Supernew preliminary version Roma, January 31, 2001

Some computations with Hecke rings

and deformation rings

Joan-C. Lario and René Schoof

(with an appendix by William Stein)

Abstract. In the proof by Wiles, completed by Taylor-Wiles, of the fact that all semi-stable elliptic curves over Q are modular, certain deformation rings play an important role. In this note we explicitly compute these rings for the elliptic curve Y^{2} + XY = X^{3} − X^{2} − X − 3 of conductor 142.

1. Introduction. In order to prove that semi-stable elliptic curves E over Q are modular, A. Wiles [17] considers the Galois representation \overline{ρ} : Gal(\overline{Q}/Q) ⟶ GL_{2}(F_{3}) provided by the 3-torsion points E[3] of a semi-stable elliptic curve E. He proves that certain rather restricted deformations of \overline{ρ} are modular. It follows then that, in particular, the deformation provided by the Galois representation ρ_{E} : Gal(\overline{Q}/Q) ⟶ GL_{2}(Z_{3}) associated to the 3-adic Tate module of E is modular. This implies that E is modular.

Wiles proceeds in two steps. First he considers certain minimal deformations of \overline{ρ}. Using the Langlands-Tunnell Theorem [9, Thm.1.3], and some so-called ‘level lowering theorems’ [8] he constructs a normalized eigenform of weight 2 and minimal level N whose associated Galois representation

ρ_{min} : Gal(\overline{Q}/Q) ⟶ GL_{2}(R)

is a minimal deformation of \overline{ρ}. Here the ring R is a finite extension of Z_{3}. R. Taylor and Wiles [15] then show that the universal minimal deformation ring is isomorphic to a ring T of Hecke operators of level N. It follows that the minimal deformations are all modular.

It may happen that the representation ρ_{E} associated to the 3-adic Tate module of E is not minimal. Therefore Wiles’s second step is to consider deformations of \overline{ρ} that are not necessarily minimal, but satisfy somewhat more relaxed conditions at the primes that divide the conductor of E. From the fact that the minimal deformation ring is a Hecke ring he then deduces that the corresponding universal deformation rings are also isomorphic to certain Hecke rings and it follows that the representation ρ_{E} is modular.

In this short note we explicitly compute the relevant universal deformation rings for a specific elliptic curve E. The main result is Theorem 3.2. Our example is the elliptic curve E of conductor 142 which is denoted by “142A” in the Antwerp Tables [1] (it is the curve 142C1 in J. Cremona’s Tables [2]). A Weierstrass equation for E is given by

Y^{2} + XY = X^{3} − X^{2} − X − 3.

As we explain below, the representation

\overline{ρ} : Gal(\overline{Q}/Q) ⟶ GL_{2}(F_{3})

provided by the 3-torsion points E[3] is unramified outside 3 and 71. Since \overline{ρ} is not ramified at the prime 2, the representation ρ_{E} associated to the 3-adic Tate module of E is not a minimal deformation of \overline{ρ}. In this case the minimal universal deformation ring is isomorphic to a ring T of Hecke operators of level 71. We compute it in section 2. It has two Z_{3}-valued points, one of which corresponds to ρ_{min} and we take as the “origin” of the deformation space Spec(T). In section 3 we study deformations ρ of \overline{ρ} that satisfy somewhat more relaxed conditions at the prime 2: they need not be unramified at 2, but should be of “type (A)”. This means that the image ρ(I_{2}) of the inertia group I_{2} at any

prime over 2 is contained in a subgroup conjugate to ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ). The deformation ρ_{E} is of

type (A). In this case the universal deformation ring is a ring of Hecke operators of level 284 = 4 · 71. We show that it is a complete intersection algebra of rank 8 over Z_{3}. Since the elliptic curve E has conductor 142, one might have expected this universal deformation ring to be isomorphic to a Hecke ring of level 2 · 71 rather than 4 · 71. Therefore we also determine the Hecke ring of level 142 in section 5. It has rank 5 over Z_{3} and it turns out not to be a complete intersection or even a Gorenstein ring.

For the three Hecke rings T of levels 71, 142 and 284, we also compute two invariants associated to the Z_{3}-algebras T and the morphisms π : T ⟶ Z_{3} provided by ρ_{min}. Writing I = ker(π), we determine the congruence ideal η = π(Ann(I)) and the cotangent space I/I^{2} of T. We have that #(I/I^{2}) = [Z_{3} : η] if and only if T is a complete intersection [5, Crit.I].

