MODULAR PARAMETRIZATIONS OF NEUMANN–SETZER ELLIPTIC CURVES

WILLIAM STEIN AND MARK WATKINS

Abstract. Suppose p is a prime of the form u^{2} + 64 for some integer u, which we take to be 3 mod 4. Then there are two Neumann–Setzer elliptic curves E_{0} and E_{1} of prime conductor p, and both have Mordell–Weil group Z/2Z. There is a surjective map X_{0}(p) ⟶^{π} E_{0} that does not factor through any other elliptic curve (i.e., π is optimal), where X_{0}(p) is the modular curve of level p. Our main result is that the degree of π is odd if and only if u ≡ 3 (mod 8). We also prove the prime-conductor case of a conjecture of Glenn Stevens, namely that that if E is an elliptic curve of prime conductor p then the optimal quotient of X_{1}(p) in the isogeny class of E is the curve with minimal Faltings height. Finally we discuss some conjectures and data about modular degrees and orders of Shafarevich–Tate groups of Neumann–Setzer curves.

1. Introduction

Let p be a prime of the form u^{2} + 64 for some integer u, which we take to be 3 modulo 4. Neumann and Setzer [Neu71, Set75] considered the following two elliptic curves of conductor p (note that Setzer chose u ≡ 1 (mod 4) instead):

(1.1)  E_{0} : y^{2} + xy = x^{3} − \frac{u + 1}{4}x^{2} + 4x − u,
(1.2)  E_{1} : y^{2} + xy = x^{3} − \frac{u + 1}{4}x^{2} − x.

For E_{1} we have c_{4} = p − 16 and c_{6} = u(p + 8) with Δ = p = u^{2} + 64, while for E_{0} we have c_{4} = p − 256 and c_{6} = u(p + 512) with Δ = −p^{2}. Thus each E_{i} is isomorphic to a curve of the form y^{2} = x^{3} − 27c_{4}x − 54c_{6} for the indicated values of c_{4} and c_{6}. The curves E_{0} and E_{1} are 2-isogenous and one can show using Lutz-Nagell and descent via 2-isogeny that

E_{0}(Q) = E_{1}(Q) = Z/2Z.

Moreover, if E is any elliptic curve over Q of prime conductor with a rational point of order 2 then E is a Neumann–Setzer curve or has conductor 17 (see [Set75]).

Let X_{0}(p) be the modular curve of level p. By [Wil95] there is a surjective map π : X_{0}(p) → E_{0}, and by [MO89, §5, Lem. 3] we may choose π to be optimal, in the sense that π does not factor through any other elliptic curve. The modular degree of E_{0} is deg(π).

We prove in Section 2 that the modular degree of E_{0} is odd if and only if u ≡ 3 (mod 8). Our proof relies mostly on results from [Maz77]. In Section 3 we show that E_{1} is the curve of minimal Faltings height in the isogeny class {E_{0}, E_{1}} of E_{1} and prove that E_{1} is an optimal quotient of X_{1}(p), which is enough to prove the prime-conductor case of a conjecture of [Ste89] (this case is not covered by the results of [Vat03]). Finally, in Section 4 we give evidence for our conjecture that there are infinitely many elliptic curves with odd modular degree, and give a conjectural refinement of Theorem 2.1. We also present some data about p-divisibility of conjectural orders of Shafarevich–Tate groups of Neumann–Setzer curves.

Date: December 2003.

1.1. Notation. Let p be a prime and n be the numerator of (p − 1)/12.

We use standard notation for modular forms, modular curves, and Hecke algebras, as in [DI95] and [Maz77]. In particular, let X_{0}(p) be the compactified coarse moduli space of elliptic curves with a cyclic subgroup of order p. Then X_{0}(p) is an algebraic curve defined over Q. Let J = J_{0}(p) be the Jacobian of X_{0}(p), and let T = Z[T_{2}, T_{3}, …] ⊂ End(J) be the Hecke algebra. Also, let X_{1}(p) be the modular curve the classifies isomorphism classes of pairs (E, P), where P ∈ E is a point of order p.

