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

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