We conclude this introduction by giving some relevant information concerning the curve 142A. Counting points on E modulo the primes p with 3 ≤ p ≤ 67 one finds that #E(F_{p}) = p + 1 − a_{p} with a_{p} as in Table 1.1.

Table 1.1 Fourier coefficients a_{p}.

\begin{matrix}p & 3 & 5 & 7 & 11 & 13 & 17 & 19 & 23 & 29 & 31 & 37 & 41 & 43 & 47 & 53 & 59 & 61 & 67 \\ a_{p} & 0 & 2 & 0 & 6 & 4 & 6 & −8 & −4 & −2 & −8 & 10 & −2 & −8 & −4 & 0 & 10 & −8 & 2 \end{matrix}

For p ≠ 3, 71, the characteristic polynomial of a Frobenius automorphism φ_{p} acting on E[3] is given by X^{2} − a_{p}X + p. As in [13, sect.5.5] one deduces from the table that \overline{ρ} is surjective and hence irreducible. Alternatively, the X-coordinates of the 3-torsion points are the zeroes of the polynomial 3X^{4} − 3X^{3} − 6X^{2} − 36X + 8. It is not difficult to show directly that the Galois group of this polynomial is isomorphic to S_{4}, and deduce that the representation \overline{ρ} is surjective.

We briefly discuss the behavior of the critical primes 2, 3 and 71 with respect to the representation \overline{ρ}. The discriminant of the curve 142A is equal to −2^{6}71.

– Since E has good reduction modulo 3, the representation \overline{ρ} is “flat at 3”, i.e. the Zariski closure of the 3-torsion points in the Néron model of E over Z_{3} is a finite flat group scheme. Since a_{3} = 0, the elliptic curve E is supersingular modulo 3 and the representation \overline{ρ} is non-ordinary.

– The curve E has multiplicative reduction modulo 71. Since the 71-adic valuation of the discriminant is not divisible by 3, the theory of the Tate curve implies that the representation \overline{ρ} is ramified at 71. Moreover, \overline{ρ} is of type (A) at 71. This means that the image \overline{ρ}(I_{71}) of the inertia group I_{71} at any prime over 71 is contained in a

subgroup conjugate to ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ).

– The curve E has multiplicative reduction at 2 as well, but since the 2-adic valuation of Δ is divisible by 3, the representation \overline{ρ} is unramified and hence flat at 2. Alternatively, one can show directly that the number field generated by a zero of the quartic polynomial above is only ramified at 3 and 71. Because of this, the Galois representation ρ_{E} associated to the 3-adic Tate module of E is not a minimal deformation of \overline{ρ} in the sense of Wiles. However, it follows from the theory of the Tate curve that ρ_{E} is a deformation of type (A) at 2. See [11] for similar octahedral calculations.

2. The Hecke ring of level 71. In this section we consider minimal deformations ρ : Gal(\overline{Q}/Q) ⟶ GL_{2}(R) of the representation \overline{ρ} : Gal(\overline{Q}/Q) ⟶ GL_{2}(F_{3}) associated to the 3-torsion points of the elliptic curve 142A mentioned in the introduction. Here R is a local Noetherian Z_{3}-algebra with residue field F_{3} and the diagram

GL_{2}(R)
ρ ↗  ↓
Gal(\overline{Q}/Q) ⟶^{\overline{ρ}} GL_{2}(F_{3})

is commutative. We consider deformations ρ of the following type:

• ρ is unramified outside {3, 71};

• ρ is flat at 3;

• ρ is of type (A) at 71;

• the determinant of ρ is the cyclotomic character. The first condition ensures that ρ is minimal. See [10, section 29]. The representation \overline{ρ} is irreducible. Therefore, by [10, sections 26, 29 and 31] there exists a universal deformation of this type:

ρ_{univ} : Gal(\overline{Q}/Q) ⟶ GL_{2}(R_{univ}).

