The Iwasawa main conjecture for an elliptic curve over an anticyclotomic extension relates a p-adic L-function (encoding arithmetic information about the curve's behavior in a tower of number fields) to the Selmer group (encoding the algebraic structure of rational points and their generalizations). The conjecture asserts equality between two ideals: the characteristic ideal of the Selmer group's Pontryagin dual, and the ideal generated by the p-adic L-function. This is the zeroth Fitting ideal — the coarsest algebraic invariant of the module.
Da Ronche, Longo, and Vigni (arXiv:2603.12357) go further: they describe the higher Fitting ideals. Where the zeroth Fitting ideal captures the “size” of the module (how many generators and relations), the higher Fitting ideals capture progressively finer structural information — the k-th Fitting ideal encodes properties visible only after quotienting by the first k relations.
The tool is bipartite Euler systems — compatible families of cohomology classes indexed by pairs of primes, one inert and one split in the imaginary quadratic field. These systems generate relations in the Selmer group, and the systematic use of multiple primes (not just one, as in the zeroth case) produces enough relations to pin down the higher Fitting ideals.
The result covers both the definite and indefinite cases — corresponding to whether the root number of the elliptic curve is +1 or -1 — under mild arithmetic conditions. In the definite case, the zeroth Fitting ideal was already known; the higher ideals are new. They reveal structure in the Shafarevich-Tate group (the mysterious part of the Selmer group whose elements correspond to locally trivial but globally nontrivial torsors) that the zeroth ideal doesn't see.