2
$\begingroup$

The Langlands correspondence over a function field matches irreducible $n$-dimensional Galois representations with cuspidal irreducible automorphic representations.

My question is: Is there any idea or conjecture around, how this could extend to reducible representations?

As automorphic representations are semisimple, they don't allow for non-trivial extension, which however on the Galois-side can happen. Maybe there are ideas from geometric Langlands (of which I don't know much) that can help?

$\endgroup$
  • 1
    $\begingroup$ Over number fields, one can construct non-semisimple Galois representations by putting a "reducible" automorphic form (e.g. Eisenstein series) into a family of "irreducible" forms (cusp forms). The original idea goes back to Ribet. However, I don't know a criterion for which extensions can be constructed in this way, or even what is expected. $\endgroup$ – Kevin Ventullo Jun 7 '13 at 23:15

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy