Appendix by J. Cremona: Verifying that $c=1$

Let $f$ be a normalised rational newform for $\Gamma_0(N)$ . Let $\Lambda_f$ be its period lattice; that is, the lattice of periods of $2\pi if(z)dz$ over $H_1(X_0(N),{\bf {Z}})$ .

We know that $E_f={\bf {C}}/\Lambda_f$ is an elliptic curve $E_f$ defined over ${\bf {Q}}$ and of conductor $N$ . This is the optimal quotient of $J_0(N)$ associated to $f$ . Our goal is two-fold: to identify $E_f$ (by giving an explicit Weierstrass model for it with integer coeffients); and to show that the associated Manin constant for $E_f$ is $1$ . In this section we will give an algorithm for this; our algorithm applies equally to optimal quotients of $J_1(N)$ .

As input to our algorithm, we have the following data:

1. a ${\bf {Z}}$ -basis for $\Lambda_f$ , known to a specific precision;
2. the type of the lattice $\Lambda_f$ (defined below); and
3. a complete isogeny class of elliptic curves $\{E_1,\dots,E_m\}$ of conductor $N$ , given by minimal models, all with $L(E_j,s)=L(f,s)$ .

So $E_f$ is isomorphic over ${\bf {Q}}$ to $E_{j_0}$ for a unique $j_0\in\{1,\dots,m\}$ .

The justification for this uses the full force of the modularity of elliptic curves defined over ${\bf {Q}}$ : we have computed a full set of newforms $f$ at level $N$ , and the same number of isogeny classes of elliptic curves, and the theory tells us that there is a bijection between these sets. Checking the first few terms of the $L$ -series (i.e., comparing the Hecke eigenforms of the newforms with the traces of Frobenius for the curves) allows us to pair up each isogeny class with a newform.

We will assume that one of the $E_j$ , which we always label $E_1$ , is such that $\Lambda_f$ and $\Lambda_1$ (the period lattice of $E_1$ ) are approximately equal. This is true in practice, because our method of finding the curves in the isogeny class is to compute the coefficients of a curve from numerical approximations to the $c_4$ and $c_6$ invariants of ${\bf {C}}/\Lambda_f$ ; in all cases these are very close to integers which are the invariants of the minimal model of an elliptic curve of conductor $N$ , which we call $E_1$ . The other curves in the isogeny class are then computed from $E_1$ . For the algorithm described here, however, it is irrelevant how the curves $E_j$ were obtained, provided that $\Lambda_1$ and $\Lambda_f$ are close (in a precise sense defined below).

Normalisation of lattices: every lattice $\Lambda$ in ${\bf {C}}$ which defined over ${\bf {R}}$ has a unique ${\bf {Z}}$ -basis $\omega_1$ , $\omega_2$ satisfying one of the following:

• Type 1: $\omega_1$ and $(2\omega_2-\omega_1)/i$ are real and positive; or
• Type 2: $\omega_1$ and $\omega_2/i$ are real and positive.

For $\Lambda_f$ we know the type from modular symbol calculations, and we know $\omega_1,\omega_2$ to a certain precision by numerical integration; modular symbols provide us with cycles $\gamma_1,\gamma_2\in H_1(X_0(N),{\bf {Z}})$ such that the integral of $2\pi if(z)dz$ over $\gamma_1,\gamma_2$ give $\omega_1,\omega_2$ .

For each curve $E_j$ we compute (to a specific precision) a ${\bf {Z}}$ -basis for its period lattice $\Lambda_j$ using the standard AGM method. Here, $\Lambda_j$ is the lattice of periods of the Néron differential on $E_j$ . The type of $\Lambda_j$ is determined by the sign of the discriminant of $E_j$ : type $1$ for negative discriminant, and type $2$ for positive discriminant.

For our algorithm we will need to know that $\Lambda_1$ and $\Lambda_f$ are approximately equal. To be precise, we know that they have the same type, and also we verify, for a specific postive $\varepsilon$ , that

$$\left\vert\frac{\omega_{1,1}}{\omega_{1,f}}-1\right\vert < \varepsilon$$   and$$\qquad \left\vert\frac{{\rm im}(\omega_{2,1})}{{\rm im}(\omega_{2,f})}-1\right\vert < \varepsilon. \tag{*}$$

Here $\omega_{1,j}$ , $\omega_{2,j}$ denote the normalised generators of $\Lambda_j$ , and $\omega_{1,f}$ , $\omega_{2,f}$ those of $\Lambda_f$ .

