Endoscopic congruences modulo adjoint $L$-values for $\mathrm{GSp}(4)$

Abstract: We establish the existence of congruences between a fixed endoscopic cuspidal automorphic representation $\Pi$ of $\mathrm{GSp}(4)$ of square-free conductor and stable cuspidal automorphic representations of the same level and weight modulo certain prime factors of the value at $1$ of the adjoint $L$-function of $\Pi$ normalized by a suitable period.