John Carlos Baez on Nostr: One last little thing. People often say something like "every motivic L-function is ...
One last little thing. People often say something like "every motivic L-function is an automorphic L-function, and vice versa", which is an equation between two functions. But how would you prove this?
Both L-functions are just summaries of information - one about a motive, another about an automorphic representation. So you have to find a way to get from a motive to an automorphic representation, and vice versa. And this is what people are trying to do.
So it's a bit like what people in combinatorics call a 'bijective proof'. You're trying to prove that two sequences of numbers are equal. You show that the first sequence counts the number of gadgets with n elements. You show the second counts the number of widgets with n elements. And then you construct a one-to-one correspondence between gadgets with n elements and widgets with n elements! That gets the job done - and it's also a deeper fact than the mere equation between sequences of numbers.
Category theorists call this 'categorification': instead of proving an equation between functions, or sequences, you go further and prove a one-to-one correspondence between things of one sort and things of another sort.
This is the attitude I instinctively take when people say "every motivic L-function is an automorphic L-function, and vice versa". But I don't understand the details yet!
(3/n, n = 3)
Both L-functions are just summaries of information - one about a motive, another about an automorphic representation. So you have to find a way to get from a motive to an automorphic representation, and vice versa. And this is what people are trying to do.
So it's a bit like what people in combinatorics call a 'bijective proof'. You're trying to prove that two sequences of numbers are equal. You show that the first sequence counts the number of gadgets with n elements. You show the second counts the number of widgets with n elements. And then you construct a one-to-one correspondence between gadgets with n elements and widgets with n elements! That gets the job done - and it's also a deeper fact than the mere equation between sequences of numbers.
Category theorists call this 'categorification': instead of proving an equation between functions, or sequences, you go further and prove a one-to-one correspondence between things of one sort and things of another sort.
This is the attitude I instinctively take when people say "every motivic L-function is an automorphic L-function, and vice versa". But I don't understand the details yet!
(3/n, n = 3)