Wiles shows [4, Thm 8] that the universal deformation ring R_{univ} is isomorphic to a certain local 3-adic ring T of Hecke operators T_{n}. By [7, section 3], the ring T is the Z_{3}-subalgebra

T ⊂ T̃ = ∏_{f} O_{f}

generated by the vectors T_{p} = (a_{p}(f))_{f} with p ≠ 3 prime. Here f runs over the normalized 3-adic eigenforms of level 71 whose Fourier coefficients a_{p}(f), p ≠ 3 are congruent to the coefficients a_{p} associated to the curve 142A. The rings O_{f} denote the rings of integers of the extension of Q_{3} generated by the Fourier coefficients a_{p}(f) of f. It follows from [7, Lemma 3.2] that the Hecke operator T_{3} is in T as well. This means that the ring T is generated by all T_{p}. By Theorem 1 of the appendix, T is then generated as a Z_{3}-module by the operators T_{n} with n ≤ 2 · 72/12 = 12.

The space S_{2}(Γ_{0}(71)) of weight 2 cusp forms of level 71 has dimension 6 and a basis can be found in [3] or can be computed with W.A. Stein’s package [14]. The Hecke action decomposes S_{2}(Γ_{0}(71)) over Q into a product of two subspaces of dimension 3 corresponding to a pair of newforms f_{71} and f′_{71}. The Fourier coefficients a_{n}(f_{71}) of f_{71} or a_{n}(f′_{71}) of f′_{71} generate the same totally real cubic field of discriminant 257, generated by u where u^{3} − 5 u + 3 = 0. For n ≤ 12 they are listed in Table 2.1.

Table 2.1 Hecke operators T_{n}.

\begin{matrix}n & a_{n}(f_{71}) & a_{n}(f′_{71}) & mod 81 \\ 1 & 1 & 1 & (1,1) \\ 2 & u & 3 − u − u^{2} & (60,69) \\ 3 & 3 − u^{2} & −3 + u + u^{2} & (48,12) \\ 4 & −2 + u^{2} & 1 + u & (34, 61) \\ 5 & −1 − u & 5 − 2u − u^{2} & (20, 11) \\ 6 & 3 − 2u & −3 − u & (45, 18) \\ 7 & −6 + 2u + 2u^{2} & −6 + 2u + 2u^{2} & (24,24) \\ 8 & −3 + u & −u & (57, 21) \\ 9 & 6 − 3u − u^{2} & u & (33,60) \\ 10 & −u − u^{2} & 6 + u − u^{2} & (66, 30) \\ 11 & 6 − 2u − 2u^{2} & 2u & (57,39) \\ 12 & −6 + 3u & −6 + 3u + 2u^{2} & (12, 3) \end{matrix}

The polynomial X^{3} − 5X + 3 ∈ Z_{3}[X] factors as a product of a linear and an irreducible quadratic factor. Mapping u to the unique root u_{0} of X^{3} − 5X + 3 in Z_{3}, we obtain a 3-adic eigenform which we also denote by f_{71} and whose Fourier coefficients a_{p}(f_{71}) are congruent to the coefficients a_{p} associated to the curve 142A. See Table 1.1. On the other hand, mapping u to a root of the irreducible quadratic divisor of X^{3} − 5X + 3, gives rise to a 3-adic eigenform whose Fourier coefficients are not congruent to the coefficients a_{p}. The same happens with the other eigenform f′_{71}.

Using the approximation u_{0} ≡ 60 (mod 3^{4}), we list the vectors T_{n} = (a_{n}(f_{71}), (a_{n}(f′_{71})) in the rightmost column of Table 2.1 with a precision of O(3^{4}). The Hecke ring T is the Z_{3}-submodule of T̃ = Z_{3} × Z_{3} generated by the vectors T_{n} for n ≤ 12.

The following Lemma enables us to do the 3-adic computations with finite precision.

Lemma 2.2. Let R be a local ring with maximal ideal 𝔪 and let M be a finitely generated R-module. Let M_{1} and M_{2} be two submodules of M and suppose there is an integer k > 0 for which – M_{1} + 𝔪^{k}M = M_{2} + 𝔪^{k}M,

– 𝔪^{k−1}M ⊂ M_{2};

then M_{1} = M_{2}.

Proof. Since 𝔪^{k−1}M ⊂ M_{2} ⊂ M_{1}+𝔪^{k}M, Nakayama’s Lemma implies that 𝔪^{k−1}M ⊂ M_{1}. This implies that 𝔪^{k}M is contained in M_{1} as well as M_{2} so that M_{1} = M_{2} as required.

