Bloch–Kato conjecture for motives

The Bloch–Kato conjecture predicts that the order of vanishing of $L(M^\vee(1),s)$ at $s=\frac{1}{2}$ equals the rank of $\mathrm{Sel}_{p^\infty}(M)$. Surveys and later work treat this as a broad conjecture with many specialized cases, not as a theorem.

Known results

• For $n=2$, theorems of this type were first proved by Flach in 1992.
• The Tamagawa-number formulation holds up to powers of $2$ for all abelian Artin motives, including Dirichlet motives (2002).
• Under strong automorphic, irreducibility, and local hypotheses, Skinner–Urban obtain one-sided implications; with global sign $+1$, the Selmer rank is at least $2$ (reported 2019).

September 2025 partial results

Peng’s reported result for polarized motives gives, in a conjugate self-dual setting and for almost all $\lambda$, that vanishing of the central $L$-value forces vanishing of the associated Bloch–Kato Selmer group. The self-dual case remains reduced to an additional conjecture involving endoscopic Rankin–Selberg periods; this does not settle the general conjecture.

Current status (as of August 2026): The general Bloch–Kato conjecture for motives remains open; several conditional and specialized cases are known, including recent results for polarized automorphic motives.