MODULARITY OF MOD 5 REPRESENTATIONS

KARL RUBIN

Introduction

The aim of this paper is to tie everything together to prove the following theorem.

Theorem A (Wiles [18] + Taylor & Wiles [16] + Diamond [3]). If E is an elliptic curve defined over Q and E is semistable at 3 and 5, then E is modular.

Our starting point will be Theorems B and C below. If E is an elliptic curve defined over a field F of characteristic zero, let G_{F} = Gal(F̄/F) and for every prime p, let

ρ̄_{F,E,p} : G_{F} → GL_{2}(F_{p})

be (the isomorphism class of) the representation of G_{F} on the p-torsion E[p] in E(F̄). We will write simply ρ̄_{E,p} for ρ̄_{Q,E,p}.

Theorem B (Wiles [18] + Taylor & Wiles [16] + Diamond [3]). Suppose E is an elliptic curve over Q, and p is an odd prime, such that

• E is semistable at p,

• ρ̄_{E,p} restricted to Gal(Q̄/Q(\sqrt{(−1)^{(p−1)/2}p})) is absolutely irreducible, and • ρ̄_{E,p} is modular.

Then E is modular.

Theorem C (Langlands [6] + Tunnell [17]). Suppose ρ : G_{Q} → GL_{2}(F_{3}) is a continuous representation satisfying

• ρ is irreducible, and

• det(ρ(complex conjugation)) = −1. Then ρ is modular.

Theorem B (for p = 3) and Theorem C together prove Theorem A under the extra hypothesis that ρ̄_{E,3} restricted to G_{Q(\sqrt{−3})} is absolutely irreducible. When ρ̄_{E,3} restricted to G_{Q(\sqrt{−3})} is not absolutely irreducible we will complete the proof of Theorem A by applying Theorem B with p = 5. We need to show that the hypotheses of Theorem B are satisfied, and this will follow from the following two theorems:

Theorem 1. Suppose E is an elliptic curve over Q such that • ρ̄_{E,3} restricted to G_{Q(\sqrt{−3})} is not absolutely irreducible,

• E is semistable at 5, and

• the j-invariant j(E) is not 11^{3}/2^{3} or −29^{3}41^{3}/2^{15}. Then ρ̄_{E,5} restricted to G_{Q(\sqrt{5})} is absolutely irreducible.

Partially supported by an NSF grant. The author also thanks the Institut des Hautes Etudes Scientifiques and the Institute for Advanced Study for their hospitality and support.

version of December 12, 1995.

Theorem 2. Suppose E is an elliptic curve over Q such that • E is semistable at 3, and

• ρ̄_{E,5} is irreducible. Then ρ̄_{E,5} is modular.

Theorem A is immediate from Theorems B, C, 1 and 2, together with the observation that the exceptional curves in Theorem 1 are modular. (More precisely, these curves are twists of the curves listed as 338E1 and 338E2 in [2]. Although it is stated in §2.14 of [2] that these models are not proved to be modular, John Cremona informs me that due to subsequent progress the computations in [2] do suffice to prove that these curves are the modular curves they appear to be.)

The fundamental hypothesis in Theorem B is that ρ̄_{E,p} is modular. When p = 3 this is known because of Theorem C, and when p = 5 we will use Theorem 2. To prove Theorem 2 we must have at our disposal a large collection of modular forms. The modular forms we use will be the ones produced by Theorems B and C when p = 3.

The proofs of Theorems 1 and 2 are essentially due to Wiles, and can be found with varying degrees of generality and detail in [18] Chapter 5 and in [3], proof of Theorem 5.4.

If F is an arbitrary field, let G_{F} = Gal(F^{s}/F) where F^{s} is a separable closure of F. If p is a prime not equal to the characteristic of F let χ_{p} : G_{F} → Aut(μ_{p}) ⟶^{∼} F_{p}^{×} be the cyclotomic character.

The following statement was first pointed out to me by Richard Taylor. Although not necessary for the proof of Theorem A, it is related to Theorem 2 (see Theorem 4 below) and we will prove it in §5.

Theorem 3. Let p be 3 or 5, and let F be a field of characteristic different from p. Suppose ρ : G_{F} → GL_{2}(F_{p}) is a representation such that det(ρ) = χ_{p}. Then there is an elliptic curve E defined over F such that ρ ≅ ρ̄_{F,E,p}.

Combining Theorems 2 and 3 (with F = Q) gives the following.

Theorem 4. Suppose ρ : G_{Q} → GL_{2}(F_{5}) is a representation satisfying • ρ is semistable at 3 (i.e., the image of an inertia group at 3 is unipotent),

• ρ is irreducible, and

• det(ρ) = χ_{5}. Then ρ is modular.