To each newform f ∈ S_{2}(Γ_{0}(p)), there is an associated abelian subvariety A = A_{f} ⊂ J_{0}(p). We call the kernel Ψ_{A} of the natural map A ↪ J → A^{∨} the modular kernel. For example, when A is an elliptic curve, this map is induced by pullback followed by push forward on divisors and Ψ_{A} is multiplication by deg(X_{0}(p) → A). The modular degree of A is the square root of the degree of Ψ_{A}. This definition makes sense even when dim(A) > 1, since the degree of a polarization is the square of its Euler characteristic, hence a perfect square (see [Mum70, §16, pg. 150]). If I ⊂ T is an ideal, let

A[I] = \{x ∈ A(\overline{Q}) : Ix = 0\}  and  A[I^{∞}] = ⋃_{n>0} A[I^{n}].

1.2. Acknowledgements. The authors would like to thank the American Institute of Mathematics for hospitality while they worked on this paper, the National Science Foundation for financial support, and Matt Baker, Frank Calegari, and Barry Mazur for helpful conversations.

2. Determination of the Parity of the Modular Degree

Let p, E_{0}, J and n be as in Section 1, and fix notation as in Section 1.1. In this section we prove the following theorem.

Theorem 2.1. The modular degree of E_{0} is odd if and only if u ≡ 3 (mod 8).

In order to prove the theorem we deduce seven lemmas using techniques and results from [Maz77].

Let m be the modular degree of E_{0}, and let

B = ker(J ⟶^{π} E_{0}).

Lemma 2.2. We have m^{2} = #(B ∩ E_{0}).

Proof. As mentioned in Section 1.1, the composition E_{0} → J → E_{0} is multiplication by the degree of X_{0}(p) → E_{0}, i.e., multiplication by the modular degree of E_{0}. The lemma follows since multiplication by m on E_{0} has degree m^{2}. □

The Eisenstein ideal I of T is the ideal generated by T_{ℓ} − (ℓ + 1) for ℓ ≠ p and T_{p} −1. By hypothesis, there is a Neumann–Setzer curve of conductor p, which implies that the numerator n of (p − 1)/12 is even (we do the elementary verification that this numerator is even in the proof of Theorem 2.1 below). As discussed in [Maz77, Prop. II.9], the 2-Eisenstein prime 𝔪 = (2) + I of T is a maximal ideal of T, with T/𝔪 ≅ Z/2Z.

Lemma 2.3. We have E_{0}[𝔪] = E_{0}[2].

Proof. By [Maz77, Prop. II.11.1, Thm. III.1.2], the Eisenstein ideal I annihilates J(Q)_{tor}, so 𝔪 annihilates J(Q)_{tor}[2]. Since J(Q)_{tor} is cyclic of order n (by [Maz77, Thm. III.1.2]), J(Q)_{tor}[2] has order 2, so J(Q)_{tor}[2] = E_{0}(Q)_{tor}[2], hence E_{0}(Q)[𝔪] ≠ 0. The Hecke algebra T acts on E_{0} through End(E_{0}) ≅ Z, so each element of T acts on E_{0} as an integer; in particular, the elements of 𝔪 all act as multiples of 2 (since E_{0}[𝔪] ≠ 0 and 2 ∈ 𝔪), so E_{0}[𝔪] = E_{0}[2] since 2 ∈ 𝔪. □

Lemma 2.4. Suppose A ⊂ J_{0}(p) is a T-stable abelian subvariety and ℘ ⊂ T is a maximal ideal such that A[℘^{∞}] ≠ 0. Then A[℘] ≠ 0. Also A[℘^{∞}] is infinite.

Proof. Arguing as in [Maz77, §II.14, pg. 112], we see that for any r, A[℘^{r}]/A[℘^{r+1}] is isomorphic to a direct sum of copies of A[℘]. If A[℘] = 0, then since A[℘^{∞}] ≠ 0, there must exist an r such that A[℘^{r}]/A[℘^{r+1}] ≠ 0. But A[℘^{r}]/A[℘^{r+1}] is contained in a direct sum of copies of A[℘] = 0, which is a contradiction.

