Supersingular analogue of the generating object

Construct a supersingular counterpart of the integral generating object used to control the cyclotomic second-jet criterion for CM elliptic curves at inert primes.

Background

The main results apply only at split ordinary primes. At inert primes the ordinary Katz-measure construction is unavailable in the required integral form, and the relevant two-variable expansion has a radius of convergence smaller than one.

The paper identifies the absence of a supersingular analogue of the generating object as an obstruction to extending the regulator scan and the rigidity arguments to inert primes.

References

Two things would have to be settled before the scan of \S\ref{ssec:scan} could be repeated at inert primes: the generic valuation of the pair of regulators for a curve of rank two, computable by the $\sigma$-function algorithms of \S 4, and the rigidity input, for which no supersingular counterpart of the generating object is known.

— Second derivatives of $p$-adic $L$-functions and the Shafarevich--Tate group of rank-two CM elliptic curves  (2609.08431 - Banwait, 8 Sep 2026) in Section “The inert half”