Perrin-Riou's local Tamagawa-number conjecture

Let $V$ be a crystalline representation of $G_\ell$ over $K_\lambda$ , and let $T\subseteq V$ be a $G_\ell$ -stable $\mathcal{O}_\lambda$ -lattice. Let $\omega$ and $\omega^*$ be lattices in

\operatorname{det}_{K_\lambda}D(V)/\operatorname{Fil}^0D(V)

and

\operatorname{det}_{K_\lambda}D(V^*(1))/\operatorname{Fil}^0D(V^*(1)),

respectively, and let $\eta(T,\omega,\omega^*)$ be the comparison-isomorphism factor defined in the source. For $j\in\mathbb{Z}$ , set

h_j(V)=\dim_{K_\lambda}\operatorname{Fil}^jD(V)/\operatorname{Fil}^{j+1}D(V),

and define $\Gamma^*(j)=(j-1)!$ for $j>0$ and $\Gamma^*(j)=(-1)^j((-j)!)^{-1}$ for $j\leq0$ . Perrin-Riou's local Tamagawa-number conjecture.

\mathcal{O}_\lambda\frac{\operatorname{Tam}_{\ell,\omega}^0(T)}{\operatorname{Tam}_{\ell,\omega^*}^0(T^*(1))}=\mathcal{O}_\lambda\prod_j\Gamma^*(-j)^{-h_j(V)}\eta(T,\omega,\omega^*)^{-1}.

This conjecturally relates the ratio of local Tamagawa factors to Hodge-filtration gamma factors and the crystalline comparison factor. It is invoked as a generalization of Perrin-Riou's conjecture from $\mathbb{Z}_\ell$ to $\mathcal{O}_\lambda$ coefficients.

Perrin-Riou's local Tamagawa-number conjecture

Let $K$ be a finite extension of $\mathbf{Q}_p$ , let $V=\mathbf{Q}_p\otimes_{\mathbf{Z}_p}T$ be a potentially semistable representation of $G_K$ , and choose bases $\omega_1\in\operatorname{det}_{\mathbf{Q}_p}t_V(K)$ , $\omega_2\in\operatorname{det}_{\mathbf{Q}_p}t_{V^*(1)}(K)$ and $\omega\in\operatorname{det}_{\mathbf{Q}_p}\mathbf{D}_{\mathrm{dR}}^K(V)$ satisfying $\omega\simeq\omega_2^{-1}\otimes\omega_1$ . Let $\alpha_{V,K}(\omega,T)$ , $\varepsilon(K,V)$ , $\Gamma^*(V)$ , $d_K$ , and the Tamagawa numbers $\operatorname{Tam}_{K,\omega_1}^0(T)$ and $\operatorname{Tam}_{K,\omega_2}^0(T^*(1))$ be as in the source.

$C_{\mathrm{EP}}(K,V)$ .

\frac{\operatorname{Tam}_{K,\omega_1}^0(T)}{\operatorname{Tam}_{K,\omega_2}^0(T^*(1))}=|d_K|_p^{\dim V/2}\left|\Gamma^*(V)\frac{\alpha_{V,K}(\omega,T)}{\varepsilon(K,V)}\right|_p.

For $L=K$ , the source reformulates the equivariant Euler–Poincaré conjecture in terms of local Tamagawa numbers. It attributes this formulation to Fukaya–Kato and states no resolution in the supplied text.

source: D. Benois and L. Berger, “Théorie d'Iwasawa des représentations cristallines II”, arXiv:math/0509623 (2005).

Perrin-Riou's local Tamagawa-number conjecture

Perrin-Riou's local Tamagawa-number conjecture

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