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Multiplicity one
Let N be a prime number.
In 1977, Mazur proved that most maximal ideals of the Hecke algebra
T
associated to J_{0}(N) satisfy "multiplicity one".
Further results in this direction were proved by many people in the course of
work on Serre's conjectures and Fermat's Last Theorem. However, even in
1999 we did not know whether or not every maximal ideal of T
satisfies multiplicity one. Finally, Lloyd Kilford, who is a student
of Kevin Buzzard, observed that the elliptic curve
431A leads to a counterexample.
History
 Mazur:
Let N be a prime number.
In Modular curves and the Eisenstein Ideal (1977),
Mazur proved that for most maximal m in T,
dim_{T/m} J[m] = 2.
"Most" means:
 m Eisenstein, or
 char(T/m) =/= 2, or
 m supersingular, or
 Im(rho_{m}(D_{2}))
not contained in scalars (Buzzard).
Example.
J_{0}(11) is an elliptic curve, and for all primes p,
J_{0}(11)[p] = Z/p x Z/p.
Temptation
 Does dim_{T/m}J[m] = 2 for all m?
Examples
 Lloyd Kilford's first counterexample to the temptation
431A
E:
y^{2} + xy = x^{3}  1
f_{E} = q  q^{2} + q^{3}  q^{4} + q^{5}  q^{6}  2q^{7} + ...
431B
F:
y^{2} + xy + y = x^{3}  x^{2}  9x  8
f_{F} = q  q^{2} + 3q^{3}  q^{4}  3q^{5}  3q^{6} + 2q^{7} + ...
Observe that f_{E} = f_{F} (mod 2).
A homology computation shows that
E /\ F = {0} in J_{0}(431).
Let m = ker(T>F_{2},
T_{p} :> a_{p}(E) mod 2).
Then
E[2] + F[2] subset J_{0}(431)[m],
but E[2] + F[2] has dimension FOUR, so "multiplicity one" fails!
 Magma log

Second example of Kilford: N=503
E=503A
F=503B
G=503C
The corresponding newforms are congruent modulo 2, but
E[2]+F[2]+G[2] has F_{2}dimension FOUR.
 Magma log
 Temptation! Does r_{A} = m_{A} for each A <===> J_{0}(N)
satisfies multiplicity one?
False, since r_{A} = m_{A} for all A
in the N=503 example.
 Kilford has found many more counterexamples to multiplicity one.
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Last modified: May 16, 2000