Theorem 2.3. There is an isomorphism between complete intersections

T ≅ Z_{3}[[X]]/(X^{2} − 9X).

Proof. It is easy to see that the row vectors in the rightmost column of Table 2.1 generate the submodule T′ of T̃ = Z_{3} ×Z_{3} with basis (1, 1) and (0, 9). Therefore 3^{2}T̃ ⊂ T′. Now we apply Lemma 2.2 to the Z_{3}-module M = T̃ and to the submodules M_{1} = T and M_{2} = T′. Let k = 3. Then 3^{k−1}T̃ ⊂ T′. Since the computations were done with an accuracy of O(3^{4}) we have that M_{1} = M_{2} modulo 3^{k}T̃. It follows that T is the Z_{3}-module generated by 1 = (1, 1) and x = (0, 9). As a Z_{3}-algebra T is therefore generated by x. Since x^{2} = 9x, the homomorphism Z_{3}[X]/(X^{2} − 9X) ⟶ T given by X ↦ x is an isomorphism. Since x^{n} → 0 in T as n → ∞, this isomorphism extends to the ring Z_{3}[[X]]/(X^{2} − 9X). This proves the Theorem.

The two Z_{3}-valued points π, π′ : Z_{3}[[X]]/(X^{2} − 9X) ⟶ Z_{3} given by π(X) = 0 and π′(X) = 9, are precisely the algebra-homomorphisms given by, say, T_{n} ↦ a_{n}(f_{71}) and T_{n} ↦ a_{n}(f′_{71}) respectively. The η-invariant with respect to the first point is the Z_{3}-ideal η = π(Ann_{T}(I)) where I = ker(π) = (X). Since T is a complete intersection, η is also the ideal for which #Z_{3}/η = #(X)/(X^{2}). Using either description one easily checks that η is the Z_{3}-ideal generated by 3^{2}. See [5, Crit.I].

3. The Hecke ring of level 284. In this section we consider deformations ρ : Gal(\overline{Q}/Q) ⟶ GL_{2}(R) of the representation \overline{ρ} that are not necessarily minimal at the prime 2. They are supposed to satisfy the following more relaxed conditions:

• ρ is unramified outside {2, 3, 71};

• ρ is flat at 3;

• ρ is of type (A) at 2 and 71;

• the determinant of ρ is the cyclotomic character. It follows from the theory of the Tate curve that the deformation given by the 3-adic Tate module of the curve 142A is of this type. According to Wiles, the universal deformation ring is isomorphic to a certain Hecke ring T of level 284 = 4 · 71 rather than 142 = 2 · 71. The reason for this is the fact that 2 divides the order of GL_{2}(F_{3}) and that therefore, deformations ρ of type (A) at 2 could have Artin conductor divisible by 4.

By [7, section 3], the Hecke operators T_{p} in the algebra T of level 284 can be represented as row vectors (a_{p}(f))_{f} with f running through the normalized 3-adic newforms of level dividing 284 whose Fourier coefficients a_{p}(f) are, for prime p ≠ 2, 3, congruent to the coefficients a_{p} associated to the elliptic curve 142A. The levels of these forms are equal to 71, 142 or 284. The forms of level 71 were already described in the previous section. The new part of the space of weight 2 cusp forms S_{2}(Γ_{0}(142)) is a product of 1-dimensional eigenspaces for the action of the Hecke algebra. The Fourier coefficients of the eigenforms can be found in the Antwerp Tables [1]. The Fourier coefficients of three of these, viz. 142A, 142F and 142G, are congruent to the coefficients a_{p} of the curve 142A. Finally, there is up to conjugacy only one newform of level 284 whose Fourier coefficients a_{p}(f_{284}) are congruent to the coefficients a_{p} of the curve 142A. Its coefficients are contained in the cubic field Q_{3}(t) where t^{3} + 3 t^{2} − 3 = 0. This totally ramified extension of Q_{3} has discriminant 81. Its ring of integers is Z_{3}[t].

