Cuts and isogenies

We consider the genus-one curves which arise in the cuts of the sunrise and in the elliptic double-box Feynman integrals. We compute and compare invariants of these curves in a number of ways, including Feynman parametrization, lightcone and Baikov (in full and loop-by-loop variants). We find that the same geometry for the genus-one curves arises in all cases, which lends support to the idea that there exists an invariant notion of genus-one geometry, independent on the way it is computed. We further indicate how to interpret some previous results which found that these curves are related by isogenies instead.

Automated summary

The study focuses on genus-one curves that appear in Feynman integrals, analyzing their invariants through various methods and finding consistent geometry across different cases. This supports the notion that there exists an invariant concept of genus-one geometry independent of the computation method. The study also offers insights on interpreting previous results related to isogenies of these curves.