Relate Galois obstructions across genera

Determine, for each fixed \(k\), the relation among the Galois obstructions \(\sigma_{2k+1}(g)\) as the genus \(g\) ranges over \(1,2,3,\ldots\).

Background

The authors define σ2k+1(g)\sigma_{2k+1}(g) as the Galois obstruction in degree $4k+2$ for genus gg. They obtain an explicit genus-dependent formula for the first obstruction when k=1k=1, but do not determine how the obstructions in a fixed degree are related as the genus varies.

References

For any fixed $k$, what is the relation among $\sigma_{2k+1}(g)$ for $g=1,2,3,\ldots$?

— The first Galois obstruction in the Johnson cokernel  (2608.17673 - Morita et al., 18 Aug 2026) in Section 7, Section “Questions,” fourth Question environment