In this way the Hecke ring T becomes the Z_{3}-subalgebra of T̃

T ⊂ T̃ = Z_{3} × Z_{3} × Z_{3} × Z_{3} × Z_{3} × Z_{3}[t]

generated by the Hecke operators T_{p} with p ≠ 2, 3 prime. Here T_{p} is identified with the row vector consisting of the p-th Fourier coefficients of the eigenforms f_{71}, f′_{71}, 142A, 142F, 142G and f_{284} as displayed in the p-th row of Table 3.1. The q-expansion of the newform of level 284 was computed using modular symbols as in [14].

It follows from [7, Lemma 3.2] that T is generated as a Z_{3}-algebra by the T_{p} with p ≠ 2 prime. By Theorem 1 of the appendix, T is then generated as a Z_{3}-module by the operators T_{n} with n odd and n ≤ 2 · 6 · 72/12 = 72. The additive group of T̃ is a free Z_{3}-module of rank 8. Choosing the Z_{3}-basis {1, t, t^{2}} for Z_{3}[t], we view the Hecke operators T_{p} as vectors in Z_{3}^{8}. For example, the Hecke operator T_{25} in Table 3.1 becomes the vector

(−4 + 2u + u^{2}, 8 − 3u − u^{2} , −1 , 11 , −1 , 5 , 9 , 2).

We do the computations modulo 3^{7}. As in section 2, let u denote the unique root in Z_{3} of the polynomial X^{3} − 5X + 3. We have that

u = 2·3 + 2·3^{3} + 3^{4} + 2·3^{5} + 2·3^{6} + O(3^{7}).

Table 3.1 Hecke operators T_{n}.

\begin{matrix}n & a_{n}(f_{71}) & a_{n}(f′_{71}) & 142A & 142F & 142G & a_{n}(f_{284}) \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 3 & 3 − u^{2} & −3 + u + u^{2} & 0 & −3 & 3 & t \\ 5 & −1 − u & 5 − 2u − u^{2} & 2 & −4 & 2 & −1 − 3t − t^{2} \\ 7 & −6 + 2u + 2u^{2} & −6 + 2u + 2u^{2} & 0 & −3 & −3 & −6 + 2t + 2t^{2} \\ 9 & 6 − 3u − u^{2} & u & −3 & 6 & 6 & −3 + t^{2} \\ 11 & 6 − 2u − 2u^{2} & 2u & 6 & 0 & −6 & 2t \\ 13 & 4 & −2 − 2u & 4 & 1 & −5 & 4 − 6t − 4t^{2} \\ 15 & −6 + 2u + u^{2} & −6 − u + u^{2} & 0 & 12 & 6 & −3 − t \\ 17 & −6 + 2u + 2u^{2} & 6 − 2u^{2} & 6 & 0 & 6 & −6 + 6t + 4t^{2} \\ 19 & 7 − u − u^{2} & 7 − u − 2u^{2} & −8 & −5 & 1 & −2 − t^{2} \\ 21 & −12 + 2u + 2u^{2} & 6 + 2u & 0 & 9 & −9 & 3 − 6t − t^{2} \\ 23 & −4 + 2u^{2} & −4 & −4 & −7 & 5 & 8 − 2t^{2} \\ 25 & −4 + 2u + u^{2} & 8 − 3u − u^{2} & −1 & 11 & −1 & 5 + 9t + 2t^{2} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 67 & −4 + 4u & −4 − 2u & 2 & −4 & 2 & −4 + 2t^{2} \\ 69 & −12 + 6u & 12 − 4u − 4u^{2} & 0 & 21 & 15 & −6 + 8t + 6t^{2} \\ 71 & 1 & 1 & 1 & 1 & 1 & 1 \end{matrix}

Applying Hermite reduction to the row vectors, we find the matrix below. We let T′ denote the Z_{3}-submodule of T̃ spanned by its rows. Incidentally, we find the same matrix using only the rows n = 1, 3, …, 17. This means that the module T′ is already generated by the T_{n} with n odd and n ≤ 17. It is easy to see that 3^{5}T̃ ⊂ T′. Now we apply Lemma 2.2 with M = T̃, M_{1} = T and M_{2} = T′. We take k = 6. Then 3^{k−1}T̃ ⊂ T′ and, since we did the computations with an accuracy of O(3^{7}), we have M_{1} + 3^{k}T̃ = M_{2} + 3^{k}T̃. It follows that T = T′. Thus, as a Z_{3}-module, T is generated by the rows of the matrix

( \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 9 & 0 & 6 & 18 & 0 & 0 & 1 \\ 0 & 0 & 3 & 0 & 54 & 0 & −1 & −1 \\ 0 & 0 & 0 & 9 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 81 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 \end{matrix} ) .