Pulling back the Néron differential on $E_{j_0}$ to $X_0(N)$ gives $c\cdot2\pi if(z)dz$ where $c\in{\bf {Z}}$ is the Manin constant for $f$ . Hence

$$c\Lambda_f = \Lambda_{j_0}.$$

Our task is now to

1. identify $j_0$ , to find which of the $E_j$ is (isomorphic to) the ``optimal'' curve $E_f$ ; and
2. determine the value of $c$ .

Our main result is that $j_0=1$ and $c=1$ , provided that the precision bound $\varepsilon$ in (*) is sufficiently small (in most cases, $\varepsilon<1$ suffices). In order to state this precisely, we need some further definitions.

A result of Stevens says that in the isogeny class there is a curve, say $E_{j_1}$ , whose period lattice $\Lambda_{j_1}$ is contained in every $\Lambda_j$ ; this is the unique curve in the class with minimal Faltings height. (It is conjectured that $E_{j_1}$ is the $\Gamma_1(N)$ -optimal curve, but we do not need or use this fact. In many cases, the $\Gamma_0(N)$ - and $\Gamma_1(N)$ -optimal curves are the same, so we expect that $j_0=j_1$ often; indeed, this holds for the vast majority of cases.)

For each $j$ , we know therefore that $a_j=\omega_{1,j_1}/\omega_{1,j}\in{\bf {N}}$ and also $b_j={\rm im}(\omega_{2,j_1})/{\rm im}(\omega_{2,j})\in{\bf {N}}$ . Let $B$ be the maximum of $a_1$ and $b_1$ .

Proposition 5.1   Suppose that (*) holds with $\epsilon=B^{-1}$ ; then $j_0=1$ and $c=1$ . That is, the curve $E_1$ is the optimal quotient and its Manin constant is $1$ .

Proof. Let $\varepsilon=B^{-1}$ and $\lambda=\frac{\omega_{1,1}}{\omega_{1,f}}$ , so $\vert\lambda-1\vert<\varepsilon$ . For some $j$ we have $c\Lambda_f=\Lambda_j$ . The idea is that ${\rm lcm}(a_1,b_1)\Lambda_1\subseteq\Lambda_{j_1}\subseteq\Lambda_j=c\Lambda_f$ ; if $a_1=b_1=1$ , then the closeness of $\Lambda_1$ and $\Lambda_f$ forces $c=1$ and equality throughout. To cover the general case it is simpler to work with the real and imaginary periods separately.

Firstly,

$$\frac{\omega_{1,j}}{\omega_{1,f}} =c\in{\bf {Z}}.$$

Then

$$c= \frac{\omega_{1,1}}{\omega_{1,f}} \frac{\omega_{1,j}}{\omega_{1,1}} = \frac{a_1}{a_j}\lambda.$$

Hence

$$0 \le \vert\lambda-1\vert = \frac{\vert a_jc-a_1\vert}{a_1} <\varepsilon.$$

If $\lambda\not=1$ , then $\varepsilon>\vert\lambda-1\vert\ge a_1^{-1}\ge B^{-1}=\varepsilon$ , contradiction. Hence $\lambda=1$ , so $\omega_{1,1}= \omega_{1,f}$ . Similarly, we have

$$\frac{{\rm im}(\omega_{2,j})}{{\rm im}(\omega_{2,f})} =c\in{\bf {Z}}$$

and again we can conclude that ${\rm im}(\omega_{2,1})= {\rm im}(\omega_{2,f})$ , and hence $\omega_{2,1}= \omega_{2,f}$ .

Thus $\Lambda_1=\Lambda_f$ , from which the result follows. $\qedsymbol$

Theorem 5.2   For all $N<60000$ , every optimal elliptic quotient of $J_0(N)$ has Manin constant equal to $1$ . Moreover, the optimal curve in each class is the one whose identifying number on the tables [Cre] is $1$ (except for class $990h$ where the optimal curve is $990h3$ ).

