The Method of Graphs. Examples and Applications

J.-F. Mestre. Tr. Andrei Jorza

February 18, 2004

1 Introduction

Let S_{k}(N, ε) be the space of cusp forms of weight k, level N and character ε, where k and N are integers ≥ 1, and ε is a Dirichlet character N. There are several ways to construct a basis. For example one can use Selberg’s trace formula. Denote by Tr(n) the trace of T_{n}, the n-th Hecke operator. The function

f = ∑_{n=1}^{∞} Tr(n)q^{n}

is in S_{k}(N, ε). The set of f_{i} = T_{i}f generate this space, and this theoretically allows us to construct a basis. For example, if N is prime, ε = 1 and k = 2, then the set of the f_{i} (1 ≤ i ≤ g where g is the genus of X_{0}(N)) is a basis of S_{2}(N, 1).

But, even in that case, which is the most favorable, the computations become hard: on an average computer we can only hope to treat N of the size of 5000 (always with N prime, weight 2 and trivial character); actually, the computation of Tr(n) requires the knowledge of many class numbers of imaginary quadratic fields of discriminant of order at most n and to obtain a basis f_{1}, f_{2}, …, f_{g} one needs to compute Tr(n) for n ≤ g^{2}.

In the following section we describe the “method of graphs”, which relies on the results of Deuring and Eichler, and developed by J. Oestrlé and myself, which allows us to obtain a basis for S_{2}(N, 1) more quickly (at least when N is a prime).

In the second section, we indicate how this method allows us to prove that certain elliptic curves defined over Q are Weil curves^{1} (which, by providing an adequate Weil curve, yields all the imaginary quadratic fields of class number at most 3, due to a result of Goldfeld and recent works of Gross and Zagier).

The third section is dedicated to the verification of a conjecture of Serre in certain particular cases; this is possible because of the method described in the first section. It is known that this conjecture, if it is true, has numerous consequences (e.g., the Shimura-Taniyama-Weil conjecture, and thus Fermat’s Last Theorem).

1It is now a theorem that every elliptic curve is a Weil curve, i.e., the Shimura-Taniyama-Weil conjecture is true. – William Stein

2 The method of graphs

2.1 Definitions and notations

In the following p is a prime number and N_{1} is a positive integer coprime to p. Set N = pN_{1}.

Let

M_{N} = ⊕_{S}Z[S]

where S is taken over all supersingular points of X_{0}(N_{1}) in characteristic p, i.e., over the set of isomorphism classes of pairs (E, C) consisting of an elliptic curve E defined over \overline{𝔽}_{p} and a cyclic group C of E of order N_{1}. Two such pairs are identified if they are, in the obvious sense, \overline{𝔽}_{p}-isomorphic.

Let

α_{S} = \frac{| Aut(S)|}{2},

where Aut(S) is the group of \overline{𝔽}_{p}-automorphisms of S. We always have α_{S} ≤ 12, and if p does not divide 6 then α_{S} ≤ 3.

Therefore we can define a scalar product on M_{N} by ⟨S, S⟩ = α_{S} and ⟨S, S′⟩ = 0 if S ≠ S′. Let Eis = ∑ α_{S}^{−1}[S] and let

M_{N}^{0} = {∑ x_{S}[S] : ∑ x_{S} = 0}

be the subspace orthogonal to Eis.

For all integers n ≥ 1 coprime with p we define an operator T_{n} on M by

T_{n}(E, C) = ∑_{C_{n}}(E/C_{n}, (C + C_{n})/C_{n})

where C_{n} runs over all the cyclic subgroups of order n such that C ∩ C_{n} = 0.

For all q | N_{1} and coprime with q′ = N_{1}/q, we define the same way the Atkin-Lehner involutions W_{q} by

W_{q}(E, C) = (E/q′C, (E_{q} + C)/q′C),

where E_{q} is the group of points of order dividing q of E.

Finally we define an involution W_{p} by W_{p} = − Frob_{p}, where Frob_{p} is the endomorphism of M_{N} that transforms (E, C) to (E^{p}, C^{p}). (The fact that it is an involution reflects that the supersingular points are defined over 𝔽_{p^{2}}.)

These operators have the following properties: the set of W_{q} and T_{n} (n coprime with N) generate an abelian semigroup of hermitian operators with respect to the scalar product ⟨·, ·⟩. The T_{n} commute with each other for all n coprime to p. If q = q_{1}q_{2} (q_{1}, q_{2} coprime) and if n = n_{1}n_{2} (n_{1}, n_{2} coprime with each other and with p) then W_{q} = W_{q_{1}}W_{q_{2}} and T_{n} = T_{n_{1}}T_{n_{2}}.

For all d | N_{1} we have a homomorphism ϕ_{d} : M_{N} ⟶ M_{N/d} that transforms (E, C) to (E, dC). This homomorphism commutes with T_{n} (n coprime with N), and with W_{q} (for q | N/d). For d | N_{1} and coprime with N_{1}/d we have

T_{d}ϕ_{d} = ϕ_{d}(T_{d} + W_{d})

2.2 An isomorphism with S_{2}(N)

We consider here the space S_{2}(N) of cusp forms of weight 2 over Γ_{0}(N), with its natural structure of T-module, where T is the Hecke algebra [?].

Theorem 2.1. There exists an isomorphism compatible with the action of the Hecke operators, between M_{N}^{0} ⊗ C and the subspace of S_{2}(N) generated by the newforms of level N and oldforms coming from the cusp forms of weight 2 and level pd, d | N_{1}.

Remark 2.2. Assume N (or without loss N_{1}) is square-free. We can determine efficiently the subspace M_{N}^{0} corresponding to newforms in S_{2}(N); it is the subspace formed by all x so that for all divisors d of N_{1} we have

ϕ_{d}(x) = ϕ_{d}(W_{d}(x)) = 0.

In particular if N = pq, q prime, it is the subspace of M_{pq} intersection of the kernel of ϕ_{q} and of ϕ_{q}W_{q}.

2.3 Relation to the quaternion algebra

The matrices of the operators T_{n} acting on M_{N} are the same as the classical Brandt matrices [?], constructed using quaternion algebras.