To see that A[℘^{∞}] is infinite, note that if ℓ is the residue characteristic of ℘ and Tate_{ℓ}(A) is the Tate module of A at ℓ, then

Tate_{℘}(A) = \underleftarrow{lim}_{r} A[℘^{r}] = Tate_{ℓ}(A) ⊗_{T} T_{℘}

is infinite. (For more details, see the proof of [RS01, Prop. 3.2].) □

The analogues of Lemmas 2.5–2.7 below are true, with the same proofs, for 𝔪 any Eisenstein prime. We state and prove them for the 2-Eisenstein prime, since that is the main case of interest to us. Let J̃^{(2)} be the 2-Eisenstein quotient of J, where J̃^{(2)} is as defined in [Maz77, §II.10]. More precisely, we have the following:

Lemma 2.5. The simple factors of J̃^{(2)} correspond to the Gal(\overline{Q}/Q)-conjugacy classes of newforms f such that A_{f}[𝔪] ≠ 0 (or equivalently, A_{f}^{∨}[𝔪] ≠ 0).

Proof. On page 97 of [Maz77] we find that the C-simple factors of J̃^{(2)} are in bijection with the irreducible components Spec(I_{f}) of Spec(T) which meet the support of the ideal 𝔪, so the I_{f} are the newform ideals contained in 𝔪. We have for any I_{f},

I_{f} ⊂ 𝔪 ⟺ T_{m}/(I_{f})_{𝔪} ≠ 0 ⟺ Tate_{𝔪}(A_{f}) ≠ 0 ⟺ A_{f}[𝔪] ≠ 0.

Note that the same argument applies to A_{f}^{∨}. □

Lemma 2.6. Suppose A and B are abelian varieties equipped with an action of the Hecke ring T and that φ : A → B is a T-module isogeny. If ℘ ⊂ T is a maximal ideal and B[℘] ≠ 0, then also A[℘] ≠ 0.

Proof. Let ψ : B → A be the isogeny complementary to φ, so ψ is the unique isogeny such that ψ∘φ is multiplication by deg(φ). Then ψ is also a T-module homomorphism (one can see this in various ways; one way is to use the rational representation on homology to view the endomorphisms as matrices acting on lattices, and to note that if matrices M and N commute, then M^{−1} and N also commute). By Lemma 2.4, the union B[℘^{∞}] is infinite, so ψ(B[℘^{∞}]) ≠ 0. Since ψ(B[℘^{∞}]) ⊂ A[℘^{∞}], Lemma 2.4 implies that A[℘] ≠ 0, as claimed. □

Lemma 2.7. Suppose B ⊂ J_{0}(p) is a sum of abelian subvarieties A_{f} attached to newforms. If B[𝔪] ≠ 0, then there is some A_{f} ⊂ B such that A_{f}[𝔪] ≠ 0.

Proof. There is something to be proved because if x ∈ B[𝔪] it could be the case that x = y + z with y ∈ A_{f} and z ∈ A_{g}, but x ∉ A_{h} for any h. Let C = ⊕A_{f}, where the A_{f} ⊂ J_{0}(p) are simple abelian subvarieties of B corresponding to conjugacy classes of newforms. Then there is an isogeny φ : C → B given by

φ(x_{1}, …, x_{n}) = x_{1} + ⋯ + x_{n},

where the sum is in B ⊂ J_{0}(p). By Lemma 2.6, C[𝔪] ≠ 0. Since C[𝔪] ≅ ⊕A_{f}[𝔪], it follows that A_{f}[𝔪] ≠ 0 for some A_{f} ⊂ B. □

Lemma 2.8. If 4 | n, then dim J̃^{(2)} > 1.

