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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)].

##### MSC:
 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11R23 Iwasawa theory
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