** Next:** Non-liftable weight one modular
** Up:** Modular Forms Modulo
** Previous:** Modular Forms Modulo
** Contents**

This section was written by Gabor Wiese.
To every mod
eigenform Deligne attaches a 2-dimensional odd "mod
" Galois representation, i.e. a continuous group homomorphism

The trace of a Frobenius element at a prime
is for almost all
given by the
-th coefficient of the (normalised) eigenform. By
continuity, the image of such a representation is a finite group.

**Problem 2.1.1**
Find group theoretic criteria that allow one (in some cases) to
determine the image computationally.

**Remark 2.1.2** (From Richard Taylor)
Problem

2.1.1 seems to me straightforward.
(Richard, Grigor, and Stein did something like this for
elliptic curves over

-- see

`http://modular.math.washington.edu/papers/bsdalg/`.)

**Problem 2.1.3**
Implement in

*SAGE* the algorithm of Problem

2.1.1.

**Problem 2.1.4**
Carry out systematic computations of mod

modular forms in order to
find ``big'' images.

Like this one can certainly realise some groups as Galois groups over
that were not known to occur before!

** Next:** Non-liftable weight one modular
** Up:** Modular Forms Modulo
** Previous:** Modular Forms Modulo
** Contents**
William Stein
2006-10-20