Bloch–Kato conjectures on special values of L-functions

Prove the Bloch–Kato conjectures relating, for a motive M and its L-function L(M,s), the order of vanishing at integers to Selmer-group or K-group ranks and the leading Taylor coefficient to arithmetic invariants including regulators, periods, and Tamagawa numbers.

Background

The paper summarizes the Bloch–Kato framework that connects special values of L-functions to deep arithmetic invariants, encompassing celebrated cases such as the Birch and Swinnerton–Dyer conjecture.

These conjectures generalize and organize a wide range of conjectural relations between analytic behavior of L-functions and arithmetic geometry.

References

Motivic L-functions and Bloch–Kato conjectures: These far-reaching conjectures relate special values of $L$-functions to arithmetic invariants.

— The Riemann Hypothesis: Past, Present and a Letter Through Time  (2602.04022 - Connes, 3 Feb 2026) in Subsubsection Motivic L-functions and Bloch–Kato conjectures

We make the following conjecture. For the motive $\mathcal{M}$ indicated in eq:curlyM, the meromorphic function $L(\mathcal{M},s)$ is holomorphic at $s=2$ and L(\mathcal{M},2) \neq 0.

eq:curlyM:

⋀2H1(X)≃Q(−1)⊕M1[a2]⊕M\bigwedge\nolimits^2 H^1(X) \simeq \mathbb{Q}(-1)^{\oplus M_1[a_2]} \oplus \mathcal{M}

— Products of point counts of higher genus curves over finite fields  (2608.18014 - Bucur et al., 18 Aug 2026) in Conjecture \ref{conj:m1a2}, Section 2.4, subsection “Sato–Tate groups”

However, the arithmeticity at critical points in the left half plane, and at $s=1$ when $F=Q$, remains open (cf.).

— Arithmeticity of critical $L$-values for Hilbert-Siegel modular forms in the left half plane and $s = 1$  (2610.23037 - Horinaga et al., 16 Oct 2026) in Section 1, Introduction