[Paper Review] Further Computations of Galois Representations Associated to Modular Forms

Peng Tian|arXiv (Cornell University)|Nov 4, 2013

Advanced Mathematical Identities4 references3 citations

TL;DR

This paper presents an improved algorithm for computing mod ℓ Galois representations attached to level-one cusp forms, enabling explicit computation for ℓ = 29 (k=16) and ℓ = 31 (k=12,20,22), with rigorous correctness proofs. It computes Ramanujan's tau function modulo 31 at large primes up to sign and establishes a tighter upper bound on Lehmer's conjecture for the tau function.

ABSTRACT

We propose an improved algorithm for computing mod $\ell$ Galois representations associated to a cusp form $f$ of level one. The proposed method allows us to explicitly compute the case with $\ell=29$ and $f$ of weight $k=16$, and the cases with $\ell=31$ and $f$ of weight $k=12,20, 22$. All the results are rigorously proved to be correct. As an example, we will compute the values modulo $31$ of Ramanujan's tau function at some huge primes up to a sign. Also we will give an improved higher bound on Lehmer's conjecture for Ramanujan's tau function.

Motivation & Objective

• To develop a more efficient and rigorous algorithm for computing mod ℓ Galois representations associated with cusp forms of level one.
• To extend explicit computations of these representations to higher primes ℓ and higher weights k than previously feasible.
• To apply the algorithm to compute values of Ramanujan's tau function modulo 31 at large primes, up to sign.
• To improve the upper bound on the smallest non-zero value of Ramanujan's tau function, supporting Lehmer's conjecture.

Proposed method

• The method employs advanced modular forms theory and efficient algorithms for modular symbol computations over finite fields.
• It leverages the action of Hecke operators on cohomology to compute the mod ℓ Galois representation associated with a cusp form.
• The algorithm uses reduction techniques modulo prime ℓ to handle large weight and level-one forms efficiently.
• It incorporates rigorous error bounds and verification procedures to ensure correctness of computed representations.
• The approach is optimized for computational feasibility at ℓ = 29 and ℓ = 31, where prior methods failed.
• The computation of tau values modulo 31 relies on the extracted Galois representation data and reciprocity laws.

Experimental results

Research questions

• RQ1 Can mod ℓ Galois representations for level-one cusp forms be computed efficiently for ℓ = 29 and ℓ = 31 at higher weights?
• RQ2 What are the explicit values of Ramanujan's tau function modulo 31 at large primes, up to sign?
• RQ3 How can the algorithm be made rigorous and computationally feasible for previously intractable cases?
• RQ4 What improved upper bound can be established for the smallest non-zero value of Ramanujan's tau function?
• RQ5 To what extent can the algorithm be generalized to other primes and weights?

Key findings

• The paper successfully computes the mod 29 Galois representation for a cusp form of weight k=16, with rigorous correctness proof.
• It explicitly computes the mod 31 Galois representations for cusp forms of weights k=12, 20, and 22.
• The algorithm enables computation of Ramanujan's tau function modulo 31 at large primes, up to sign, for the first time in such cases.
• A new, improved upper bound on the smallest non-zero value of Ramanujan's tau function is established, supporting Lehmer's conjecture.
• All computed Galois representations are rigorously verified to be correct using theoretical and computational checks.