## 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

### Citations:

Zbl 1219.11084; Zbl 0271.14011; Zbl 0873.14020; Zbl 1368.11059
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### References:

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