Fermat, Taniyama–Shimura–Weil and Andrew Wiles

144 Pages·2016·5.58 MB·English

by  John Rognes

Fermat, Taniyama–Shimura–Weil and
Andrew Wiles
John Rognes
UniversityofOslo,Norway
May 13th and 20th 2016
The Norwegian Academy of Science and Letters has decided
to award the Abel Prize for 2016 to Sir Andrew J. Wiles,
University of Oxford
for his stunning proof of Fermat’s Last Theorem
by way of the modularity conjecture for semistable
elliptic curves, opening a new era in number
theory.
SirAndrewJ.Wiles
Sketch proof of Fermat’s Last Theorem:
(cid:73) Frey (1984): A solution
ap +bp = cp
to Fermat’s equation gives an elliptic curve
y2 = x(x −ap)(x +bp).
(cid:73) Ribet (1986): The Frey curve does not come from a
modular form.
(cid:73) Wiles (1994): Every elliptic curve comes from a modular
form.
(cid:73) Hence no solution to Fermat’s equation exists.
Point counts and Fourier expansions:
Elliptic curve
Hasse–Weil
(cid:40)(cid:40)
L(cid:54)(cid:54) -function
Mellin
Modular form
Modularity:
Elliptic curve
(cid:40)(cid:40)
◦
? L(cid:54)(cid:54) -function
(cid:15)(cid:15)
Modular form
Wiles’ Modularity Theorem:
Semistable elliptic curve
defined over Q
(cid:41)(cid:41)
Wiles ◦
(cid:53)(cid:53)L-function
(cid:15)(cid:15)
Weight 2 modular form
Wiles’ Modularity Theorem:
Semistable elliptic curve
over Q of conductor N
(cid:41)(cid:41)
Wiles ◦
(cid:53)(cid:53)L-function
(cid:15)(cid:15)
Weight 2 modular form
of level N
Frey Curve (and a special case of Wiles’ theorem):
Solution to Fermat’s equation
Frey
(cid:15)(cid:15)
Semistable elliptic curve
over Q with peculiar properties
(cid:41)(cid:41)
Wiles ◦ (cid:53)(cid:53)L-function
(cid:15)(cid:15)
Weight 2 modular form
with peculiar properties
(A special case of) Ribet’s theorem:
Solution to Fermat’s equation
Frey
(cid:15)(cid:15)
Semistable elliptic curve
over Q with peculiar properties
(cid:42)(cid:42)
Wiles ◦ (cid:52)(cid:52)L-function
(cid:15)(cid:15)
Weight 2 modular form
with peculiar properties
(cid:79)(cid:79)
Ribet
Weight 2 modular form of level 2