Some Modular Degree and Congruence Modulus Computations

William A. Stein

July 23, 2001

Contents

1 The Definitions 1

2 The History 2

3 The Naive Algorithms 3 3.1 A way to compute m_{E} . . . . . . . . . . . . . . . . . . . . . . 3 3.2 A way to compute c_{E} . . . . . . . . . . . . . . . . . . . . . . . 3

4 The Examples 3

5 The Future 4

1 The Definitions

Let E/Q be an elliptic curve that is an optimal quotient of J_{0}(N_{E}), where N = N_{E} is the conductor of E. Here J_{0}(N) is the Jacobian of the algebraic curve X_{0}(N) and a deep theorem implies that there is a surjective morphism π : X_{0}(N) → E. The condition that E is optimal means that the induced map π_{∗} : J_{0}(N) → E has (geometrically) connected kernel.

Definition 1.1. The modular degree of E is

m_{E} = deg(π).

One reason that the modular degree is well worth thinking about is that an assertion about how m_{E} grows relative to N_{E} is equivalent to the ABC Conjecture.

Let f = f_{E} = ∑ a_{n}q^{n} ∈ S_{2}(Γ_{0}(N)) be the newform attached to E.

Definition 1.2. The congruence modulus of E is

c_{E} = # (\frac{S_{2}(Γ_{0}(N), Z)}{Zf + (Zf)^{⊥}}) ,

where (Zf)^{⊥} is the unique T = Z[… T_{n} …]-module complement of Zf in S_{2}(Γ_{0}(N), Z). Equivalently,

c_{E} = max{c : f ≅ g (mod c) for some g ∈ (Zf)^{⊥} }.

2 The History

• <1984: ??

• 1984: Don Zagier wrote the often-cited paper Modular parametrizations of elliptic curves (1985), in which he gave an algorithm to compute m_{E} (sometimes?). The paper incluced

– A result of Ribet:

Theorem 2.1 (Ribet). If N_{E} is prime, then

m_{E} = c_{E}.

– It also said

c_{E} | m_{E}.

• 1998: Frey and Müller published a wonderful survey: Arithmetic of modular curves and applications.

– They ask: Question 4.4: Let E be an optimal quotient of any conductor. Does m_{E} = c_{E}?

– They remark that c_{E} | m_{E} and give two references [Ribet 83, Inventiones] and [Zagier 1985].

• 1995: Cremona wrote a Math. Comp. paper, and computed m_{E} for every curve of conductor ≤ N, where N is a few thousand.

• 2001: Mark Watkins, who did a Ph.D. on the class number problem of Gauss, computed m_{E} for some curves with N_{E} HUGE, using an algorithm he created from a formula of M. Flach.

3 The Naive Algorithms

3.1 A way to compute m_{E}

Use the (not-exact!) sequence:

H_{1}(E, Z) → H_{1}(X_{0}(N), Z) → H_{1}(E, Z).

The composition map from H_{1}(E, Z) → H_{1}(E, Z) is multiplication by m_{E}, and H_{1}(E, Z) can be computed because its image in H_{1}(X_{0}(N), Z) is saturated, as E is optimal. This algorithm is described in detail in [Kohel-Stein, ANTS IV], and amounts to finding “left and right eigenvectors” and taking their dot product.

3.2 A way to compute c_{E}

Compute S_{2}(Γ_{0}(N), Z) ⊂ Z[[q]] to precision [SL_{2}(Z) : Γ_{0}(N)]/6 using, e.g., modular symbols, then use a Smith Normal Form algorithm.

4 The Examples

These examples were computed by myself and Amod Agashe.

• 54B: Let E be the elliptic curve y^{2} + xy + y = x^{3} − x^{2} + x − 1. Then m_{E} = 2 and c_{E} = 6. In fact, it’s easy to see that 3 | c_{E} “by hand” by writing down the form f corresponding to 54B and the form g corresponding to X_{0}(27) and noting that f(q) ≡ g(q) + g(q^{2}) (mod 3). (Because of the “Sturm Bound”, it suffices to check this up to O(q^{19}).) Hey c_{E} ≠ m_{E}!! In fact, c_{E} |̸ m_{E}!! When we first did this computation, Ribet had already mentioned to us that he had really proved that m_{E} | c_{E}, not vice-versa. We were, however, extremely surprised to find so quickly an example in which c_{E} ≠ m_{E}.

• T-shirt: My t-shirt has 243A and 243B on it. For 243A, we have m_{E} = 9 and c_{E} = 27. For 243B, we have m_{E} = 6 and c_{E} = 54. I designed the t-shirt many months before I knew that question 4.4 had a negative answer.

• 242B: N = 2 · 11^{2}.

m_{E} = 2^{4} ≠ c_{E} = 2^{4} · 11

The failure is probably not just a “small primes” phenomenon.

Moral: A little computation sometimes greatly cleans the air.

5 The Future

Based on computations, Amod and I conjectured and Ribet proved the following theorem.

Theorem 5.1 (Ribet, 2001). Let E be an elliptic curve of conductor N. If p^{2} ∤ N then ord_{p}(m_{E}) = ord_{p}(c_{E}).

New Version of “Question 4.4. For all N_{E} ≤ 539, we have

ord_{p}(c_{E}/m_{E}) ≤ ord_{p}(N_{E}) − 1.

In particular, for p ≥ 5, do we have

ord_{p}(c_{E}/m_{E}) ≤ 1?

Is this true in general?