On the quantitative variation of congruence ideals and integral periods of modular forms

We prove the conjecture of Pollack and Weston on quantitative analysis level lowering congruence à la Ribet for modular forms higher weight. It was formulated studied in context integral Jacquet–Langlands correspondence anticyclotomic Iwasawa theory weight two square-free first time. use a completely different method based $$R={\mathbb {T}}$$ theorem established by Diamond–Flach–Guo Dimitrov an explicit comparison adjoint L-values. briefly discuss arithmetic applications our main result at end.