January 25, 2016: Explicitly computing the mod-5 representation attached to 11a
William Stein
Motivating problem: Galois representations attached to elliptic curves
As a motivating problem for explicitly computing (1) prime factorizations, (2) rings of integers, (3) -maximal orders, and (4) maps to and from finite fields, we will compute Galois representations attached to elliptic curves.
Let be an elliptic curve over and let be a prime of good reduction.
Consider the group of elements of order dividing . Fix a basis for .
The mod Galois representation attached to is the homomorphism got by letting the Galois group act on .
The number field got by adjoining all and coordinates of elements of to is the fixed field in for the subgroup . The field is ramified at most at and the primes of bad reduction for . Note that is a Galois extension.
Quick Exercise: Give a quick example of and in which is unramified at all primes?
Let be a prime number. Let be the ring of integers of and let be a prime of over , which means that . The map induces a map from to the finite field , of characteristic .
Let be the decomposition group of in , i.e., the subgroup of automorphisms that send to itself, and let be the inertia group. We have an exact sequence
Let be a choice of lift of . Note that is well defined when , which is the case for all unramified primes (in particular, for ).
For a prime of good reduction for , let .
Theorem: For , the characteristic polynomial of is .
Goal: Understand the details of how to explicitly compute the matrix . Use the above theorem as a consistency check.
Rest of today: compute one example. Later: talk about how to factor , compute the map explicitly, etc....
We will compute with the mod-5 representation attached to the elliptic curve 11a (this is the one we were thinking about at lunch last week in response to Ralph Greenberg's question.)
Let's compute .
I don't know how in Sage, given a set of elements of QQbar, to get the number field they generate easily.
But we can just take a random linear combination and it is likely to give us that field.
Then we can do a computation to check if it worked.
These are generators as a -module:
Next step: let's factor the prime 2.
Compute the residue class field explicitly and reduction map:
So we can compute the matrix of on .
We have the following (arbitrary choice of) basis for :
Clearly acts trivially on , since is already rational, hence reduces to a point in .
Reduce the points and modulo :
Now we need to figure out what linear combination of P1 and P2 reduces to Frob2P2.
We'll just brute force it for now:
Conclusion: sends to and to .
Double check: Is ?
YEP.
Final note: Computing Frob_p for other primes is not more difficult. The difficulty is entirely a function of the original choice of .