Proof. This follows from the remark on page 163 of [Maz77]. Since the proof is only sketched there, we give further details for the convenience of the reader. Because 4 | n, the cuspidal subgroup C, which is generated in J_{0}(p) by (0) − (∞) and is cyclic of order n, contains an element of order 4. Let C(2) be the 2-primary part of C, and let D = ker(J_{0}(p) → J̃^{(2)}). If there is a nonzero element in the kernel of the homomorphism C(2) → J̃^{(2)}, then D[𝔪] ≠ 0, where 𝔪 is the 2-Eisenstein prime. But then by Lemma 2.7, there is an A_{f} ⊂ D such that A_{f}[𝔪] ≠ 0. By Lemma 2.5, A_{f}^{∨} is a quotient of J̃^{(2)}, so A_{f} ⊂ (J̃^{(2)})^{∨} so A_{f} cannot be in D. This contradiction shows that the map C(2) → J̃^{(2)} is injective, so J̃^{(2)} contains a rational point of order 4. However, as mentioned in the introduction, E_{0}(Q) has order 2, so J̃^{(2)} ≠ E_{0}. Thus J̃^{(2)} has dimension bigger than 1. □

Having established the above lemmas, we are now ready to deduce the theorem.

Proof of Theorem 2.1. It seems more straightforward to prove the equivalent statement that the modular degree is even if and only if u ≡ 7 (mod 8), so we will prove this instead.

(⟹) u ≡ 7 (mod 8) implies that the modular degree is even: Writing u = 8k + 7 we see that p = (8k + 7)^{2} + 64 ≡ 1 (mod 16), so 16 | (p − 1) hence 4 | n. By Lemma 2.3 and Lemma 2.5, E_{0} is a factor of J̃^{(2)}. By Lemma 2.8, the dimension of J̃^{(2)} is bigger than 1, so by Lemma 2.5 there is an A_{f} distinct from E_{0} such that A_{f}[𝔪] ≠ 0. Since A_{f} ⊂ B = ker(J_{0}(p) → E_{0}), it follows that B[𝔪] ≠ 0. As discussed on page 38 of [Maz77], J[𝔪] has dimension 2 over F_{2} so E_{0}[𝔪] = J[𝔪], hence B[𝔪] ⊂ E_{0}[𝔪]. It follows that 2 | #(B ∩ E_{0}), so E_{0} has even modular degree.

(⟸) Modular degree even implies that u ≡ 7 (mod 8): Suppose that the modular degree m of E_{0} is even. Letting B = ker(J_{0}(p) → E_{0}), we have

E_{0} ∩ B ≅ ker(E_{0} → J_{0}(p) → E_{0}),

so Ψ := E_{0} ∩ B = E_{0}[m]. Lemma 2.3 and our assumption that m is even imply that

E_{0}[𝔪] = E_{0}[2] ⊂ E_{0}[m] = Ψ,

so Ψ[𝔪] ≠ 0. Since Ψ[𝔪] ≠ 0, and Ψ ⊂ B, we have B[𝔪] ≠ 0. By Lemma 2.7, there is some A_{f} ⊂ B such that A_{f}[𝔪] ≠ 0. Then by Lemma 2.5 we see that A_{f} is an isogeny factor of J̃^{(2)}. Thus J̃^{(2)} has dimension bigger than 1. If u = 8k + 3, then p = (8k + 3)^{2} + 64 ≡ 9 (mod 16), so that 2 || n. However, when 2 || n, [Maz77, Prop. III.7.5] implies that J̃^{(2)} = E_{0}, which is false, so u ≡ 7 (mod 8). □

Remark 2.9. Frank Calegari observed that Lemma 2.8 and its converse also follow from conditions (i) and (v) of Théorème 3 of [Mer96].

3. The Stevens Conjecture for Neumann–Setzer Curves is True

Let E be an arbitrary elliptic curve over Q of conductor N. Stevens conjectured in [Ste89] that the optimal quotient of X_{1}(N) in the isogeny class of E is the curve in the isogeny class of E with minimal Faltings height. In this section we explain why this conjecture is true when N is prime.

Let p = u^{2}+64 be prime and E_{1} and E_{0} be as in Section 1. In this section we verify that the curve E_{1} has smaller Faltings height than E_{0}, then show that E_{1} is X_{1}(p)- optimal. The Stevens conjecture asserts that the X_{1}(p)-optimal curve is the curve of minimal Faltings height in an isogeny class, so our results verify the conjecture for Neumann–Setzer curves. In fact, the Stevens conjecture is true for all isogeny classes of elliptic curves of prime conductor. For if E is an elliptic curve of prime conductor, then by [Set75] there is only one curve in the isogeny class of E, unless E is a Neumann–Setzer curve or the conductor of E is 11, 17, 19, or 37. When the isogeny class of E contains only one curve, that curve is obviously both X_{1}-optimal and of minimal Faltings height. The conjecture is also well-known to be true for curves of conductor 11, 17, 19, or 37 (see [Ste89]). We note that Vatsal [Vat03] has recently extended results of Tang [Tan97] that make considerable progress toward the Stevens conjecture, but his work is not applicable to Neumann–Setzer curves.

