[Paper Review] Computing Congruences of Modular Forms and Galois Representations Modulo Prime Powers

Xavier Taixés i Ventosa, Gabor Wiese|arXiv (Cornell University)|Sep 15, 2009

Algebraic Geometry and Number Theory7 references4 citations

TL;DR

This paper develops algorithms to compute congruences of modular forms and Galois representations modulo prime powers ℓⁿ, using the congruence number and Newton polygon methods to determine when two Hecke eigenforms are congruent modulo ℓⁿ. A key result is that level raising modulo ℓⁿ does not necessarily follow from divisibility of congruence numbers, challenging a naive generalization of Ribet's theorem.

ABSTRACT

This article starts a computational study of congruences of modular forms and modular Galois representations modulo prime powers. Algorithms are described that compute the maximum integer modulo which two monic coprime integral polynomials have a root in common in a sense that is defined. These techniques are applied to the study of congruences of modular forms and modular Galois representations modulo prime powers. Finally, some computational results with implications on the (non-)liftability of modular forms modulo prime powers and possible generalisations of level raising are presented.

Motivation & Objective

• To develop computational methods for determining congruences between modular forms and Galois representations modulo ℓⁿ, where ℓ is prime and n ≥ 1.
• To address the challenge of working over non-reduced, non-factorial rings like ℤ/ℓⁿℤ, which complicates standard congruence definitions.
• To generalize level-raising phenomena from modulo ℓ to higher powers ℓⁿ, particularly in the context of Hecke eigenforms.
• To provide algorithmic tools for computing congruences between newforms via characteristic polynomials of Hecke operators and their roots.
• To investigate the non-liftability of modular forms modulo ℓⁿ and the behavior of Galois representations in this setting.

Proposed method

• Define congruence modulo ℓⁿ between algebraic integers via the congruence number, a derived invariant from coprime monic integral polynomials.
• Use the Newton polygon of the polynomial formed by differences of roots to refine congruence detection when the congruence number is insufficient.
• Apply the Sturm bound to determine the finite number of q-expansion coefficients needed to uniquely identify modular forms modulo ℓⁿ.
• Construct algorithms that compute the maximal ℓⁿ such that two newforms are congruent modulo ℓⁿ by analyzing common roots of their Hecke operator characteristic polynomials.
• Implement degeneracy maps modulo ℓⁿ to compare forms of different levels by reducing them to a common level.
• Use the congruence number and Newton polygon method to compute upper and lower bounds for congruences with Eisenstein series modulo ℓⁿ.

Experimental results

Research questions

• RQ1 Does the condition that ℓⁿ divides the congruence number of the characteristic polynomials of T_p imply level raising modulo ℓⁿ for newforms?
• RQ2 Can the classical level-raising theorem of Ribet be generalized to congruences modulo ℓⁿ, and if so, under what conditions?
• RQ3 What is the maximal ℓⁿ such that a newform f is congruent modulo ℓⁿ to an Eisenstein series in the same level and weight?
• RQ4 Is the sum of the ℓ-adic valuations of the highest congruence powers between f and newforms in level Np equal to the ℓ-adic valuation of the congruence number involving (p+1)²?
• RQ5 To what extent are modular forms modulo ℓⁿ non-liftable, and how does this affect the structure of associated Galois representations?

Key findings

• The congruence number provides a good upper bound for the largest ℓⁿ such that two monic coprime integral polynomials have roots congruent modulo ℓⁿ.
• For the newform on Γ₀(17) of weight 2, no level-raising occurs modulo 9 to level 17·59, despite ℓ² dividing the congruence number, indicating that divisibility of the congruence number is not sufficient for level raising modulo ℓⁿ.
• The algorithm computes congruences between newforms modulo ℓⁿ more efficiently than naive methods using full number field embeddings.
• The method detects congruences with Eisenstein series modulo ℓⁿ by analyzing common roots of characteristic polynomials of T_p and those in the Eisenstein subspace, yielding upper bounds on the congruence level.
• The paper observes that the sum of ℓ-adic valuations of congruence powers between f and newforms in level Np may not equal the ℓ-adic valuation of the congruence number involving (p+1)², suggesting a more refined invariant is needed.
• A counterexample shows that the naive generalization of Mazur’s theorem on congruences with Eisenstein series modulo ℓ to higher powers ℓⁿ fails, as congruences modulo ℓⁿ do not always extend from modulo ℓ.