Computations of Galois representations associated to modular forms of level one

• Autorzy: Peng Tian
• Języki publikacji: EN

Abstrakty

EN

We propose an improved algorithm for computing mod ℓ Galois representations associated to a cusp form f of level one. The proposed method allows us to explicitly compute the case with ℓ = 29 and f of weight k = 16, and the cases with ℓ = 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.

Kategorie tematyczne

• 11G30: Curves of arbitrary genus or genus ≠ 1 over global fields
• 11Fxx: Discontinuous groups and automorphic forms
• 11Y40: Algebraic number theory computations
• 11-04: Explicit machine computation and programs (not the theory of computation or programming)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2014
• Tom: 164
• Numer: 4
• Strony: 399-411

Daty

wydano

2014

Twórcy

autor

Peng Tian

• Department of Mathematics, Nanjing University, 210093, Nanjing, P.R. China
• Dipartimento di Matematica, Università di Roma "Tor Vergata", 00133 Roma, Italy

Identyfikatory

DOI

10.4064/aa164-4-5