Remark. The proof of Theorem 1 depends in a crucial way on the fact that the genus of the modular curve X_{0}(3·5) is greater than zero, and the proof of Theorem 2 on the fact that the genus of X(5) is equal to zero. For primes p (see [12] §1.6),

genus(X_{0}(3p)) > 0 ⇔ p ≥ 5,
genus(X(p)) = 0 ⇔ p ≤ 5.

Thus 5 is the only prime that will work for Wiles’ argument.

Acknowledgment. I would like to thank Alice Silverberg for helpful conversations.

1. Preliminaries: group theory

Lemma 5. Suppose E is an elliptic curve over a field F and p > 2 is prime. If F has an embedding into R and ρ̄_{F,E,p} is irreducible, then ρ̄_{F,E,p} is absolutely irreducible.

Proof. Fix a complex conjugation τ ∈ G_{F} and write E[3]^{+} (resp. E[3]^{−}) for the subspace of E[3] where τ acts via +1 (resp. −1). The characteristic polynomial of ρ̄_{F,E,p}(τ) is x^{2} − 1, so dim_{F_{3}} E[3]^{+} = dim_{F_{3}} E[3]^{−} = 1.

Suppose ρ̄_{F,E,p} is not absolutely irreducible, i.e., there is a one- dimensional subspace W of E[3] ⊗ F̄_{p} which is stable under G_{F}. Then W = E[3]^{±} ⊗ F̄_{p} for some choice of sign, since no other one-dimensional subspace is stable under τ. Hence E[3]^{±} = W ∩ E[3] is stable under G_{F}, so ρ̄_{F,E,p} is reducible.

Proposition 6. Suppose E is an elliptic curve over Q such that • ρ̄_{E,3} is irreducible, and

• the restriction of ρ̄_{E,3} to G_{Q(\sqrt{−3})} is not absolutely irreducible. Then there are distinct subgroups C_{3}, C′_{3} of order 3 of E[3] such that the unordered set {C_{3}, C′_{3}} is fixed by G_{Q}.

Proof. Fix a complex conjugation τ ∈ G_{Q}. Our hypotheses on ρ̄_{E,3} imply (using Lemma 5) that there is a one-dimensional subspace W of E[3] ⊗ F̄_{3} which is stable under G_{Q(\sqrt{−3})} but not under G_{Q}. Then W^{τ} is also stable under G_{Q(\sqrt{−3})}, and E[3] ⊗ F̄_{3} = W ⊕ W^{τ}.

It follows that ρ̄_{E,3}(G_{Q(\sqrt{−3})}) is a cyclic subgroup of SL_{2}(F_{3}), of order prime to 3, and so the image of G_{Q(\sqrt{−3})} in PGL_{2}(F_{3}) has order at most 2. Let C denote the set of 4 subgroups of order 3 of E[3], and let E[3]^{+} ∈ C be the subgroup of points fixed by complex conjugation. Then G_{Q} acts on C by a subgroup of S_{4} of order at most 4. Since τ fixes E[3]^{+} but acts nontrivially on C, the G_{Q}-orbit of E[3]^{+} has order at most 2. The assumption that ρ̄_{E,3} is irreducible implies that this orbit has order exactly 2, so this orbit is the desired set {C_{3}, C′_{3}}.

Proposition 7. Suppose E is an elliptic curve over Q such that • ρ̄_{E,5} is irreducible and

• E is semistable at 5. Then the restriction of ρ̄_{E,5} to G_{Q(\sqrt{5})} is absolutely irreducible.

Proof. Let I_{5} ⊂ G_{Q} denote the inertia group of some prime above 5, and fix σ ∈ I_{5} such that σ ∉ G_{Q(\sqrt{5})} (possible since 5 ramifies in Q(\sqrt{5})/Q). Suppose that E satisfies the hypotheses of the proposition but the restriction of ρ̄_{E,5} to G_{Q(\sqrt{5})} is not absolutely irreducible. By Lemma 5, this restriction is then reducible, so there is a subgroup C_{5} of E[5] which is stable under G_{Q(\sqrt{5})} but not under G_{Q}. Then C_{5}^{σ} is also stable under G_{Q(\sqrt{5})}, and E[5] = C_{5} ⊕ C_{5}^{σ}. With a basis of E[5] chosen from C_{5} and C_{5}^{σ}, since

det(ρ̄_{E,5}(G_{Q(\sqrt{5})})) = χ_{5}(G_{Q(\sqrt{5})}) = \{±1\} and det(ρ̄_{E,5}(σ)) = χ_{5}(σ) ∉ \{±1\}

we see that

ρ̄_{E,5}(G_{Q(\sqrt{5})}) ⊂ \{( \begin{matrix}a & 0 \\ 0 & ±a \end{matrix} ) : a ∈ F_{5}^{×}\} and ρ̄_{E,5}(σG_{Q(\sqrt{5})}) ⊂ \{( \begin{matrix}0 & a \\ ±2a & 0 \end{matrix} ) : a ∈ F_{5}^{×}\}.

