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# The modular degree and the congruence modulus

In 1985, Zagier published a proof due to Ribet that the modular degree of an optimal elliptic curve of prime conductor equals its congruence modulus. Over a decade later, Frey and Muller asked whether or not this assertion generalizes to arbitrary optimal elliptic curves (Question 4.4 of Frey and Muller, Arithmetic of Modular Curves and Applications). It does not, as the elliptic curve 54B demonstrates.

In another direction, Amod Agashe asked whether or not Ribet's theorem generalized to optimal quotients of J0(N) of dimension greater than 1 and prime level. In one formulation it does not, as 431F shows.

## History

1. Zagier:
• Zagier wrote Modular Paramaterizations of Elliptic Curves (1985).
• He gave an algorithm to compute the modular degree mE of any optimal prime conductor elliptic curve:
```         X0(N)
|
|
| phi
|
\|/
E

mE := deg(phi)
```
• Partial Table:  NE mE 11 1 17 1 179 9 197 10

2. Doi-Ohta:
• Computed congruence modulus rE (1977, SLNM 601):
rE := # S2(N) / (Z fE + (Z fE)perp)
= max { r : f = g (mod r), some g in S2(N), g not a multiple of f}
• Partial Table:  NE rE 11 1 17 1 179 9 197 10

3. van der Geer: Hey look!!

4. Ribet:
Theorem: If NE is prime, then rE = mE.
Main ingredients:
• T = End(J0(N)/Q), when N is prime
• dim E = 1 and E is self dual.

## Temptations

1. Frey-Mueller: Does rE = mE for every optimal elliptic curve E?
2. Agashe-Stein: Does rA = mA for every newform optimal abelian variety quotient A at prime level?

The quantities rA and mA are defined as follows:
Fix a newform f in S2(N;C). Let

If := AnnT(f).

Then
Af := J0(N) / IfJ0(N)
mA  :=   sqrt(deg(Afv --> Af))
rA  :=  #(S2(N) / (S2(N)[If] + S2(N)[If]v)).

## Examples

1. First counterexample to temptation 1
54B

E:      y2 + xy + y = x3 - x2 + x -1

2. First counterexample to temptation 2
431F is an abelian variety of dimension 24. (Note: 431 is prime.)
• mA = 211 * 6947
• rA = 210 * 6947
• Note; mA doesn't even divide rA.
• Ribet really proves a statement about annihilators. When dim A = 1 this is equivalent to proving statements about mA and rA.
• Agashe proved that at square-free level that

p divides mA <===> p divides rA.

• Magma log

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