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Compatibility of arithmetic and algebraic local constants (the case \(\ell \neq p\)). (English) Zbl 1335.11044

Let \(F\) be a number field with absolute Galois group \(G:=\mathrm{Gal}(\overline{F}/F)\) and, for any prime \(v\) of \(F\), let \(G_v\) be the decomposition group of \(v\) in \(G\). Fix \(p\neq 2\) and let \(\mathcal{K}\) be a \(p\)-adic field (i.e., a finite extension of \(\mathbb{Q}_p\)) with residue field \(k\): denote by \(V\) a finite dimensional \(\mathcal{K}\)-vector space with a continuous \(G_v\)-action and a non-degenerate \(G_v\)-equivariant skew-symmetric pairing \[ \langle\,,\,\rangle : V\times V \rightarrow \mathcal{K}(1):=\mathcal{K}\otimes_{\mathbb{Z}_p} \mathbb{Z}_p(1). \] Assume \(v\nmid p\) and let \(T\) be a \(G_v\)-stable self-dual (with respect to \(\langle\,,\,\rangle\)) lattice in \(V\). Define \[ H^1_f(T):=\mathrm{Ker}\left\{ H^1(G_v,T)\rightarrow H^1(G_v,V)/H^1_{\mathrm{ur}}(G_v,V) \right\}, \] where \(H^1_{\mathrm{ur}}(G_v,V):=H^1(G_v/I_v,V^{I_v})\) is the unramified cohomology, i.e., \(I_v\) is the inertia subgroup of \(G_v\) and \[ \mathfrak{F}(T):=\mathrm{Im}\left\{ H^1_f(T)\rightarrow H^1(G_v,T/\pi)\right\}, \] where \(\pi\) is a uniformizer for \(\mathcal{K}\) and all maps are the natural ones. For a pair of lattices \(T\), \(T'\) isomorphic modulo \(\pi\), B. Mazur and K. Rubin [Ann. Math. (2) 166, No. 2, 579–612 (2007; Zbl 1219.11084)] showed that the \(\mathfrak{F}(T)\) are Lagrangian subspaces of \(H^1(G_v,T/\pi)\) and introduced an arithmetic local constant \[ d(\mathfrak{F}(T),\mathfrak{F}(T')):= \dim_k \left(\mathfrak{F}(T)/\mathfrak{F}(T)\cap\mathfrak{F}(T')\right) \pmod{2}. \] In the paper under review, the author shows a deep relation between this constant and the one defined by P. Deligne [Lect. Notes Math. 349, 501–597 (1973; Zbl 0271.14011)] (denoted by \(\varepsilon(V)\,\)), i.e., \[ (-1)^{d(\mathfrak{F}(T),\mathfrak{F}(T'))} = \varepsilon(V)/\varepsilon(V'). \] The same relation is shown to hold in a couple of cases when \(v\mid p\) (adjusting the definition of the \(H^1_f(T)\) à la Bloch-Kato and using the local constant \(\varepsilon(V)\) associated to the representation of the Weil-Deligne module defined in [J.-M. Fontaine, in: Périodes \(p\)-adiques. Séminaire du Bures-sur-Yvette, France, 1988. Paris: Société Mathématique de France. 321–347 (1994; Zbl 0873.14020)]), in particular under hypotheses which guarantee that \(V\) and \(V'\) are crystalline representations.
The author then applies this local equalities to the global situation in which the \(V\)’s are \(v\)-adic realizations of some motive \(M\), the \(\mathfrak{F}(T)\) define Selmer structures and the global constants are product of (finitely many) local ones. In the particular case of elliptic curves with CM defined over a totally real field, this provides new cases of the \(p\)-parity conjecture which serve as a complement to the ones already appeared in [the author, Algebra Number Theory 7, No. 5, 1101–1120 (2013; Zbl 1368.11059)].

MSC:

11G05 Elliptic curves over global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11S40 Zeta functions and \(L\)-functions
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