Let B_{p,∞} be the quaternion algebra over Q ramified exactly at p and infinity, and let O is an Eichler order of level N_{1} (defined by Eichler [?] in the case when N_{1} is square-free, and defined in general by Pizer [?]), and let I_{1}, I_{2}, …, I_{h} be representatives of the left ideal classes of O.

Let O_{i} be the right order (i.e., right normalizers) of the ideals I_{i}, and e_{i} be the number of units of O_{i}. The Brandt matrix B(n) = (b_{i,j}^{(n)}) has i, j entry

b_{i,j}^{(n)} = e_{j}^{−1} · |{α : α ∈ I_{j}^{−1}I_{i}, Nor(α) Nor(I_{j})/ Nor(I_{i}) = n}|

where Nor is the norm over B_{p,∞} (the norm of an ideal being the gcd of the norms of its nonzero elements).

In the language of supersingular curves of characteristic p, we may give these matrices (actually their transposes) the following interpretation:

Let S be a supersingular point as in I.1, i.e., a supersingular elliptic curve E defined over \overline{𝔽}_{p} together with a cyclic group C of order N_{1}. The ring of endomorphisms O_{1} of S is an Eichler order of level N_{1}. To all the other supersingular points S′ = (E′, C′) we associate the set I_{S,S′} of homomorphisms from S to S′, i.e. the set of all homomorphisms α from E to E′ that send C to C′. This is obviously a left ideal over O_{1}, and its inverse ideal is I_{S′,S}. We can prove that all the right ideals of O_{1} are obtained in this way, and the whole Eichler order of level N_{1} if the rign of endomorphisms of a supersingular point S. It is clear that the general term B_{i,j}^{(n)} of the n-th Brandt matrix is the number of isogenies of S_{i} to S_{j} (the supersingular points being conveniently indexed,) two such isogenies being identified is different by an automorphism of S_{j}. We can retrieve the matrix of the operator T_{n} acting over M_{n}.

On the other hand if for all pairs of supersingular points (S, S′) we associate the function

θ_{S,S′}(q) = ∑_{α} q^{deg α}

where α goes through all the homomorphisms of S to S′, we retrieve the functions θ classically associated with the ideals of the orders of the quaternions, or, if one prefers, associated with the positive integer quadratic forms in 4 variables.

It is therefore easy to prove that if ∑ x_{S}[S] is an elements of M_{N} ⊗ C eigenvector of all the Hecke operators and if f(q) is the corresponding modular form, we have, for all S′

x_{S′}f(q) = ∑_{S} x_{S}θ_{S,S′}

which allows, in theory, to find the coefficients a_{n} of f, using the x_{S}. In practice, unfortunately, the computation of a_{n} demands the knowledge of all the isogenies of degree n to S′, and there doesn’t seem to be a simple algorithm for that.

Nevertheless, in certain cases, there exists a different method to calculate the coefficients of f, which is easy as far as computation is concerned. Suppose that N is a prime (thus equal to p), or N is a product of primes pq and X_{0}(q) is of genus 0 (thus q = 2, 3, 5, 7 or 13).

In the appendix, we give for each such case an equation of X_{0}(q) of the form xy = p^{k}, thus the action of the Hecke operators T_{2} and T_{3} over X_{0}(q), which is given by an equation much simpler than the equation of modular polynomials Φ_{2}(j, j′), Φ_{3}(j, j′) (which give the action of T_{2}, T_{3} on X_{0}(1), parametrized by the modular invariant j; cf. section 2.4).

Let u = x if N = pq and u = j if N = p. The Fourier expansion of u at infinity is 1/q + · · ·. Let f(q) = ∑ a_{n}q^{n} a normalized newform of level N and weight 2 corresponding to a vector ∑ x_{S}[S] of M_{N}^{0} ⊗K, where K is the extension of Q generated by the a_{n}. Therefore there exists a prime ideal ℘ of K over p so that

(∑ x_{S} · u(S)) f(q)\frac{dq}{q} ≡ ∑ x_{S} \frac{du}{u − u(S)} (mod ℘).  (1)

(it is about the congruence between Laurent series in q).

Suppose for example that f corresponds to a Weil curve of conductor N, so that a_{n} are in Z. The x_{S} are in Z and one can prove that ∑ x_{S}u(S) ≠ 0. Thus we know a_{n}mod p for all n. Hasse’s inequality |a_{l}| < 2\sqrt{l} for l prime proves that we know the a_{n} for n < p^{2}/16.

2.4 Explicit construction of the net M_{N}

In this section we suppose that N is odd. Suppose that given an explicit model of the curve X_{0}(N_{1}), and so the action of the Hecke operator T_{2} on that model (cf. Appedix).

First we need to find a supersingular points. Note that they are defined over 𝔽_{p^{2}}. For example suppose that N = p. First we check to see if p is inert in one of the 9 imaginary quadratic fields of class number 1. If yes, then one can take for the initial value of j the modular invariant of the curve of complex multiplication by the ring of integers of corresponding fields. If not, one can know a list of minimal polynomials of modular invariants of elliptic curves of complex multiplication by imaginary quadratic fields of small class numbers, and apply the same method. One needs here to solve over 𝔽_{p^{2}} a polynomial equation, which can be done in log p operations – at least probabilistically. Finally suppose that all these attempts fail. There remains the possibility to enumerate all the values of 𝔽_{p} until finding a supersingular value. We know there must exist a supersingular j-invariant in 𝔽_{p}, but unfortunately only a very small number—on the order of the size of the class group of Q(\sqrt{−p}), or approximately \sqrt{p}.

So assume we know a supersingular point S_{1}. Knowing the action of T_{2} on the model given by X_{0}(N) allows us to obtain the three supersingular points S_{2}, S_{3}, S_{4} (not necessarily distinct) related to S_{1} by a 2-isogeny. It comes down to solving a degree 3 polynomial over 𝔽_{p^{2}}, which needs extracting cubic and square roots, operations that need O(log p) operations. Sometimes we may as well exlude this computation. Suppose that n = p and that we have, say p ≡ 2(mod 3). Thus p is inert in Q(\sqrt{−3}), so j = 0 is a supersingular value, and we know that the three isogenies of degree 2 send the curve of the invariant to the curve of complex multiplication by Z[\sqrt{−3}], for which the invariant is j = 54000.

