Optimal Elliptic Curve Quotients

Let $N$ be a positive integer and let $X_0(N)$ be the modular curve over ${\bf {Q}}$ that classifies isomorphism classes of elliptic curves with a cyclic subgroup of order $N$ . The Hecke algebra ${\bf {T}}$ of level $N$ is the subring of the ring of endomorphisms of $J_0(N)={\mathrm{Jac}}(X_0(N))$ generated by the Hecke operators $T_n$ for all $n\geq 1$ . Suppose $f$ is a newform of weight $2$ for  $\Gamma_0(N)$ with integer Fourier coefficients, and let $I_f$ be kernel of the homomorphism ${\bf {T}}\to {\bf {Z}}[\ldots, a_n(f), \ldots]$ that sends $T_n$ to $a_n(f)$ . Then the quotient $E = J_0(N)/I_f J_0(N)$ is an elliptic curve over ${\bf {Q}}$ . We call $E$ the optimal quotient associated to $f$ . Composing the embedding $X_0(N)\hookrightarrow J_0(N)$ that sends $x$ to $(\infty) -(x)$ with the quotient map $J_0(N) \rightarrow E$ , we obtain a surjective morphism of curves $\phi_{\scriptscriptstyle E}: X_0(N) \rightarrow E$ . The modular degree $m_{\scriptscriptstyle E}$ of $E$ is the degree of  $\phi_{\scriptscriptstyle E}$ .

Let $E_{{\bf {Z}}}$ denote the Néron model of $E$ over ${\bf {Z}}$ . A general reference for Néron models is [BLR90]; for the special case of elliptic curves, see, e.g., [Sil92, App. C, §15], and [Sil94]. Let $\omega$ be a generator for the rank $1$ ${{\bf {Z}}}$ -module of invariant differential $1$ -forms on  $E_{{\bf {Z}}}$ . The pullback of $\omega$ to $X_0(N)$ is a differential $\phi_{\scriptscriptstyle E}^*\omega$ on $X_0(N)$ . The newform $f$ defines another differential  $2 \pi i f(z) dz = f(q)dq/q$ on $X_0(N)$ . Because the action of Hecke operators is compatible with the map $X_0(N)\to E$ , the differential $\phi_{\scriptscriptstyle E}^*\omega$ is a ${\bf {T}}$ -eigenvector with the same eigenvalues as $f(z)$ , so by [AL70] we have $\phi_{\scriptscriptstyle E}^*\omega = c \cdot 2 \pi i f(z) dz$ for some $c \in {\bf {Q}}^*$ (see also [Man72, §5]). The Manin constant $c_{\scriptscriptstyle{E}}$ of $E$ is the absolute value of the rational number $c$ defined above.

The following conjecture is implicit in [Man72, §5].

Conjecture 2.1 (Manin)   We have $c_{\scriptscriptstyle{E}}= 1$ .

Significant progress has been made towards this conjecture. In the following theorems, $p$ denotes a prime and $N$ denotes the conductor of $E$ .

Theorem 2.2 (Edixhoven [][Prop. 2)   edix:manin] The constant $c_{\scriptscriptstyle{E}}$ is an integer.

Edixhoven proved this using an integral $q$ -expansion map, whose existence and properties follow from results in [KM85]. We generalize his theorem to quotients of arbitrary dimension in Theorem 3.4.

Theorem 2.3 (Mazur, [][Cor. 4.1)   maziso] If  $p\mid c_{\scriptscriptstyle{E}}$ , then $p^2 \mid 4N$ .

Mazur proved this by applying theorems of Raynaud about exactness of sequences of differentials, then using the ``$q$ -expansion principle'' in characteristic $p$ and a property of the Atkin-Lehner involution. We generalize Mazur's theorem in Corollary 3.7.

The following two results refine the above results at $p=2$ .

Theorem 2.4 (Raynaud [][Prop. 3.1)   abbull] If $4\mid c_{\scriptscriptstyle{E}}$ , then $4\mid N$ .

Theorem 2.5 (Abbes-Ullmo [][Thm. A)   abbull] If $p\mid c_{\scriptscriptstyle{E}}$ , then $p \mid N$ .

We generalize Theorem 2.4 in Theorem 3.10. However, it is not clear if Theorem 2.5 generalizes to dimension greater than $1$ . It would be fantastic if the theorem could be generalized. It would imply that the Manin constant is $1$ for newform quotients $A_f$ of $J_0(N)$ , with $N$ odd and square free, which be useful for computations regarding the conjecture of Birch and Swinnerton-Dyer.

B. Edixhoven also has unpublished results (see [Edi89]) which assert that the only primes that can divide  $c_{\scriptscriptstyle{E}}$ are $2$ , $3$ , $5$ , and $7$ ; he also gives bounds that are independent of $E$ on the valuations of  $c_{\scriptscriptstyle{E}}$ at $2$ , $3$ , $5$ , and $7$ . His arguments rely on the construction of certain stable integral models for $X_0(p^2)$ .

See Section 5 for more details about the following computation:

Theorem 2.6 (Cremona)   If $E$ is an optimal elliptic curve over ${\bf {Q}}$ with conductor at most $130000$ , then $c_E=1$ .

theorem_type[defi][lem][][definition][][] [Congruence Number] The congruence number tex2html_wrap_inline$r_E$ of tex2html_wrap_inline$E$ is the largest integer tex2html_wrap_inline$r$ such that there exists a cusp form tex2html_wrap_inline$g&isin#in;S_2(&Gamma#Gamma;_0(N))$ that has integer Fourier coefficients, is orthogonal to tex2html_wrap_inline$f$ with respect to the Petersson inner product, and satisfies tex2html_wrap_inline$g &equiv#equiv;f r$. The congruence primes of tex2html_wrap_inline$E$ are the primes that divide tex2html_wrap_inline$r_E$.

To the above list of theorems we add the following:

Theorem 2.7   If $p\mid c_{\scriptscriptstyle{E}}$ then $p^2\mid N$ or $p\mid m_{\scriptscriptstyle E}$ .

This theorem is a special case of Theorem 3.11 below. In view of Theorem 2.3, our new contribution is that if $m_{\scriptscriptstyle E}$ is odd and ${\mathrm{ord}}_2(N)=1$ , then $c_{\scriptscriptstyle{E}}$ is odd. This hypothesis is very stringent--of the optimal elliptic curve quotients of conductor $\leq 120000$ , only $56$ of them satisfy the hypothesis. In the notation of [Cre], they are

14a, 46a, 142c, 206a, 302b, 398a, 974c, 1006b, 1454a, 1646a, 1934a, 2606a, 2638b, 3118b, 3214b, 3758d, 4078a, 7054a, 7246c, 11182b, 12398b, 12686c, 13646b, 13934b, 14702c, 16334b, 18254a, 21134a, 21326a, 22318a, 26126a, 31214c, 38158a, 39086a, 40366a, 41774a, 42638a, 45134a, 48878a, 50894b, 53678a, 54286a, 56558f, 58574b, 59918a, 61454b, 63086a, 63694a, 64366b, 64654b, 65294a, 65774b, 71182b, 80942a, 83822a, 93614a

Each of the curves in this list has conductor tex2html_wrap_inline$2p$ with tex2html_wrap_inline$p&equiv#equiv; 34$ prime. The situation is similar to that of [SW04, Conj. 4.2], which suggests there are infinitely many such curves. See also [CE05] for a classification of elliptic curves with odd modular degree.