functor7

Hey, I have a question specific to Langlands Program. I have rough familiarity with it in the arithmetic context, but not so much in the geometric context. From what I've seen, it looks like a lot of the arithmetic stuff would work out, if only we had a more geometric interpretation of these objects (hence arithmetic geometry). So it seems like the geometric side of Langlands has better, more complete tools to deal with things. My question is, what are some things that Geometric Langlands has that Arthrimetic Langlands wishes it did, and what questions can we ask only on the geometric side using these tools?

Just checked out your Memory Alpha Page, and you wrote some of my favorite TNG and DS9 episodes! The Defector is probably my favorite Romulan story, it cloaks them in a mystery that not even invisibility devices can while having a very emotional progression as well. First Duty, Tapestry, The Pegasus all get the mind engaged and the heart pumping. Thank you!

For my question: My favorite villains are the Romulans, ruthless, mysterious, philosophical and also, in a way, human. They have some truly amazing moments. Since you can't have a good hero without a good villian, out of all the races you've written for, what is your favorite "evil" species and why?

Again, thank you for your contributions to TNG, DS9 and BSG, some of my favorite TV series!

Great, thanks!

Your book mentions that the analog to automorphic representations in the geometric context is automorphic sheaves. In the arithmetic context, Galois Representations have nice functorial properties under change of field, but it is very hard to find a corresponding result for automorphic forms. But when we do find such a correspondence, we can get some nice results relatively easily, eg Base Change for GL(2) gives Langlands-Tunnell.

I am not familiar with Automorphic Sheaves (yet...), but from your comment it seems that they come with these kinds of internal relations, that are difficult to prove when looking at Automorphic Forms, already built in. Is this impression in the right direction or do they have other symmetries that we can exploit? Thank you for your response.