In any case, we have at most one time when we need to solve a 3rd degree equation: once S_{2} is known, we search from S_{i} (i ≥ 2) the three supersingular points which are related, but we already know one, so we only need to solve a second degree equation, which comes down to square roots over 𝔽_{p^{2}} which is fast (probabilistic methods require O(log p) operations using an algorithm that is very simple to implement).

To prove that we can find, step by step, all the supersingular points of M_{N} it is enough to prove that the graph of T_{2} (and more generally of T_{n}) is connected. But, as Serre remarked, the eigenvalue a_{2} = 3 of T_{2} over M_{N} has multiplicity equal to the number of connected components of the graph of T_{2}. But in M_{N}, the space M_{N}^{0} corresponding to the cusp forms of codimension 1, so 3 is a simple eigenvalue in M_{N} (because for a cusp form we have |a_{2}| < 2\sqrt{2}), so the graph of T_{2} is connected.

In conclusion, an algorithm in O(N log N) operations gives all the supersingular points and the Brandt matrix B_{2} associated to them. One of the advantages of this matrix is that it is very sparse; on each line and column there are at most 3 nonzero terms, which are integers whose sum is 3. This allows, given an eigenvalue, to find very quickly, if N is large, the corresponding eigenvectors.

2.5 Examples

1. Take for example N = p = 37. Since 37 is inert in Q(\sqrt{−2}), one can take as the first vertex of our graph the curve E_{1} of complex multiplications by Z[\sqrt{−2}], for which the modular invariant is j_{1} = 8000 ≡ 8mod 37. We need to find now all the invariants of curves 2-isogenous to this, i.e., to solve the equation Φ_{2}(x, 8000) ≡ 0(mod 37). But \sqrt{−2} is an endomorphism of degree 2 of E_{1}, so j_{1} is a root (over Q) of the polynomial Φ_{2}(x, 8000). Dividing this polynomial by x − 8000 we get a second degree polynomials with roots j_{2}, j_{3}, the invariants of the other two curves, E_{2}, E_{3} related to E_{1} by a degree 2 isogeny. Let ω ∈ 𝔽_{p^{2}} so that ω^{2} = −2. One gets that then j_{2} = 3+14ω, j_{3} = 3−14ω.

Another method to find j_{2}, j_{3} consists in remarking that 37 is equally inert in the field K = Q(\sqrt{−15}), for which the class number is 2. The second degree polynomial giving the values of the modular invariants of 2 curves of complex multiplication by the ring of integers of K is x^{2} + 191025x − 121287375, whose roots generate Q(\sqrt{5}), so modulo 37 are conjugate in 𝔽_{37^{2}}. We can thus find j_{2}, j_{3}.

For N prime congruent to 1 mod 12, the number of supersingular curves mod N is (N − 1)/12. For N = 37 we get 3 supersingular curves. It remains to show that the action of T_{2} on E_{2} (by conjugations we get the action on E_{3}). It is not possible to have 2 isogenies of E_{2} on E_{1}, because then we would have 5 isogenies of degree 2 starting in E_{1}. Therefore there is one 2-isogeny of E_{2} over E_{2}.

Actually, if there is a 2-isogeny of an elliptic curve of invariant j on itself, this invariant is the root of the equation Φ_{2}(x, x) = 0, a fourth degree equation that can be written as

(x − 1728)(x − 8000)(x + 3375)^{2}

. (To see this, one can make the computation of the equation of Φ_{2}(j, j′) above. One can also search which are the curves of complex multiplication that admit a degree 2 endomorphism, i.e., which are the imaginary quadratic fields that contains an element of norm 2. One finds, by multiplication by the units of the (“corps pres?”) the elements 1 + i, \sqrt{−2}, \frac{1+\sqrt{−7}}{2} and \frac{1−\sqrt{−7}}{2} that are the endomorphisms of degree 2 of the curves of invariant j = 1728, j = 8000 and for the last two, j = −3375.)

By order, mod p, the graph of T_{2} cannot contain a loop of a supersingular curve on itself – although this curve is defined over 𝔽_{p} (and, more precisely, it is one of 3 curves described above). Therefore, there are 2 isogenies relating E_{2} to E_{3} and the graph of T_{2} acting on M_{37} is completely determined.

To compute the corresponding eigenvectors, one can evidently diagonalize the matrix (3, 3) of T_{2} but there is a simpler method:

the involution W_{37} = − Frob_{37} separates M_{37} in an obvious way into two orthogonal proper subspaces, one generated by u_{1} = [E_{2}] − [E_{3}], associated with the eigenvalue 1, and the other associated with the eigenvalue -1, generates by Eis = [E_{1}] + [E_{2}] + [E_{3}] and the vector product of u_{1} and Eis, let it be u_{2} = 2[E_{1}] − [E_{2}] − [E_{3}]. One can deduce, without recourse to T_{2}, that there exist 2 newforms for which the q-expansion has rational coefficient, and thus that J_{0}(37), the jacobian of X_{0}(37) is isogenous to the product of 2 elliptic curves (which is well-known, see for example [?]). Formula (1) above allows us to obtain the first 83 terms of their function L.

2. p − 37, N = 2 · 37.

To study X_{0}(74) one uses the homomorphism ϕ_{2} of M_{74} to M_{37} defined previously. The fibres of reach of the three supersingular points [E_{1}], [E_{2}] and [E_{3}] of X_{0}(1)mod 37 are formed by three distinct supersingular points of X_{0}(2)mod 2. In a general way, write that if S_{1}, S_{2}, …, S_{k} are the supersingular points of X_{0}(qM)mod p above a supersingular point S of X_{0}(M)mod p (p, q coprime and coprime with M), one has the formula

\frac{q + 1}{Aut S} = ∑_{1}^{k} \frac{1}{Aut S_{i}} .

