\) and \(h^0 (F,M_{\mathfrak p}^*(1))\) of the generalized Selmer groups \(H_f^1 (F,M_{\mathfrak p}^*(1))\) and \(H^0 (F,M_{\mathfrak p}^*(1))\).

In the special case when the motive \(M\) is self-dual and pure, the parity of \(\text{ord}_{s=0} L(\iota M,s)\) should then be determined by the global \(\varepsilon\)-factor \(\varepsilon(M)\), and the parity conjecture predicts that \[ \text{ord}_{s=0} L(\iota M,s) \equiv h_f^1 (F,M_{\mathfrak p}^*(1)) \bmod 2, \] or, equivalently, that \[ (-1)^{h_f^1 (F,M_{\mathfrak p}^*(1))} = \varepsilon(M) = \prod_v \varepsilon_v(M_{\mathfrak p}). \] These local \(\varepsilon\)-factors can be expressed in terms of the Galois representation alone, and one is led to the conjecture that \[ (-1)^{h_f^1 (F,M_{\mathfrak p}^*(1))} = \prod_v \varepsilon_v(V) \] for any symplectically self-dual geometric pure representation \(V\) of \(G_{F,S}\).

The author shows that this conjecture is invariant under \(p\)-adic deformation, that is: if \(V\) and \(V'\) are representations of \(G_{F,S}\) as above belonging to the same \(p\)-adic family which, in addition, satisfies the Panchishkin condition at all primes \(v \mid p\), then \[ (-1)^{h_f^1 (F,V)} /\varepsilon(V) = (-1)^{h_f^1 (F,V')} /\varepsilon(V'). \]

In the Erratum, the erroneous treatment of the archimedean \(\varepsilon\)-factors is corrected.

For Parts I and II, see [Asian J. Math. 4, No. 2, 437–497 (2000; Zbl 0973.11066)] and [C. R. Acad. Sci., Paris, Sér. I, Math. 332, No. 2, 99–104 (2001; Zbl 1090.11037)]. Part IV has been published in [Compos. Math. 145, No. 6, 1351–1359 (2009; Zbl 1221.11150)].

In fact, let \(F\) and \(L\) be number fields inside some fixed algebraic closure of the rationals, and let \(M\) be a motive over \(F\) with coefficients in \(L\). The \(L\)-function of \(M\), if it is well defined, is a Dirichlet series \(\sum a_n n^{-s}\). For each embedding \(\iota\) of the algebraic closure of the rationals into the complex numbers, the \(L\)-function \[ L(\iota M, s) = \sum_{n \geq 1} \iota(a_n) n^{-s} \] converges absolutely for all \(s\) with \(\text{Re } s \gg 0\). It is expected to admit a meromorphic continuation to the whole complex plane and to satisfy a functional equation.

Let \(p\) be a prime number, \({\mathfrak p}\) a prime in \(L\) above \(p\), and \(F_S\) the maximal extension of \(F\) unramified outside a suitable finite set \(S\) of primes of \(F\). The \({\mathfrak p}\)-adic realization \(M_{\mathfrak p}\) of \(M\) is a finite-dimensional \(L_{\mathfrak p}\)-vector space, on which the Galois group \(G_{F,S}\) acts continuously.

By conjectures due to Bloch and Kato, generalized by Fontaine and Perrin-Riou, the order of vanishing of \(L(\iota M,s)\) at \(s = 0\) is equal to the difference of the \(L_{\mathfrak p}\)-dimensions \(h_f^1 (F,M_{\mathfrak p}^*(1))\) and \(h^0 (F,M_{\mathfrak p}^*(1))\) of the generalized Selmer groups \(H_f^1 (F,M_{\mathfrak p}^*(1))\) and \(H^0 (F,M_{\mathfrak p}^*(1))\).

In the special case when the motive \(M\) is self-dual and pure, the parity of \(\text{ord}_{s=0} L(\iota M,s)\) should then be determined by the global \(\varepsilon\)-factor \(\varepsilon(M)\), and the parity conjecture predicts that \[ \text{ord}_{s=0} L(\iota M,s) \equiv h_f^1 (F,M_{\mathfrak p}^*(1)) \bmod 2, \] or, equivalently, that \[ (-1)^{h_f^1 (F,M_{\mathfrak p}^*(1))} = \varepsilon(M) = \prod_v \varepsilon_v(M_{\mathfrak p}). \] These local \(\varepsilon\)-factors can be expressed in terms of the Galois representation alone, and one is led to the conjecture that \[ (-1)^{h_f^1 (F,M_{\mathfrak p}^*(1))} = \prod_v \varepsilon_v(V) \] for any symplectically self-dual geometric pure representation \(V\) of \(G_{F,S}\).

The author shows that this conjecture is invariant under \(p\)-adic deformation, that is: if \(V\) and \(V'\) are representations of \(G_{F,S}\) as above belonging to the same \(p\)-adic family which, in addition, satisfies the Panchishkin condition at all primes \(v \mid p\), then \[ (-1)^{h_f^1 (F,V)} /\varepsilon(V) = (-1)^{h_f^1 (F,V')} /\varepsilon(V'). \]

In the Erratum, the erroneous treatment of the archimedean \(\varepsilon\)-factors is corrected.

For Parts I and II, see [Asian J. Math. 4, No. 2, 437–497 (2000; Zbl 0973.11066)] and [C. R. Acad. Sci., Paris, Sér. I, Math. 332, No. 2, 99–104 (2001; Zbl 1090.11037)]. Part IV has been published in [Compos. Math. 145, No. 6, 1351–1359 (2009; Zbl 1221.11150)].

Reviewer: Franz Lemmermeyer (Jagstzell)

##### MSC:

11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |

11R23 | Iwasawa theory |