For later use we observe that the determinant of the matrix and hence the index [T̃ : T] are equal to 3^{12}. The first row of the matrix is the unit element 1 ∈ T. We let x = (0, 9, 0, 6, 18 , t^{2}) be the element corresponding to the second row and y = (0, 0, 3, 54, −t − t^{2}) be the one corresponding to the third row. The morphism φ : Z_{3}[X, Y] ⟶ T given by f(X, Y) ↦ f(x, y) extends to a Z_{3}-algebra morphism

φ̂: Z_{3}[[X, Y]] ⟶ T.

This follows from the fact that x^{n}, y^{n} → 0 in T̃ as n → ∞.

Theorem 3.2. The ring homomorphism φ̂ induces an isomorphism of Z_{3}-algebras

Z_{3}[[X, Y]]/(F_{1}, F_{2}) ⟶^{≅} T,

where

\begin{aligned}F_{1} & = 29412Y − 9804Y^{2} − 91158XY − 1641XY^{2} + 11618X^{2}Y − 787(X^{3} − 15X^{2} + 54X), \\ F_{2} & = 8514Y − 477Y^{2} − 8204XY + 1741XY^{2} + 2369X^{2}Y − 787Y^{3}. \end{aligned}

Proof. We compute the first few monomials in x and y in the subring T of T̃:

\begin{matrix}1 = ( 1, & 1, & 1, & 1, & 1, & 1 & ), \\ x = ( 0, & 9, & 0, & 6, & 18, & t^{2} & ), \\ y = ( 0, & 0, & 3, & 0, & 54, & −t − t^{2} & ), \\ x^{2} = ( 0, & 81, & 0, & 36, & 324, & −9 + 3t + 9t^{2} & ), \\ xy = ( 0, & 0, & 0, & 0, & 972, & 6 − 3t − 6t^{2} & ), \\ y^{2} = ( 0, & 0, & 9, & 0, & 2916, & −3 + 3t + 4t^{2} & ), \\ x^{2}y = ( 0, & 0, & 0, & 0, & 17496, & 45 − 18t − 39t^{2} & ), \\ xy^{2} = ( 0, & 0, & 0, & 0, & 52488, & −27 + 12t + 24t^{2} & ). \end{matrix}

Using the Z_{3}-basis {1, t, t^{2}} for the ring Z_{3}[t], we write these eight vectors as row vectors in Z_{3}^{8}. The determinant of the resulting 8 × 8 matrix is equal to −2^{2}3^{12}787. This implies that the Hecke ring T is equal to the Z_{3}-span of the eight monomials above. It follows that φ and hence φ̂ are surjective. In order to determine the kernels of φ and φ̂, we express x^{3}, y^{3} and x^{2}y^{2} as Z_{3}-linear combinations of the eight monomials above. Solving the corresponding linear systems of eight equations in eight unknowns one finds that

\begin{aligned}787x^{3} & = 29412y − 9804y^{2} − 91158xy − 1641xy^{2} + 11618x^{2}y − 787(54x − 15x^{2}), \\ 787y^{3} & = 8514y − 477y^{2} − 8204xy + 1741xy^{2} + 2369x^{2}y, \\ 787x^{2}y^{2} & = 15444y − 5148y^{2} − 15870xy + 12657xy^{2} + 6219x^{2}y. \end{aligned}

In other words, F_{1}(x, y) = F_{2}(x, y) = F_{3}(x, y) = 0 where F_{1} and F_{2} are as above and

F_{3} = 15444Y − 5148Y^{2} − 15870XY + 12657XY^{2} + 6219X^{2}Y − 787X^{2}Y^{2}.