The equation of X_{0}(2) used here is that described in the appendix: uv = 2^{12}, the involution W_{2} switching u and v. Recall that W_{37} = − Frob_{37} and that j = (u+16)^{3}/u (where j is the invariant of the curve E, image of the point (E, C) of X_{0}(2) via the homomorphism “oubli – oblivion?” of X_{0}(2) on X_{0}(1).) From the equation j = j_{1} = 8 one gets the values of the three supersingular points of E_{1}, of coordinates u_{1} = (−1 + ω)/2, u_{2} = (−1 − ω)/2 = W_{2}(u_{1}) and u_{3} = 27 = W_{2}(u_{2}). (Here again, it is possible to guess the value of u_{3}, because it is clear by the action of T (2) on X_{0}(1)mod 37 done previously that one of the above E_{1} must be invariant relative to W_{2}; or the two solutions of u^{2} = 2^{12} are u_{1}, −u_{1}. Replacing them in the equation that gives j one can see that it is about u_{1}. To get u_{2}, u_{3} it is enough to solve a second degree equation.) One can compute that u_{4} = W_{2}(u_{1}) = 2^{12}/u_{1} = −5 − 5ω, and one finds that the corresponding invariant j(u_{4}) is j_{2} = 3 + 14ω. One solves the second degree equation given 2 other points above by j_{2} and so u_{5} = 15 + 17ω, u_{6} = 16 − 12ω. Note that u_{7} = W_{2}(u_{2}) = ū_{4}, u_{8} = W_{2}(u_{5}) = ū_{5} and u_{9} = W_{2}(u_{6}) = ū_{6} the x-coordinates of three supersingular points over E_{3} (x ⟶ x̄ being the nontrivial automorphism of 𝔽_{p^{2}}.) We get the list of all supersingular points of X_{0}(2)mod 37. As said above, the space M_{74}^{new} corresponding to the newforms is the intersection of the kernel of ϕ_{2} and the kernel of ϕ_{2}W_{2}. If we write [u_{i}], i = 1, …, 9) the generators of M_{74} corresponding to the supersingular points of x-coordinate u_{i}, an examination of the action of W_{37} and W_{2} prove that M_{74}^{new} is the direct sum of two 2-dimensional subspaces, one G_{1}, generated by e_{1} = [u_{1}] − [u_{2}] − [u_{4}] + [u_{7}] − [u_{9}] and e_{2} = [u_{5}] − [u_{6}] − [u_{8}] + [u_{9}], on which W_{37} = −W_{2} = 1 and the other, G_{2}, generated by e_{3} = [u_{1}] + [u_{2}] − 2[u_{3}] + [u_{4}] − [u_{6}] + [u_{7}] − [u_{9}], on which W_{2} = −W_{37} = 1. Using the equation of T_{3} acting on X_{0}(2) (cf. appendix), one can prove that the matrix of T_{3} acting on G_{1} (respectively G_{2}) in the basis (e_{1}, e_{2}) (respectively (e_{3}, e_{4})) is

( \begin{matrix}−1 & 1 \\ 1 & 0 \end{matrix} ), of characteristic polynomial x^{2} + x − 1 (respectively ( \begin{matrix}3 & 1 \\ 1 & 0 \end{matrix} ), of charac-

teristic polynomial x^{2} − 3x − 1). One deduces that J_{0}^{new}(74) is isogenous to the product of two abelian simple varieties, A_{1} (resp. A_{2}), of real multiplication by the ring of integers of Q(\sqrt{5}), (respectively Q(\sqrt{13}).)

If λ = \frac{−1+\sqrt{5}}{2}, μ = \frac{3+\sqrt{13}}{2}, then the vectors v_{1} = e_{1} + (λ + 1)e_{2}, v_{2} = e_{1} − λe_{2}, v_{3} = μe_{3} + e_{4}, v_{4} = (3 − μ)e_{3} + e_{4} corresponding to the 4 newforms f_{1}, f_{2}, f_{3}, f_{4} of weights 2 and level 74. Using (1) one gets the first 83 values of the coefficients of these newforms. For example for f_{1} the list of the first values of a_{l} is

\begin{matrix}l & 2 & 3 & 5 & 7 & 11 & 13 \\ a_{l} & 1 & \frac{−1+\sqrt{5}}{2} & \frac{1−3\sqrt{5}}{2} & −1 + \sqrt{5} & \frac{−5−\sqrt{5}}{2} & \frac{1+3\sqrt{5}}{2} \end{matrix}

and for f_{3} one gets

\begin{matrix}l & 2 & 3 & 5 & 7 & 11 & 13 \\ a_{l} & −1 & \frac{3+\sqrt{13}}{2} & −1 − \sqrt{13} & \frac{1−\sqrt{13}}{2} & \frac{−1−\sqrt{13}}{2} & \frac{−1+\sqrt{13}}{2} \end{matrix}

3 Application to the study of Weil curves

Let f = ∑ a_{n}q^{n} be a newform of weight 2 and level N with integer coefficients. They correspond to a strong Weil curve E of conductor N. Unfortunately the coefficients a_{n} don’t give too much information on E and do not allow us to obtain a simple equation for E. (In [?] there is a method due to Serre that sometimes allows us to get such an equation, but that method is not systematic.) Here we give a method that at least when N = p is a prime, one can solve the problem.

From now on let N be a prime. According to the last section, for a newform f there is associated a vector v_{f} = ∑ x_{S}[S], x_{S} ∈ Z, an eigenvector of the Hecke operators defined in 2.1. Theorem 1 doesn’t describe the isomorphism (which is not canonical) between S_{2}(N) and M_{N}^{0} ⊗ C. But suppose known the terms a_{n} of f (a_{2} is sufficient in general). The construction of section 2.4 gives us both the supersingular values mod N and the graph of T_{2} acting on M_{N}. We can determine the eigenspace V_{2} associated with the eigenvalue a_{2}. If it is of dimension 1, we also have v_{f}, or at least the space it generates. Otherwise, we apply T_{3} to V_{2} (which is of tentatively small dimension—for conductors < 80000, dim V_{2} doesn’t goes beyond 6), until finding a 1-dimensional space, corresponding to the same eigenvalues of the operators T_{l} as f. Choose in this space a vector r_{f} = ∑ x_{E}[E] with integer x_{E} coprime in pairs^{2}; then r_{f} is determined up to sign.

