[Paper Review] On the quantitative variation of congruence ideals and integral periods of modular forms

Chan-Ho Kim, Kazuto Ota|arXiv (Cornell University)|May 8, 2019

Advanced Algebra and Geometry34 references4 citations

TL;DR

This paper proves a conjecture by Pollack and Weston on the quantitative behavior of congruence ideals in higher-weight modular forms, establishing a precise formula relating the congruence ideal at full level to its $N^{-}$-new part via Tamagawa exponents. Using a novel combination of $R=\mathbb{T}$ theorems, Galois cohomology, and explicit adjoint $L$-value comparisons, the authors resolve the level-lowering congruence problem without relying on geometric methods, extending prior results to non-square-free levels and higher weights with weakened ramification assumptions.

ABSTRACT

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 $R=\mathbb{T}$ 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.

Motivation & Objective

• To resolve the conjecture of Pollack and Weston on the quantitative variation of congruence ideals in higher-weight modular forms.
• To generalize level-lowering congruence results beyond the weight 2 and square-free level cases.
• To establish a precise formula linking the congruence ideal at full level to its $N^{-}$-new part via Tamagawa exponents.
• To overcome limitations of prior geometric approaches by developing a cohomological and analytic method based on adjoint $L$-values and $R=\mathbb{T}$ theorems.
• To provide a framework applicable to anticyclotomic Iwasawa theory and integral Jacquet–Langlands correspondence in higher weights.

Proposed method

• Utilizes the $R=\mathbb{T}$ theorem for quaternion algebras, established by Diamond–Flach–Guo and Dimitrov, to relate congruence ideals to Selmer groups.
• Applies Galois cohomology to compute the size of adjoint Selmer groups with relaxed and 'new' local conditions at primes dividing $N^{-}$.
• Performs a comparison between two Selmer groups—one with relaxed and one with 'new' local conditions—to isolate the difference in congruence ideals.
• Introduces an analytic correction via Euler factors of adjoint $L$-values to reconcile the discrepancy between the Selmer computation and the desired congruence formula.
• Employs explicit computation of adjoint $L$-values to interpret the difference in congruence ideals as a ratio of $L$-values, completing the proof.
• Avoids fixed quaternion algebras and relies solely on classical $R=\mathbb{T}$ theorems from [DFG04, Dim09] to maintain generality.

Experimental results

Research questions

• RQ1 How does the congruence ideal of a higher-weight modular form vary under level lowering, particularly when $N^{-}$ is not square-free or the form is not of weight 2?
• RQ2 What is the precise arithmetic interpretation of the difference between the full-level congruence ideal and the $N^{-}$-new congruence ideal?
• RQ3 Can the $R=\mathbb{T}$ method be extended to compute congruence ideals when the residual representation is unramified at some $q \mid N^{-}$, i.e., when $t_f(q) > 0$?
• RQ4 How do adjoint $L$-values encode the local behavior of congruence ideals at primes dividing $N^{-}$?
• RQ5 To what extent can the freeness of Hecke modules over $\mathbb{T}$-algebras be related to the structure of congruence ideals in higher weights?

Key findings

• The paper proves the formula $\mathrm{ord}_{\lambda}\eta_f(N) = \mathrm{ord}_{\lambda}\eta_f(N^+,N^-) + \sum_{q|N^{-}} t_f(q)$, which quantifies level-lowering congruences for modular forms of weight $k \geq 2$.
• The result holds under weakened assumptions: $N^{-}$ need not be square-free, $k$ can be greater than 2, and ramification at $q \mid N^{-}$ is not required.
• The authors establish a new method that combines Galois cohomology and adjoint $L$-value computations to bypass the limitations of geometric approaches in higher weights.
• The difference between the full-level and $N^{-}$-new congruence ideals is shown to be encoded in the Euler factors of the adjoint $L$-function at primes dividing $N^{-}$.
• The method enables a comparison of Petersson inner products and canonical periods, leading to a new formula for the $\lambda$-adic valuation of their ratio.
• Under suitable assumptions, the paper confirms that $\mathrm{ord}_{\lambda}(\langle f,f\rangle / \langle f_B,f_B\rangle) = \sum_{q|N^{-}} t_f(q)$, linking arithmetic invariants to congruence data.