Therefore J = (F_{1}, F_{2}, F_{3}) ⊂ ker(φ). Since Z_{3}[X, Y]/J has Z_{3}-rank at most 8, it follows that ker(φ) = J. Writing J′ for the Z_{3}[[X, Y]]-ideal (F_{1}, F_{2}, F_{3}), there is a commutative diagram

Z_{3}[X, Y]/J ⟶^{≅} T.
↓  ↗
Z_{3}[[X, Y]]/J′

Here the diagonal arrow is the morphism induced by φ̂. It is surjective. Reducing everything modulo 3, we obtain the diagram

F_{3}[X, Y]/\overline{J} ⟶^{≅} T/3T,
↓  ↗ \overline{φ}
F_{3}[[X, Y]]/\overline{J}′

where \overline{J} and \overline{J}′ denote the ideals generated by the reduced polynomials \overline{F}_{1}, \overline{F}_{2} and \overline{F}_{3} in the rings F_{3}[X, Y] and F_{3}[[X, Y]] respectively and \overline{φ} : F_{3}[[X, Y]]/\overline{J}′ ⟶ T/3T denotes the surjective morphism induced by φ̂. Since F_{3}[X, Y]/\overline{J} is an Artin ring, the natural map to the product of the completions at its maximal ideals is an isomorphism. Since the vertical map is the natural map from F_{3}[X, Y]/\overline{J} to its completion at the maximal ideal (X, Y), it is surjective. We conclude that \overline{φ} is an isomorphism

F_{3}[[X, Y]]/\overline{J}′ ⟶^{≅} T/3T.

From

\begin{aligned}\overline{F}_{1} & = −X^{3} − X^{2}Y, \\ \overline{F}_{2} & = −Y^{3} + X Y + X Y^{2} − X^{2}Y, \\ \overline{F}_{3} & = −X^{2}Y^{2}, \end{aligned}

we obtain the relation

−Y (X − 1)\overline{F}_{1} + X^{2}\overline{F}_{2} = (1 + X + Y)\overline{F}_{3}

in F_{3}[[X, Y]]. Since 1 + X + Y is a unit in F_{3}[[X, Y]], the monomial \overline{F}_{3} belongs to the ideal (\overline{F}_{1}, \overline{F}_{2}) so that \overline{J}′ = (\overline{F}_{1}, \overline{F}_{2}). We claim that the ideal Ĵ = ker(φ̂) ⊂ Z_{3}[[X, Y]] is generated by F_{1} and F_{2}. Indeed, since the Hecke algebra T is free over Z_{3}, the exact sequence

0 ⟶ Ĵ ⟶ Z_{3}[[X, Y]] ⟶ T ⟶ 0.

remains exact after tensoring with F_{3}. It follows that Ĵ/3Ĵ = \overline{J}′ = (\overline{F}_{1}, \overline{F}_{2}). Therefore

Ĵ ⊂ (F_{1}, F_{2}) + 3 Ĵ ⊂ (F_{1}, F_{2}) + 𝔪 Ĵ

where 𝔪 = (3, X, Y) denotes the maximal ideal of the local ring Z_{3}[[X, Y]]. Nakayama’s Lemma implies then that Ĵ = (F_{1}, F_{2}). This proves the Theorem.

The “points” of the Hecke algebra T are (X, Y) = (0, 0), (9, 0), (0, 3), (6, 0), (18, 54) and (t^{2}, −t − t^{2}) and its conjugates. The first two correspond to the newforms of level 71, the next three to eigenforms of level 142 and the three conjugates of (t^{2}, −t − t^{2}) to an eigenform of level 284. The point (0, 3) corresponds to the elliptic curve 142A of the introduction. The point (0, 0) is our origin. The canonical morphism from T to the minimal universal deformation ring of section 2 is given by X ↦ X and Y ↦ 0. Since T is a complete intersection, the η-invariant associated to it is equal to the Z_{3}-ideal generated by #I/I^{2} where I is the ideal (X, Y). The order of I/I^{2} is given by the determinant of the matrix of the coefficients of the linear terms of F_{1} and F_{2}. Since

F_{1} = −787·54X + 29412Y + terms of deg ≥ 2,
F_{2} =  8514Y + terms of deg ≥ 2,

it is the Z_{3}-ideal generated by

η = det ( \begin{matrix}−787·54 & 29412 \\ 0 & 8514 \end{matrix} ) = 3^{5} · (unit).

This is 3^{3} times the η-invariant of the minimal deformation ring computed in section 2. This agrees with [7, Lemma 4.4] because the contribution of the Euler factors at 2 is equal to (1 − 2)((1 + 2)^{2} − a_{2}(f_{71})^{2}) = −3^{2} + u^{2} and has 3-adic valuation equal to 3.

