Computations of Galois representations associated to modular forms of level one

Peng Tian¹

¹ Department of Mathematics Nanjing University 210093, Nanjing, P.R. China and Dipartimento di Matematica Università di Roma “Tor Vergata” 00133 Roma, Italy

Acta Arithmetica, Tome 164 (2014) no. 4, pp. 399-411.

Résumé

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.

Détail

DOI : 10.4064/aa164-4-5

Keywords: propose improved algorithm computing mod ell galois representations associated cusp form level proposed method allows explicitly compute ell weight cases ell weight results rigorously proved correct example compute values modulo ramanujans tau function huge primes sign improved uper bound lehmers conjecture ramanujans tau function