Case I: E is ordinary or multiplicative at 5. Since σ ∈ I_{5}, σ stabilizes a proper subgroup of E[5] (see [9], Proposition 13 and the remarks before Proposition 11). But the description above shows this is impossible.

Case II: E is supersingular at 5. Proposition 12 of [9] shows that ρ̄_{E,5}(I_{5}) is cyclic of order 24. Again, the description above shows that ρ̄_{E,5}(G_{Q}) has no such subgroup.

2. Preliminaries: modular curves

Write H for the complex upper half plane. If Γ is a congruence subgroup of SL_{2}(Z), Ĥ/Γ will denote the compactification of H/Γ, obtained by adjoining a finite set of cusps ([12] §1.3).

We will need the congruence groups

Γ(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) : ( \begin{matrix}a & b \\ c & d \end{matrix} ) ≡ ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) (mod N)\}
Γ_{0}(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) : c ≡ 0 (mod N)\}
Γ_{split}(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) : either b ≡ c ≡ 0 or a ≡ d ≡ 0 (mod N)\}.

Lemma 8. As in [12] §1.6 define μ(Γ) = [SL_{2}(Z) : {±1}Γ], ν_{∞}(Γ) = number of cusps of H/Γ, ν_{i}(Γ) = number of elliptic points of order i of H/Γ, i = 2 or 3.

Then we have the following table:

Γ  μ(Γ) ν_{∞}(Γ) ν_{2}(Γ) ν_{3}(Γ) genus of H/Γ
Γ_{0}(15)  24  4  0  0  1
Γ_{0}(5) ∩ Γ_{split}(3) 36  4  4  0  1
Γ(5) ∩ Γ_{0}(3)  240  24  0  0  9
Γ(5) ∩ Γ_{split}(3) 360  24  0  0  19

Proof. Let Γ denote one of the four groups in the table. Then Γ(15) ⊂ Γ ⊂ SL_{2}(Z), so μ(Γ) is the index of the image of Γ in PSL_{2}(Z/15Z), which is easily computed.

The cusps of H/Γ(15) are described in [12] §1.6: there are 96 of them, and they can be identified with the coset space PSL_{2}(Z/15Z)/U where U is the subgroup generated by ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ). From this it is not difficult to work out the number of cusps for each of the four groups (for Γ_{0}(15) it is done in Proposition 1.43 of [12]).

For elliptic points, Propositions 1.43 and 1.39 of [12] cover all of the 0 entries in the table, and ν_{2}(Γ_{0}(5) ∩ Γ_{split}(3)) can be worked out with some effort using the methods of [12] §1.6.

The last column follows from Proposition 1.40 of [12], which says that the genus of H/Γ is 1 + μ(Γ)/12 − ν_{2}(Γ)/4 − ν_{3}(Γ)/3 − ν_{∞}(Γ)/2.

2.1. The modular curve X_{0}(15). Let X_{0}(15) be the modular curve over Q which parametrizes isomorphism classes of pairs (E, C_{15}) where E is an elliptic curve and C_{15} is a subgroup of E of order 15.

Lemma 9. (i) X_{0}(15) is a curve of genus one with four cusps, all of which are

rational over Q. (ii) #(X_{0}(15)(Q)) = 8. (iii) The four rational points of X_{0}(15) which are not cusps correspond to four

pairs (E_{i}, C_{15}^{(i)}) where j(E_{i}) ∈ {−25/2, −5^{2}241^{3}/2^{3}, −5·29^{3}/2^{5}, 5·211^{3}/2^{15}}. (iv) If E is an elliptic curve over Q with a rational subgroup of order 15, then E

is not semistable at 5.

Proof. Since X_{0}(15)(C) = Ĥ/Γ_{0}(15), the genus and the number of cusps of X_{0}(15) are given in Lemma 8. The fact that the four cusps are rational follows from the action of G_{Q} on the cusps (see for example [5] §VI.5).

