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Computations of Galois representations associated to modular forms of level one

Volume 164 / 2014

Peng Tian Acta Arithmetica 164 (2014), 399-411 MSC: 11-04, 11Fxx, 11G30, 11Y40. DOI: 10.4064/aa164-4-5

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 uper bound on Lehmer's conjecture for Ramanujan's tau function.

Authors

  • Peng TianDepartment of Mathematics
    Nanjing University
    210093, Nanjing, P.R. China
    and
    Dipartimento di Matematica
    Università di Roma “Tor Vergata”
    00133 Roma, Italy
    e-mail

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