A fixed algebraic number controlling all unit-condition failures

Construct or identify a fixed algebraic number, independent of the split prime, whose prime divisors contain every split prime at which the cyclotomic unit condition fails for a fixed rank-two CM elliptic curve.

Background

The paper observes that the zeroth jet of the relevant p-adic L-function is governed by a fixed algebraic quantity through Gillard’s theorem, whereas the second-jet criterion involves algebraic combinations of critical values whose weights vary with p.

The authors ask whether an analogous fixed algebraic object exists for the failures of the unit condition, which would provide a uniform explanation of the exceptional primes.

References

Is there a fixed algebraic number whose prime divisors contain every split prime at which the unit condition fails?

— 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 Question 3 (label q:fixed), Section 1 subsection “The questions”; repeated in Section “Questions”