The Bloch–Kato formula for modular forms

For each prime $p$, let $V_p:=V_{f,\mathfrak{p}}$ be the $p$-adic Galois representation attached to a newform $f$ of weight $2k-2$, let $T_p:=T_{f,\mathfrak{p}}$ be a stable lattice, and put $W_p:=V_p/T_p$. Let $W_p(j)$ denote its $j$th Tate twist. Define

$$\operatorname{\textcyr{Sh}}(j)=\bigoplus_{\ell}\operatorname{H}^1_f(\mathbb{Q},W_{\ell}(j))/\pi_*\operatorname{H}^1_f(\mathbb{Q},V_{\ell}(j))$$

and

$$\Gamma_{\mathbb{Q}}(j)=\bigoplus_{\ell}\operatorname{H}^0(\mathbb{Q},W_{\ell}(j)).$$

Here $c_p(j)$ are Tamagawa factors and $\operatorname{vol}_{\infty}(k)$ is a real period. Bloch–Kato formula. With this notation,

$$L(k,f)=\frac{\left(\prod_{\ell}c_{\ell}(k)\right)\operatorname{vol}_{\infty}(k)\#\operatorname{\textcyr{Sh}}(1-k)}{\#\Gamma_{\mathbb{Q}}(k)\#\Gamma_{\mathbb{Q}}(k-2)}.$$

This is the explicit Tamagawa-number formulation of the Bloch–Kato conjecture; the paper discusses it as the arithmetic relation motivating its congruence results, while the normalization of the real period requires Deligne's treatment.