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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 J0(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

1. Mazur: Let N be a prime number. In Modular curves and the Eisenstein Ideal (1977), Mazur proved that for most maximal m in T,

dimT/m J[m] = 2.

"Most" means:
• m Eisenstein, or
• char(T/m) =/= 2, or
• m supersingular, or
• Im(rhom(D2)) not contained in scalars (Buzzard).

Example. J0(11) is an elliptic curve, and for all primes p,

J0(11)[p] = Z/p x Z/p.

## Temptation

1. Does dimT/mJ[m] = 2 for all m?

## Examples

1. Lloyd Kilford's first counterexample to the temptation

431A        E:      y2 + xy = x3 - 1
fE = q - q2 + q3 - q4 + q5 - q6 - 2q7 + ...

431B        F:      y2 + xy + y = x3 - x2 - 9x - 8
fF = q - q2 + 3q3 - q4 - 3q5 - 3q6 + 2q7 + ...

Observe that fE = fF (mod 2). A homology computation shows that

E /\ F = {0} in J0(431).

Let m = ker(T--->F2,     Tp :--> ap(E) mod 2). Then

E + F subset J0(431)[m],

but E + F has dimension FOUR, so "multiplicity one" fails!
2. Magma log

3. Second example of Kilford: N=503
E=503A
F=503B
G=503C
The corresponding newforms are congruent modulo 2, but E+F+G has F2-dimension FOUR.
4. Magma log

5. Temptation! Does rA = mA for each A <===> J0(N) satisfies multiplicity one?

False, since rA = mA for all A in the N=503 example.

6. Kilford has found many more counterexamples to multiplicity one.

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