To go further, we need a geometric interpretation of the x_{E}. Let Δ = ±N^{δ}, the discriminant of the minimal Weierstrass model of E, let ϕ : X_{0}(N) ⟶ E be a minimal cover of E of degree n = deg ϕ.

According to Deligne-Rapoport [?], there exists a model X_{0}(N)_{/Z} of X_{0}(N) defined over Z for which reduction mod N is the union of two projective lines, one C_{∞} classifying the elliptic curves of characteristic N provided with the group scheme kernel of the Frobenius (this corresponding to inseparable isogenies), the other one, C_{0}, classifying the curves provided with Verschiebung. These two lines intersect at supersingular points. As far as the curve E is concerned, reduction mod N of its Neron model has identity component E_{/𝔽_{N}}^{0} isomorphic to 𝔽_{N^{2}} of the multiplicative group G_{m}. One can prove that the cover extends to X_{0}(N)_{/Z} −§ where § is the set of all supersingular points of characteristic N, and define by restriction a regular “application ?” of C_{∞} on E_{/\overline{𝔽}_{N}}^{0} , of a rational function ϕ over C_{∞}, for which the poles and zeros are in E. The divisor ∑ λ_{E}[E] of ϕ, E going through all the supersingular curves mod N, and thus an element of M_{N}^{0}, defined up to sign (depending on the choice of isomorphism of E_{/𝔽_{N}}^{0} over G_{m}.)

Proposition 3.1. In the above notation the divisor (Φ) = ∑ λ_{E}[E] is equal to ±r_{f}.

2That seems to strong to me; do we just mean that the gcd of coefficients is 1?

It is not difficult to see that (Φ) is proportional to r_{f}. By contradiction, the fact that the l_{E} are coprime with one another is obtained from the result of Ribet which says that if l is a prime different from 2, 3 then all cusp forms mod l of weight 2 and level Np (where Np is square-free) for which the associated representation mod l is irreducible and not ramified at p, comes from a cusp form mod l of weight 2 and level N (this result was conjectured by Serre in 1985. This also shows that the Taniyama-Weil conjecture implies the Fermat theorem.)

To prove the previous theorem, one proves first that δ is related to λ_{E} by δ = gcd(λ_{E}ω_{E} − λ_{F} ω_{F}) where ω_{E} is the number of automorphisms of E. Suppose that a prime number l divides the gcd is λ_{E}. It also divides δ, and one deduces from here that p is not ramified in the field of points of order l or E. If l is coprime with 6 Ribet’s ^{3} theorem shows that the modular form f associated to E is congruent mod l to a modular form of weight 2 and level 1, which cannot be but the Eisenstein series. The curve E is semi-stable, which implies ([?], p.306) that E or a curve Q-isogenous to it has a point of finite order l. If l = 2, 3 we get the same result due to [?], Appendix. Now, we know explicitly the curves of prime conductor with torsion [?] namely the curves 11A and 11B of [?], which have a point of order 5, curves 17A,17B,17C (point of order 4), 17D (point of order 2), 19A and 19B (point of order 3), 37B, 37C (point of order 3) and the curves of Setzer-Neumann [?], which have a point of order 2. In each of these cases, we know δ, which is equal to the number of finite points rational over Q of the considered curves, and one can verify that the λ_{E} are coprime with one another. This proves the proposition. Note that along the proof we showed that Ribet’s theorem implies the following

Theorem 3.2. Let E be a strong Weil curve of prime conductor N. The valuation of its discriminant in N is equal to the number of torsion points of E(Q).

We state without proof the theorem that allows us to get an explicit equation for E once we know the λ_{E}.

Theorem 3.3. Let E be a strong Weil curve of prime conductor N, and ∑ λ_{E}[E] the element of M_{N}^{0} associated to E via the constructions above. There exists an equation of E

y^{2} = x^{3} − \frac{c_{4}}{48}x − \frac{c_{6}}{864}

with c_{4}, c_{6} ∈ Z so that, if H = max(\sqrt{|c_{4}|}, \sqrt[3]{|c_{6}|}) we have:

1. H ≤ \frac{8n}{\sqrt{N}−2}(log(H^{6}/1728) + b), where b = (Γ(1/3)/Γ(2/3))^{3} = 7.74316962 . . ..

2. Let Δ′ = (c_{4}^{3} − c_{6}^{2})/1728. Then Δ′ = Δ if E is supersingular in characteristic 2, and Δ′ = Δ or 2^{12}Δ otherwise.

3. c_{4} ≡ (∑ λ_{E}j_{E})^{4}mod N.

4. c_{6} ≡ −(∑ λ_{E}j_{E})^{6}mod N.

3K.Ribet, Lectures on Serre’s conjectures, MSRI, Fall 1986

5. nδ = λ_{E}^{2}ω_{E}.

If the λ_{E} are known then 5 allows us to get n and 1 allows us to find a bound on H, thus on c_{4}, c_{6}. By 2 we have c_{4}^{3} − c_{6}^{2} = 1728Δ′, which allows us to find c_{4}, c_{6}. The congruences 3 and 4 allow us to reduce the number of computations significantly. Thus we have found an equation of a strong Weil curve corresponding to the initial newform f.

This method also allows us to prove that an elliptic curve of small prime conductor is a Weil curve. Suppose that we are given such a curve by its equation. Then we may compute the number of its points N_{l} mod l for l = 2, 3, . . .. Next we search, by the method of graphs, whether a_{2} = 3 − N_{2} is the eigenvalue of T_{2} acting on M_{N}. If not then the Taniyama-Weil conjecture is false. If yes, then continue with T_{3} acting on the found eigenspace, if it is not of dimension 1, until we get an eigenspace of dimension 1 for the Hecke operators, with integers eigenvalues. If there is no such thing, then we get a counterexample to the Taniyama-Weil conjecture. If there is one, we compute the equation of a corresponding Weil curve. If this curve is isogenous to the initial curve, we are done. Otherwise, the initial curve is not a Weil curve.

In particular, this allows us to prove that the elliptic equation

y^{2} + y = x^{3} − 7x + 6

of conductor 5077, is a Weil curve.

