MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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. – Kevin Ventullo Jun 7 '13 at 23:15

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.