8
$\begingroup$

I know in number theorythere are loads of conditional results, dependant on RH for instance. On the other hand, Langland's programme is supposed to provide some understanding of the absolute Galois group of $\mathbb Q$, but most of it is conjectural. So if Langland's programme is to do its job, one might expect conditional results. Are there some instance of it?

$\endgroup$
5
  • 3
    $\begingroup$ Well for instance I think the modularity theorem is seen as a special case of the Langlands program, so "Fermat's last theorem" could be considered a "consequence" of the Langlands program (not conditional though). $\endgroup$ Oct 1, 2018 at 14:48
  • $\begingroup$ If I'm not mistaken, under Langlands' conjectures, the Rankin-Selberg convolution of any two automorphic representations is itself automorphic. I don't think such a result has been established unconditionally. $\endgroup$ Oct 1, 2018 at 15:10
  • 5
    $\begingroup$ One important consequence would be meromorphic continuation for motivic $L$-functions (simply because meromorphic continuation is known for automorphic $L$-functions). This is a fairly elementary conjecture for which, as far as I know, no one has any real ideas, Langlands aside. $\endgroup$ Oct 1, 2018 at 15:33
  • 4
    $\begingroup$ Another well-known consequence of Langlands functoriality is the generalised Ramanujan conjecture, which is known for classical holomorphic cusp forms but not in more general settings. $\endgroup$ Oct 1, 2018 at 18:00
  • $\begingroup$ Possible duplicate of: mathoverflow.net/questions/78247/… $\endgroup$ Feb 28, 2019 at 21:34

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.