This curve seems to be the smallest curve (ordering the curves by their conductors) having a Mordell-Weil rank ≥ 3 [?]. The interest in it is the following:

Let f(z) = ∑ a_{n}q^{n} (q = e^{2πiz}), a newform of weight 2 and conductor N, and let L(s) = ∑ a_{n}n^{−s}, the associated L function. If the order of L in 1 is ≥ 3 then Goldfeld proved that there exists a computable constant C_{f} so that

log p < C_{f}h(−p),

where p ≡ 3(mod 4) is a prime number coprime with N and h(−p) is the number of classes of imaginary quadratic fields of discriminant −p. We have other formulas, but more complicated, in the case of imaginary quadratic fields of non-prime discriminant (see [?] for example).

If the Birch and Swinnerton-Dyer conjecture is true, all the Weil curves for which the Mordell-Weil group over Q is of rank ≥ 3 have to be given by such modular forms, but until the work of Gross and Zagier [?], there was no way to verify that the derivative at 1 of the L function of a Weil curve is indeed 0. The results of Gross and Zagier allow to write L′(1) as the product of a non-zero factor easily computable and the Néron-Tate height of a Heegner point (cf. [?] for more details.) It is therefore possible, by decreasing the height of rational points on the curve and increasing L′(1) by a careful computation, to prove that L is of order ≥ 3 at s = 1. (In all the previous, we considered odd Weil curves, i.e., for which the L function has an odd order at 1 – or if one prefers for which the sign of the functional equation is -1.)

One has several method to construct Weil curves for which the Mordell-Weil group is of rank ≥ 3 (and which are good candidates for the preceding question: by the method of Gross-Zagier, one may compute L′(1). If it is zero, one has an L function which allows to obtain an increase of the absolute value of the discriminant of imaginary quadratic fields of given class numbers; if it is non-zero, the conjecture of Birch and Swinnerton-Dyer is false.) One can, for example, search for curves of complex multiplication of rank 3 (we know that they are Weil curves), but the constant C_{f} is very large. One can deform^{4} a Weil curve (for example the curve 37C of [?] until getting a rank 3 curve (for the curve 37C, one can deform by Q(\sqrt{−139}), as shown by Gross and Zagier [?].) This leads to a constant C_{f} of order of 7000

One may choose some elliptic curve defined over Q, or rank 3, and try to prove that it is a Weil curve. This was done in [?] for the mentioned curve of conductor 5077, using the trace formula. But the computation is very long. The method of graphs allows us to do it in about 5 seconds an a computer that needed 5 hours with the mentioned method.

For this curve, one has C_{f} < 50: all imaginary quadratic curves of discriminant d with |d| > e^{150} therefore has a class number ≥ 4. On the other hand, there is no imaginary quadratic field of discriminant d and class number 3 for 907 < |d| < 10^{2500} [?]. Therefore (after an examination of a table of class numbers of the first quadratic fields):

Theorem 3.4. The imaginary quadratic fields of class number 3 are the 16 fields of discriminant: −23, −31, −59.−83, −107, −139, −211, −283, −307, −331, −379, −499, −547, −643, −883, −907.

4 Application to a conjecture of Serre

Let ρ be a continuous representation of Gal(\overline{Q}/Q) in GL_{2}(V) where V is a dimension 2 vector space over a finite field 𝔽_{q} of characteristic p. Assume this is an odd representation, i.e., that ρ(c) the image of the complex conjugation, seen as an element of Gal(\overline{Q}/Q) has eigenvalues 1 and -1. In that case put G = Imρ.

In [?] Serre defines the level, the character and the weight of such a representation:

1. The level.

Let l be a prime number different from p. Write G_{i} (i = 0, …) the groups of ramifications of ρ at l. Let

n(l) = ∑_{i=0}^{∞} \frac{g_{i}}{g_{0}} codimV^{G_{i}},

where g_{i} = |G_{i}|.

The conductor of the representation ρ is defined as

N = ∏_{l≠p} l^{n(l)}.

2. The character.

4Twist?

The determinant of ρ yields a character of Gal(\overline{Q}/Q) in 𝔽_{q}^{∗}, for which the conductor divides pN. Therefore, one can write

det ρ = εχ^{k−1},

where χ is the cyclotomic character of conductor p and ε is the character (Z/NZ)^{∗} ⟶ 𝔽_{q}^{∗}. The integer k is defined mod (p − 1), and the fact that the representation is odd implies that ε(−1) = (−1)^{k}.

By definition, ε is the character of the representation ρ.

3. The weight.

The integer k above is defined mod (p − 1). Read Serre’s article for the definition of the weight k ∈ Z of the representation ρ. As the conductor N depends only on the behavior of ρ ar places coprime with p, the definition of weight only uses the local properties at p of the representation ρ.

Then Serre’s conjecture is:

Conjecture 4.1. Let ρ be a representation as above, of weight k, level N and character ε. Assume this representation is irreducible. Then it comes from a cusp form mod p of weight k, level N and character ε.

This conjectures, if true, has numerous consequences: it implies the Taniyama-Weil conjecture and Fermat’s theorem.

Many such representations ρ are modular, either by construction, or because they are part of classical conjectures (Langlands, Artin, …) that carry on the conjecture (but sometimes in a weak form, i.e., with a weight or conductor bigger than those defined in [?].)

In order to verify (or contradict) Serre’s conjecture, we need to find the extensions K/Q of Galois group subgroup of GL_{2}(𝔽_{q}) of odd determinant and p ≠ 2. It is in general not difficult to calculate, for l prime and not too large, the trace a_{l} of Frob_{l} in GL_{2}(𝔽_{q}): if P (x) is a polynomial whose roots generate K the decomposition of P mod l usually will suffice.

It is however, much harder to find modular forms mod p, if they exist, that correspond to the representation ρ given by the field K: the discriminant of K is usually large, thus so is the conductor of ρ, which is related to it, so it is not easy to make the computations.

4.1 The case SL_{2}(𝔽_{4})

A troubling case is that of p = 2, because, since −1 ≡ 1(mod 2) all representations are odd.

The representations of Gal(\overline{Q}/Q) in GL_{2}(𝔽_{2}) = S_{3} (although altogether real, cf. [?]) come from weight 1 modular forms; the group S_{3} can be realized as a subgroup of GL_{2}(C). One can hope that by multiplication with convenient Eisenstein series, one can obtain a modular form of weight and level predicted by the Serre conjecture (cf. [?] for examples.)

