A team of number theorists has proved a long‑standing conjecture that connects the sign in the functional equation of certain Galois representations to the size of their Selmer groups. The result applies to *symplectic self‑dual* $p$‑adic de Rham representations that satisfy the Panchishkin condition at an odd prime $p$ and are pure of weight –1. They show that the ratio of the global root numbers of two such representations equals the parity of the difference of specific local invariants. These invariants are defined over the cyclotomic $oldsymbol{ extbf{Z}}_p$‑extension: at $p$ they are dimensions of strict Selmer spaces; at other primes they are local Iwasawa $oldsymbol{eta}$‑invariants. By proving that Bloch–Kato and strict local conditions agree in the cyclotomic tower, the authors can express the global root‑number ratio in terms of these local data. The theorem yields new cases of the $p$‑parity conjecture, including situations with rank greater than two, such as Artin twists of CM Hida families, ordinary modular forms, symmetric cubes in Hida families, and abelian surface families.

The authors emphasize that the theorem requires the Selmer group of the first representation over the cyclotomic extension to be $oldsymbol{O}$‑torsion with vanishing $oldsymbol{O}$‑μ–invariant. If this μ–invariant is non‑zero, the parity argument and comparison of Selmer coranks used in the proof break down.

Overall, the work expands the class of representations for which the $p$‑parity conjecture is known and introduces new local invariants that link analytic and arithmetic properties of Galois representations.