Lemma 3.1. The curve E_{1} has smaller Faltings height than E_{0}.

Proof. By [Ste89, Thm. 2.3, pg. 84] it is enough to exhibit an isogeny from E_{1} to E_{0} whose extension to Néron models is étale. Let φ be the isogeny E_{0} → E_{1} of degree 2 whose kernel is the subgroup generated by the point whose coordinates are (u/4, −u/8) in terms of the Weierstrass equation (1.1) for E_{0}, which is a global minimal model for E_{0}. The kernel of φ does not extend to an étale group scheme over Z, since its special fiber at 2 is not étale (it has only one \overline{F}_{2}-point), so the morphism on Néron models induced by E_{0} → E_{1} cannot be étale, since kernels of étale morphisms are étale. By [Ste89, Lemma 2.5] the dual isogeny E_{1} → E_{0} extends to an étale morphism of Néron models. □

Proposition 3.2. The curve E_{1} is X_{1}(p)-optimal.

Proof. By [MO89, §5, Lem. 3], E_{0} is an optimal quotient of X_{0}(p), so we have an injection E_{0} ↪ J_{0}(p). As in [Maz77, pg. 100], let Σ be the kernel of the functorial map J_{0}(p) → J_{1}(p) induced by the cover X_{1}(p) → X_{0}(p). By [Maz77, Prop. II.11.6], Σ is the Cartier dual of the constant subgroup scheme U, which turns out to equal J_{0}(p)(Q)_{tor}. Because #(E_{0} ∩ U) = 2 and E_{0}[2] is self dual, we have #(E_{0} ∩ Σ) = 2. Thus the image of E_{0} in J_{1}(p) is the quotient of E_{0} by the subgroup generated by the rational point of order 2 (note that the Cartier dual of Z/2Z is μ_{2} = Z/2Z). This quotient is E_{1}, so E_{1} ⊂ J_{1}(p), which implies that E_{1} is an optimal quotient of X_{1}(p), as claimed. □

Remark 3.3. The above proposition could also be proved in a slightly different manner. The Faltings height of an elliptic curve is \sqrt{2π/Ω} where Ω is the volume of the fundamental parallelogram associated to the curve. When the conductor is prime, we have by [AL96] that the Manin constants for X_{0}(p) and X_{1}(p) are 1; this says that for a G-optimal curve E, the period lattice generated by G has covolume equal to Ω_{E}. Since the lattice generated by Γ_{1}(p) is contained in the lattice generated by Γ_{0}(p) (and thus has larger covolume), the Faltings height of the X_{1}(p)-optimal curve must be less than or equal to that of the X_{0}(p)-optimal curve. So if these two curves differ, the X_{1}(p)-optimal curve must have smaller Faltings height.

Remark 3.4. On page 12 of [Maz98], there is a “To be removed from the final draft” comment that asks (in our notation) whether E_{0} is X_{0}(p)-optimal when p ≡ 1 (mod 16). This is already answered by [MO89], whereas here we go further and show additionally that E_{1} is X_{1}(p)-optimal.

4. Conjectures

4.1. Refinement of Theorem 2.1. The following conjectural refinement of Theorem 2.1 is supported by the experimental data of [Wat02]. It is unclear whether the method of proof of Theorem 2.1 can be extended to prove this conjecture.

Conjecture 4.1. If u ≡ 7 (mod 8), then 2 exactly divides the modular degree of E_{0} if and only if u ≡ 7 (mod 16).