In order to obtain the most interesting case for characteristic 2, one considers the representations with values in GL_{2}(𝔽_{4}). The isomorphism A_{5} ≃ SL_{2}(𝔽_{4}) allows us to obtain several examples. Let K be an extension of Q of Galois group A_{5}. Since A_{5} “immerses ?” into P GL_{2}(C), if the field is not completely real, the associated representation ρ comes from a weight 1 modular form (module Artin’s conjecture, cf. [?]). Suppose now that K is real. None of the classical conjectures allow us to suspect that ρ comes from a modular form, even if of higher weight or level. It is this case that we will study in what follows. The method of graphs here is indispensable, the modular forms that we look at having a conductor too large to be studied with the Eichler-Selberg trace formula.

Let P (x) = x^{5} +a_{1}x^{4} +a_{2}x^{3} +a_{3}x^{2} +a_{4}x+a_{5} be a rational polynomial of discriminant D. In order that the field of roots of P be A_{5} it is sufficient and necessary that P be irreducible, that D be square-free, and that there exist a prime number l not dividing D so that P mod l having exactly two roots in 𝔽_{l} (this last condition assuring that the group is all of A_{5}).

It is clear that ε = 1. If p | D, p coprime with 30, n(p) = 1 if it “seulment si l’inertie en p ==?” is of order 2, and thus the polynomial P has at most double roots mod p. As far as the weight k is concerned, it is either 2 or 4 according to the ramification of K at 2. To simplify the computation, we have limited to searching examples among the representations of prime level and weight 2.

On the other hand, since it is about representations in SL_{2}(𝔽_{4})m the coefficient a_{2} of the sought modular form, if it exists, cannot be in 𝔽_{4}, but in 𝔽_{16}. This comes from the fact that the coefficient a_{l} of a modular form mod l is equal to an eigenvalue of Frob_{l}, and not to its trace. Now, if a matrix in SL_{2}(𝔽_{4}) is of order 5, its eigenvalues are in 𝔽_{16} not in 𝔽_{4}.

The examples treated above were obtained by making a systematic search on a computer of convenient polynomials (totally real, of type A_{5}, for which the conductor of the associated representation is a prime N, and for which the weight is 2).

Thereafter, for each such polynomial P, one computes the corresponding eigenvalue a_{2} (in 𝔽_{16}), and one tries to find whether there exists a modular form mod 2 of level N and weight 2 so that T_{2} has a_{2} as an eigenvalue. In all the cases considered, we have thereafter found an eigenspace of dimension 1 or 2. Using the operators T_{3}, T_{5}, one calculates the coefficients a_{3}, a_{5}, and verifies that they correspond to the values predicted by the decomposition of P in 3 and 5.

Clearly, this doesn’t really prove that the representation ρ associated to P is modular: we have only exhibited a modular form mod 2 of proper level and weight for which the terms a_{2}, a_{3}, a_{5} are convenient. But there is a good indication of the truthfulness of the conjecture of Serre in the considered cases: an exhaustive search over numerous primes N of the coefficients a_{2} of modular forms of weight 2 and level N proves that it is rare that there are fields of small degree. (Actually, is seems that 2, and in general the small primes, are the most “inert” possible in the fields that appear in the Hecke algebra of modular forms, fields which themselves in general appear to have the largest degree possible, taking into account constraints such as the Atkin-Lehner involutions, primes of Eisenstein, etc. One gets that one has small factors, – corresponding for example to elliptic curves with prime conductor – but this is apparently rare.)

4.2 A few examples

1. P (x) = x^{5} − 10x^{3} + 2x^{2} + 19x − 6.

The discriminant is (2^{3}887)^{2}. This polynomial is irreducible mod 5, thus irreducible over Q. Its roots are all real (apply Sturm’s algorithm). One has that

P (x) ≡ x(x − 1)(x^{3} + x^{2} − 1)mod 3,

which gives a cycle of order 3; the Galois group of K, the field of roots of P, is thus A_{5}.

From P (x) ≡ (x − 462)(x − 755)^{2}(x − 788)^{2}mod 887 one gets that the conductor N of the associated representation is N = 887. One can also prove that 2 is “little ramified” in the sense of [?], thus ρ has weight 2. Examining the reduction mod 2 of P proves that the coefficients a_{2}, a_{3}, a_{5} of the modular form mod 2 of level 887 (which must correspond to ρ via the Serre conjecture) are 1, 1, j (where j ∈ 𝔽_{4} has the property that j^{2} + j + 1 = 0).

One therefore applies the method of graphs: the space of modular forms mod 2 of weight 2 and level 887 has dimension 73, and computation shows that the eigenspace G_{1} of T_{2} corresponding to the eigenvalue 1 has dimension 2; T_{3} acts as the identity on G_{1}, and j, j^{2} are the eigenvalues of T_{5} acting on G_{1}, from where get a basis of G_{1} formed by f_{1} = q + q^{2} + q^{3} + q^{4} + jq^{5} + ⋯ and f_{2} = q + q^{2} + q^{3} + q^{4} + j^{2}q^{5} + ⋯, eigenvectors of Hecke operators. These corroborate the conjecture.

2. P (x) = x^{5} − 23x^{3} + 55x^{2} − 33x − 1.

Then D = 13613^{2}, P (x) ≡ (x − 6308)(x − 2211)^{2}(x − 8248)^{2}mod 13613, N = 13613; P being irreducible mod 2, Frob_{2} is a cycle of order 5, and a_{2} = ζ_{5} is a fifth root of unity, viewed as an element of 𝔽_{16}. Computation also shows that in the space of modular forms mod 2 of level 13613 and weight 2, which has dimension 1134, ζ_{5} is a simple eigenvalue of T_{2}. The coefficients a_{3}, a_{5} are respectively equal to 1 + ζ_{5}^{2} + ζ_{5}^{3} = j and ζ_{5}^{2} + ζ_{5}^{3} = j^{2}, which are the traces of Frob_{3}, Frob_{5} in SL_{2}(𝔽_{4}).

