January 29, 2016: Explicitly compute the mod-2 representation attached to 11a
As a warm-up before returning to , let's compute for .
This is more straightforward to do without getting stuck with really difficult computations that force us to learn things more deeply.
Remark: Suppose is very small, e.g., or so we can explicitly compute (I hope). Here's a reflection of "something deep". Suppose is a HUGE PRIME. Consider the quantities:
A:
B: .
These two quantities are equal (by the theorem I mentioned -- it is more generally called the "Eichler-Shimura relation".) In particular, .
The first is hard to compute, since for HUGE, computing is VERY difficult. Even computing anything at all in is at least pretty hard.
On the other hand, to compute , after doing some big initial computation, you just need to find a small prime such that , and you're done. Finding is purely a computation in the number field . It's a lot like quadratic reciprocity, except the "congruence" there is replaced by "defining the same element of ".
Schoof: Incidentally, a key idea used in Schoof's groundbreaking algorithm for computing for large , which is at the foundation of elliptic curve cryptograph, is the relation above, which reduces computing to the problem of computing for sufficiently many primes .