Failure of the Hasse principle

Construct perfect evolution algebras over the rationals that are mutually isomorphic over the reals and over every p-adic field but are not isomorphic over the rationals.

Background

Forms of a perfect evolution algebra are governed by nonabelian Galois cohomology of its automorphism group. The paper notes that the Hasse principle holds for the relevant two-dimensional cyclic examples, while more complicated diagonal or nonabelian automorphism groups may permit failures. No explicit counterexample is known.

References

There exist perfect evolution $\mathbb Q$-algebras $A,B$ which are isomorphic over $\mathbb R$ and over $\mathbb Q_p$ for every prime $p$, but not over $\mathbb Q$.

— Evolution algebras: a survey of results and a list of open problems  (2610.22248 - Kaygorodov et al., 15 Oct 2026) in Conjecture c:hasse, Section 2