S M Nazmuz Sakib Modularity Certificate Principle

We formalize a finite, effectively-checkable certificate intended to certify the paramodular
modularity of a large class of ordinary abelian surfaces over Q. The S M Nazmuz Sakib
Modularity Certificate Principle asserts that for an abelian surface A/Q, ordinary at
the primes 3 and 5, with suitable big-image and local hypotheses, the existence of a genus-2
Siegel eigenform F of paramodular level N for which a finite collection of Hecke congruences
Tr(ρA,ℓ(Frobp)) ≡ Tr(ρF,ℓ(Frobp)) (mod ℓ) (ℓ ∈ {3, 5}, p ∈ Sℓ)
holds for an explicit, effectively computable set of good primes Sℓ, implies that A is Q-
isogenous to the abelian variety AF associated to F . The proposal is motivated by, and
conditional on, ordinary modularity-lifting techniques and the recent breakthrough of Boxer–
Calegari–Gee–Pilloni (2025) on modularity for ordinary abelian surfaces, together with
Faltings–Serre methods adapted to the paramodular setting. We present the precise statement,
conditional reductions, examples, and an algorithm to search for certificates. We do not
claim an unconditional proof; instead we offer a concrete, testable framework whose validity
would algorithmize recent advances.