4. The Hecke ring of level 142. In this section we study the 3-adic Hecke ring T of level 142. In contrast to the Hecke rings of levels 71 and 284, we show that it is not a Gorenstein ring. The Z_{3}-algebra T is generated by the vectors T_{p} = (a_{p}(f))_{f} for p ≠ 2. Here f runs over the 3-adic normalized eigenforms of level 142 whose Fourier coefficients are congruent to the coefficients a_{p} of the curve 142A. The algebra is obtained by simply omitting the last component of the row vectors that generate the Hecke ring of level 284. Therefore the results of the previous section imply that modulo 3^{7}T̃, the ring T is equal to the Z_{3}-submodule T′ of T̃ = Z_{3}×Z_{3}×Z_{3}×Z_{3}×Z_{3} generated by the rows of the matrix

( \begin{matrix}1 & 1 & 1 & 1 & 1 \\ 0 & 9 & 0 & 6 & 18 \\ 0 & 0 & 3 & 0 & 54 \\ 0 & 0 & 0 & 9 & 0 \\ 0 & 0 & 0 & 0 & 81 \end{matrix} ) .

We see that 3^{4}T̃ ⊂ T′ so by Lemma 2.2 we have that T = T′. The determinant of the matrix and hence the index [T̃ : T] are equal to 3^{9}. We let x ∈ T be the second and y ∈ T be the third row of the matrix. The morphism φ : Z_{3}[X, Y] ⟶ T given by f(X, Y) ↦ f(x, y) extends to a Z_{3}-algebra morphism φ̂: Z_{3}[[X, Y]] ⟶ T.

Theorem 4.1. The morphism φ̂ induces an isomorphism

Z_{3}[[X, Y]]/(F_{1}, F_{2}, F_{3}) ≅ T,

where

\begin{aligned}F_{1} & = 17XY − 6Y^{2} + 18Y, \\ F_{2} & = 17(X^{3} − 15X^{2} + 54X) − 12Y^{2} + 36Y, \\ F_{3} & = Y^{3} − 57Y^{2} + 162Y. \end{aligned}

Proof. We have that

\begin{matrix}1 = ( 1, & 1, & 1, & 1, & 1 & ), \\ x = ( 0, & 9, & 0, & 6, & 18 & ), \\ y = ( 0, & 0, & 3, & 0, & 54 & ), \\ x^{2} = ( 0, & 81, & 0, & 36, & 324 & ), \\ y^{2} = ( 0, & 0, & 9, & 0, & 2916 & ). \end{matrix}

Since the determinant of the matrix whose rows are the above five row vectors is equal to 3^{9} times a unit, the Hecke ring T is generated by 1, x, x^{2}, y, y^{2} as a module over Z_{3}. Therefore the morphism ψ : Z_{3}[X, Y] ⟶ T given by ψ(X) = x and ψ(Y) = y, is surjective. We have the following relations in T:

\begin{aligned}17xy & = 6y^{2} − 18y, \\ 17(x^{3} − 15x^{2} + 54x) & = 12y^{2} − 36y, \\ y^{3} & = 57y^{2} − 162y. \end{aligned}

Using the same arguments as in proof of Theorem 3.2, the result follows.

It follows that T/3T ≅ F_{3}[X, Y]/(X^{3}, XY, Y^{3}). This is a finite local F_{3}-algebra with maximal ideal 𝔪 = (X, Y). The annihilator of 𝔪 is (X^{2}, Y^{2}). Since the F_{3}-dimension of this ideal is 2 rather than 1, the ring T is not Gorenstein by [16, Prop.1.4]. For the sake of completeness we compute the two invariants. Let π : T ⟶ Z_{3} be the point given by π(X) = π(Y) = 0 and let I = (X, Y) = ker(π). It is easy to see that the group I/I^{2} is isomorphic to (Z/9Z) × (Z/27Z). On the other hand, a somewhat more cumbersome linear algebra computation shows that Ann_{T}(I) is zero so that η = 0.

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Appendix by W. Stein.

Theorem. The ring T of Hecke operators acting on the space of cusp forms of weight k and level N is generated as an abelian group by the Hecke operators T_{n} with

n ≤ \frac{kN}{12} ∏_{p|N}(1 + \frac{1}{p})

Proof.