Fix a prime p and let
&rho&Delta : Gal(F/Q) → GL2(Fp)be the mod p Galois representation which is the reduction of the p-adic Galois representation associated to the modular form &Delta. A p-adic Galois reprsentation ρ with reduction equal to &rho&Delta is called a deformation of &rho&Delta. (More honestly, a deformation is a certain equivalence class of such liftings.) The set of deformations of &rho&Delta contains the Galois representations attached to infinitely many modular forms congruent to &Delta, but also contains many more Galois representations which are so unnatural that they could not possibly arise from modular forms.
Nevertheless, for most primes p
it is actually possible to completely describe the
deformation space (i.e., the set of deformations)
of &rho&Delta.
Specifically, it is proved in
Unobstructed modular
deformation problems and
Explicit unobstructed
primes for modular deformation problems of squarefree level
that for p different from 2,3,5,7,11 and
691 the deformation space of
&rho&Delta is
simply a three-dimensional p-adic box.
Simple as this box appears, it actually contains quite complicated objects.
For example, for p = 2411, it contains the
infinite fern of
Fernando
Gouvêa
and
Barry
Mazur.