The Bloch–Kato Tamagawa number conjecture for the standard representation of a Siegel modular form

Let $\rho^\theta_\blambda$ be the $p$-adic realization of the motive associated with the Siegel modular form, let $\rho^\theta_\blambda atural\theta$ denote its twist by $\theta$, and let $H_f^1(\mathbb{Q},\rho^\theta_\blambda atural\theta)$ be the Bloch–Kato Selmer group. Write $\mathrm{Tam}(\rho^\theta_\blambda atural\theta)$ for its Tamagawa number and $P^{\mathrm{mot}}_\varphi$ for the corresponding Deligne period. Then the $p$-part of the Bloch–Kato Tamagawa number conjecture predicts, up to units in $\mathcal{O}$,

$$\frac{L^{Np\infty}(1,\mathrm{St}(\pi)\otimes\xi) \widetilde{L}_{Np\infty}(1, \mathrm{St}(\pi)\otimes\xi)}{P^\mathrm{mot}_\varphi}=\chi(H_f^1(\mathbb{Q},\rho_\mathrm{st}\circ\rho_\pi\otimes\xi))\,\mathrm{Tam}(\rho^\mathrm{st}_\pi\otimes\xi).$$

This is the conjectural Bloch–Kato interpretation of the special standard $L$-value in terms of a Selmer group, Tamagawa factors, and a motivic period. The paper's main theorem proves an analogous identity with an automorphic period under Taylor–Wiles-type hypotheses, but the displayed motivic formula itself is presented as conjectural.