Proof. For all $N<60000$ we used modular symbols to find all newforms $f$ and their period lattices, and also the corresponding isogeny classes of curves. In all cases we verified that (*) held with the appropriate value of $\varepsilon$ . (The case of $990h$ is only exceptional on account of an error in labelling the curves several years ago, and is not significant.) $\qedsymbol$

Remark 5.3   In the vast majority of cases, the value of $B$ is $1$ , so the precision needed for the computation of the periods is very low. For $N<60000$ , out of $258502$ isogeny classes, only $136$ have $B>1$ : we found $a_1=2$ in $13$ cases, $a_1=3$ in $29$ cases, and $a_1=4$ and $a_1=5$ once each (for $N=15$ and $N=11$ respectively); $b_1=2$ in $93$ cases; and no larger values. Class $17a$ is the only one for which both $a_1$ and $b_1$ are greater than $1$ (both are $2$ ).

Finally, we give a slightly weaker result for $60000<N<130000$ ; in this range we do not know $\Lambda_f$ precisely, but only its projection onto the real line. (The reason for this is that we can find the newforms using modular symbols for $H_1^{+}(X_0(N),{\bf {Z}})$ , which has half the dimension of $H_1(X_0(N),{\bf {Z}})$ ; but to find the exact period lattice requires working in $H_1(X_0(N),{\bf {Z}})$ .) The argument is similar to the one given above, using $B=a_1$ .

Theorem 5.4   For all $N$ in the range $60000<N<130000$ , every optimal elliptic quotient of $J_0(N)$ has Manin constant equal to $1$ .

Proof. We continue to use the notation above. We do not know the lattice $\Lambda_f$ but only (to a certain precision) a positive real number $\omega_{1,f}^{+}$ such that either $\Lambda_f$ has type $1$ and $\omega_{1,f}=2\omega_{1,f}^{+}$ , or $\Lambda_f$ has type $2$ and $\omega_{1,f}=\omega_{1,f}^{+}$ . Curve $E_1$ has lattice $\Lambda_1$ , and the ratio $\lambda=\omega_{1,1}^{+}/\omega_{1,f}^{+}$ satisfies $\vert\lambda-1\vert<\varepsilon$ . In all cases this holds with $\varepsilon=\frac13$ , which will suffice.

First assume that $a_1=1$ .

If the type of $\Lambda_f$ is the same as that of $\Lambda_1$ (for example, this must be the case if all the $\Lambda_j$ have the same type, which will hold whenever all the isogenies between the $E_j$ have odd degree) then from $c\Lambda_f=\Lambda_j$ we deduce as before that $\lambda=1$ exactly, and $c=a_1/a_j=1/a_j$ , hence $c=a_j=1$ . So in this case we have that $c=1$ , though there might be some ambiguity in which curve is optimal if $a_j=1$ for more than one value of $j$ .

Assume next that $\Lambda_1$ has type $1$ but $\Lambda_f$ has type $2$ . Now $\lambda=\omega_{1,1}/2\omega_{1,f}$ . The usual argument now gives $ca_j=2$ . Hence either $c=1$ and $a_j=2$ , or $c=2$ and $a_j=1$ . To see if the latter case could occur, we look for classes in which $a_1=1$ and $\Lambda_1$ has type $1$ , while for some $j>1$ we also have $a_j=1$ and $\Lambda_j$ of type $2$ . This occurs 28 times for $60000<N<130000$ , but for 15 of these the level $N$ is odd, so we know that $c$ must be odd. The remaining 13 cases are

$$62516a, 67664a, 71888e, 72916a, 75092a, 85328d, 86452a, 96116a,$$

$$106292b, 111572a, 115664a, 121168e, 125332a;$$

we have been able to eliminate these by carrying out the extra computations necessary as in the proof of Theorem 5.2. We note that in all of these 13 cases, the isogeny class consists of two curves, $E_1$ of type 1 and $E_2$ of type 2, with $[\Lambda_1:\Lambda_2]=2$ , so that $E_2$ rather than $E_1$ has minimal Faltings height.

Next suppose that $\Lambda_1$ has type $2$ but $\Lambda_f$ has type $1$ . Now $\lambda=2\omega_{1,1}/\omega_{1,f}$ . The usual argument now gives $2ca_j=1$ , which is impossible; so this case cannot occur.

Finally we consider the cases where $a_1>1$ . There are only three of these for $60000<N<130000$ : namely, $91270a$ , $117622a$ and $124973b$ , where $a_1=3$ . In each case the $\Lambda_j$ all have the same type (they are linked via $3$ -isogenies) and the usual argument shows that $ca_j=3$ . But none of these levels is divisible by $3$ , so $c=1$ in each case. $\qedsymbol$