3. We write the other found polynomials; in each case there exists a modular form of weight 2 and appropriate level, for which the first terms a_{n} correspond to those values predicted by the Serre conjecture.

P (x) = x^{5} + x^{4} − 16x^{3} − 7x^{2} + 57x − 35, N = 8311, \sqrt{D} = N
P (x) = x^{5} + 2x^{4} − 43x^{3} + 29x^{2} + 2x − 3, N = 8447, \sqrt{D} = 2^{2}N
P (x) = x^{5} + x^{4} − 13x^{3} − 14x^{2} + 18x + 14, N = 15233, \sqrt{D} = 2N
P (x) = x^{5} + x^{4} − 37x^{3} + 67x^{2} + 21x + 1, N = 24077, \sqrt{D} = 2^{2}N

5 Appendix: The curves X_{0}(p) of genus 0

In [?], it is proven that if p is a prime number then the curve X_{0}(p) over Z_{p} is formally isomorphic to the curve of equation xy = p^{k}, in the neighborhood of each point reducing mod p to a supersingular point S, k being one half the number of automorphisms of S.

If X_{0}(p) has genus 0 (i.e., p = 2, 3, 5, 7, 13) one has such a model over Z, given by the function

x = (\frac{η(z)}{η(pz)})^{\frac{24}{p−1}} ,  (2)

where η(z) = q^{1/24} ∏_{i=1}^{∞}(1 − q^{n}) and q = e^{2πiz}.

This results from Fricke [?], who gives for each of the above p’s an expression of the “oubli ?” homomorphism j : X_{0}(p) ⟶ X_{0}(1), which associates to each point (E, C) of X_{0}(p) the point (E) of X_{0}(1), parametrized by the modular invariant j.

In the following we recall these equations and give the expressions of the correspondences T_{2}, T_{3} over these curves. The variable x is the one given by equation (2), the involution W_{p} switches x and y and the divisor of x is (0) − (∞), where 0 and ∞ are two points of X_{0}(p).

1. p = 2 The equations given by Fricke (modified to give the model of X_{0}(2) over Z) are:

xy = 2^{12}

j = \frac{(x + 16)^{3}}{x}

T_{2} is given by

y^{2} − y(x^{2} + 2^{4}3x) − 2^{12}x = 0

(to each point x is associated by T_{2} the formal sum of points of coordinate y that are roots of this polynomial.)

T_{3} is given by

x^{4}+y^{4}−x^{3}y^{3}−2^{3}3^{2}x^{2}y^{2}(x+y)−2^{2}3^{2}5^{2}xy(x^{2}+y^{2})+2·3^{2}1579x^{2}y^{2}−2^{15}3^{2}xy(x+y)−2^{24}xy = 0

2. p = 3.

xy = 3^{6}

j = \frac{(x + 27)(x + 3)^{3}}{x}

T_{2} : x^{3} + y^{3} − 2^{3}3xy(x + y) − x^{2}y^{2} − 3^{6}xy = 0
T_{3} : y^{3} − y^{2}(x^{3} + 2^{2}3^{2}x^{2} + 2 · 3^{2}5y) − 3^{6}yx(x + 2^{2}3^{2}) − 3^{12}x = 0

3. p = 5.

xy = 5^{3}

j = \frac{(x^{2} + 10x + 5)^{3}}{x}

T_{2} : x^{3} + y^{3} − x^{2}y^{2} − 2^{3}xy(x + y) − 7^{2}xy = 0

T_{3} : x^{4}+y^{4}−x^{3}y^{3}−2·3^{2}x^{2}y^{2}(x+y)−3^{4}xy(x^{2}+y^{2})−2·3^{2}23x^{2}y^{2}−2250xy(x+y)−5^{6}xy = 0 4. p = 7.

xy = 7^{2}

j = \frac{(x^{2} + 13x + 49)(x^{2} + 5x + 1)^{3}}{x}

T_{2} : x^{3} + y^{3} − x^{2}y^{2} − 2^{3}xy(x + y) − 7^{2}xy = 0

T_{3} : x^{4}+y^{4}−x^{3}y^{3}−2^{2}3x^{2}y^{2}(x+y)−2·3·7xy(x^{2}+y^{2})−3·53x^{2}y^{2}−2^{2}3·7^{2}xy(x+y)−7^{4}xy = 0

5. p = 13.

xy = 13

j = \frac{(x^{2} + 5x + 13)(x^{4} + 7x^{3} + 20x^{2} + 19x + 1)^{3}}{x}

T_{2} : x^{3} + y^{3} − x^{2}y^{2} − 2^{2}xy(x + y) − 13xy = 0

T_{3} : x^{4}+y^{4}−x^{3}y^{3}−2·3x^{2}y^{2}(x+y)−3·5xy(x^{2}+y^{2})−3·11x^{2}y^{2}−2·3·13xy(x+y)−13^{2}xy = 0

The polynomials above that give T_{2}, T_{3} are of simpler form than the classical modular equations Φ_{2}(j, j′) and Φ_{3}(j, j′) (that correspond to the action of T_{2} and T_{3} on X_{0}(1)). For comparison, we recall their expressions:

Φ_{2}(j, j′) = j^{3} + j^{′3} − j^{2}j^{′2} + 2^{4}3 · 31jj′(j + j′) − 2^{4}3^{4}5^{3}(j^{2} + j^{′2})
+3^{4}5^{3}4027jj′ + 2^{8}3^{7}5^{6}(j + j′) − 2^{12}3^{9}5^{9}

Φ_{3}(j, j′) = j^{4} + j^{′4} − j^{3}j^{′3} − 2^{2}3^{3}9907jj′(j^{2} + j^{′2}) + 2^{3}3^{2}31j^{2}j^{′2}(j + j′)

−2^{16}5^{3}3^{5}17 · 263jj′(j + j′) + 2^{15}3^{2}5^{3}(j^{3} + j^{′3}) + 2 · 3^{4}13 · 193 · 6367j^{2}j^{′2}

−2^{31}5^{6}22973jj′ + 2^{30}3^{3}5^{6}(j^{2} + j^{′2}) + 2^{45}3^{3}5^{9}(j + j′)