We can note that the pattern seems to end here; for curves with u ≡ 15 (mod 16) the data give no further information about the 2-valuation of the modular degree. For instance, with u = −17 we have that [1, 1, 1, −2, 16] has modular degree 2^{3} · 3, while with u = 175 the curve [1, 1, 1, −634, −6484] has modular degree 2^{2} · 3^{3} · 5 · 23. Similarly, we have that u = −33 gives the curve [1, −1, 1, −19, 68] with modular degree 2^{5} · 3, while u = 127 gives the curve [1, 1, 1, −332, −2594] of modular degree 2^{2} · 3^{2} · 5 · 43.

4.2. The Parity of the Modular Degree. According to Cremona’s tables [Cre], of the 29755 new optimal elliptic curve quotients of J_{0}(N) with N < 8000, a mere 89 have odd modular degree, which is less than 0.3%. There are 52878 non Neumann– Setzer curves in the database of [BM90] with prime conductor N ≤ 10^{7}; of these curves 4592, or 8%, have odd modular degree (see [Wat02]). One reason that curves tend to have even modular degree is that for many curves the modular parametrization factors through an Atkin-Lehner quotient. Note that the method of [Wat02] used to compute the modular degree is rigourous when the level is prime because by [AL96] the Manin constant is 1 when the level is odd and square-free.

If f(x) = (8x + 3)^{2} + 64, then it is a well-known conjecture (see [HL22] and e.g., [Guy94, §A1]) that there are infinitely many primes of the form f(n) for some integer n, thus we make the following conjecture.

Conjecture 4.2. There are infinitely many elliptic curves over Q with odd modular degree.

Our data suggest the following conjecture:

Conjecture 4.3. If E is an optimal elliptic curve quotient of J_{0}(p) with p ≢ 3 (mod 8) and E is not a Neumann–Setzer curve then the modular degree of E is even or p = 17.

There are 23442 Brumer-McGuinness (see [BM90]) curves of conductor 37 ≤ p ≤ 10^{7} with p ≡ 3 (mod 8), of which 11815 have even functional equation, of which 7322 have rank 0, and 4589 have odd modular degree. The significance of the data concerning the rank is that the second author has conjectured that 2^{r} divides the modular degree, where r is the rank.

Remark 4.4. Instead of asking about divisibility by 2, one could ask about divisibility by p. The first author and Frank Calegari make a conjecture about discriminants of Hecke algebras in [CS03] that implies that the modular degree of an elliptic curve of prime conductor p is not divisible by p. This conjecture agrees with our data.

4.3. Shafarevich–Tate Groups of Neumann–Setzer Curves. We consider the distribution of � in the Neumann–Setzer family (and note that similar phenomena occur in the related families listed in [SW02]). We look at u with u^{2} + 64 prime and less than 2 · 10^{12}. We now take u to be positive, which thus replaces the restriction that u be 3 mod 4. The heuristics of [Del01] would seem to give us an idea of how

Table 1. Frequency of a prime dividing �

restriction  number  p = 3  p = 5  p = 7  p = 11
u ≡ 1 (mod 8)  25559  33.2%  16.9%  9.2%  3.0%
u ≡ 3 (mod 8)  25557  39.7%  20.3%  14.3%  8.4%
u ≡ 5 (mod 8)  25584  36.2%  18.5%  11.5%  5.0%
u ≡ 7 (mod 8)  25612  34.3%  20.3%  14.3%  8.2%
u ≡ 0 (mod 3)  34009  36.0%  18.7%  12.1%  6.0%
u ≡ 1 (mod 3)  34032  35.2%  18.6%  11.5%  5.6%
u ≡ 2 (mod 3)  34271  36.3%  19.7%  13.3%  6.9%
u ≡ 0 (mod 5)  34208  33.1%  18.0%  11.4%  5.4%
u ≡ 2 (mod 5)  33879  37.1%  19.5%  12.8%  6.5%
u ≡ 3 (mod 5)  34225  37.3%  19.5%  12.7%  6.5%
total  102312  35.8%  19.0%  12.3%  6.2%
Delaunay  36.1%  20.7%  14.5%  9.2%

often we expect a given prime to divide � . For instance, since Neumann–Setzer curves have rank 0, the prime 3 should divide � about 36.1% of the time. However, Table 1 gives a slightly different story with effects seen that depend on the various congruential properties of u.

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