He writes until the chalk dust erases his fingerprints.
The object does not change:
odd, absolutely irreducible, semistable. Determinant
.
Everything lives inside this cage. He demands deformation theory to
carry the weight.
Minimal deformation problem
Fix
, the finite set of places.
For each
he prescribes a local deformation condition
.
Finite flat at
, unramified outside
, ordinary if necessary.
Define the functor:
This functor is representable by a complete Noetherian local
-algebra
.
He stares at the hull:
Each is an obstruction. Each generator is a wall between him and Fermat.
Hecke algebra comparison
He moves to the Hecke side.
For modular forms of level
, weight
, let
denote the Hecke algebra generated by
for
and
for
.
Localize at the maximal ideal
corresponding to
:
The universal property gives a natural surjection:
He knows: if
, then every semistable elliptic curve is
modular.
If every semistable elliptic curve is modular, Fermat’s Last Theorem is
dead.
Tangent space calculation
He presses into the tangent spaces.
Compute the Zariski tangent space:
where denotes the adjoint representation of trace zero matrices.
He compares to the cotangent space on the Hecke side:
The numerical criterion demands equality of length:
Every time he calculates, the inequality is off by one.
The criterion resists.
Cohomology labyrinth
He calculates local conditions.
For
not dividing
:
At :
The global Selmer group:
Every dimension formula fails to reconcile.
Iwasawa interlude
He mutters through Iwasawa theory.
Take
.
.
The Iwasawa algebra:
The Selmer group over
is
-torsion.
Characteristic power series:
with , invariants encoded.
He tests congruences between modular forms by comparing
-invariants.
The data refuse to line up cleanly.
Non-minimal deformations
The compulsive shift: abandon minimality.
Allow ramification at auxiliary primes
.
Define the enlarged deformation problem with local conditions
loosened.
The surjection extends:
He then patches across .
Patching modules
Define:
cohomology of modular curves at auxiliary level.
Construct inverse system:
becomes a balanced module: depth equals
dimension.
He proves Gorenstein property, then complete intersection:
Now the numerical criterion is satisfied.
Congruence modules
He isolates the congruence module:
Its annihilator controls the failure of
.
But patching forces
to vanish.
The alignment is exact.
Final collapse
He writes it as the theorem he cannot deny:
Isomorphism of complete intersections.
Every semistable elliptic curve is modular.
Fermat’s Last Theorem collapses into corollary.
Comments
the contrast between the fire writing and the hard math is so comical. loving this website