From the tables [1] 15C or [2] 15A1 one can read off that X_{0}(15) has 8 rational points. In the same tables one finds 4 elliptic curves over Q (50A1, 50A2, 50A3, and 50A4 in [2]), with j-invariants −25/2, −5^{2}241^{3}/2^{3}, −5·29^{3}/2^{5}, and 5·211^{3}/2^{15}, respectively, each isogenous to E_{0} : y^{2} + xy + y = x^{3} − x − 2 and having a rational subgroup of order 15. Thus these four curves represent all the non-cusp rational points of X_{0}(15). This proves (iii).

Suppose E is defined over Q and has a rational subgroup C_{15} of order 15. Then (E, C_{15}) ∈ X_{0}(15)(Q), so by (iii), E is isogenous to a quadratic twist of E_{0}. The equation above for E_{0} is minimal and is not semistable at 5. Since it has discriminant −2·5^{4}, this equation remains minimal over any quadratic extension of Q (see for example [15] Exercise 7.1). It follows that E is not semistable at 5.

2.2. The modular curve X_{0,split}(5, 3). Let X_{0,split}(5, 3) be the modular curve over Q which parametrizes triples (E, C_{5}, {C_{3}, C′_{3}}) where E is an elliptic curve, C_{5} is a subgroup of order 5, and {C_{3}, C′_{3}} is an unordered set of distinct subgroups of order 3.

Lemma 10. (i) X_{0,split}(5, 3) is a curve of genus one with four cusps, all of which

are rational over Q. (ii) #(X_{0,split}(5, 3)(Q)) = 8. (iii) The four rational points of X_{0,split}(5, 3) which are not cusps correspond to

four triples (E_{i}, …) where j(E_{i}) ∈ {11^{3}/2^{3}, −29^{3}41^{3}/2^{15}}.

Proof. The classical theory of modular curves shows that

X_{0,split}(5, 3)(C) = Ĥ/(Γ_{0}(5) ∩ Γ_{split}(3)).

Thus the genus and the number of cusps of X_{0,split}(5, 3) are given in Lemma 8. The fact that the four cusps are rational follows from the action of G_{Q} on the cusps (see for example [5] §VI.5).

Define a map f from X_{0,split}(5, 3) to the jacobian J_{0}(15) of X_{0}(15) by

f : (E, C_{5}, {C_{3}, C′_{3}}) ↦ (E, C_{5} + C_{3}) + (E, C_{5} + C′_{3}) − 2[∞]

where [∞] denotes the infinity cusp on X_{0}(15). Then f is not constant, because if E is an elliptic curve without complex multiplication by Q(i) and C_{3}, C′_{3}, C′′_{3} are three distinct subgroups of order 3, then

f(E, C_{5}, {C_{3}, C′_{3}}) − f(E, C_{5}, {C_{3}, C′′_{3}}) = (E, C_{5} + C′_{3}) − (E, C_{5} + C′′_{3})

which is not zero in J_{0}(15) since (E, C_{5} +C′_{3}), (E, C_{5} +C′′_{3}) represent distinct points of X_{0}(15) and X_{0}(15) has genus 1 (Lemma 9(i)). Therefore X_{0,split}(5, 3) is isogenous to X_{0}(15) and so #(X_{0,split}(5, 3)(Q)) ≤ 8 (see [1] or [2]).

In [2], one finds that the elliptic curve denoted 338E1, E : y^{2} + xy + y = x^{3} + x^{2} + 3x − 5 has a rational subgroup of order 5. Changing variables to the model Y^{2} = X^{3} + 5X^{2} + 56X − 304, one computes that the X- coordinates of the points of order 3 are the roots of (X^{2} + 12X + 192)(3X^{2} − 16X − 48). Thus E gives rise to two of the non-cusp rational points of X_{0,split}(5, 3), corresponding to the two choices of G_{Q}-stable pairs of subgroups of order 3. (These two points are distinct because Aut(E) = ±1.) One computes easily that j(E) = 11^{3}/2^{3}. The quotient of E by its rational subgroup of order 5 (denoted 338E2 in [2]) gives rise to the other two non-cusp points of X_{0,split}(5, 3)(Q) and has j-invariant −29^{3}41^{3}/2^{15}.

2.3. Twists of modular curves. Suppose p > 2 is a prime. As in [14] §1 or [7] §1, there is an open modular curve Y_{p} over Q whose points correspond to isomorphism classes of pairs (E, ϕ) where E is an elliptic curve and ϕ : E[p] → F_{p} × μ_{p} is an isomorphism such that det(ϕ) : ∧^{2}E[p] → μ_{p} is the Weil pairing. Define X_{p} to be the compactification of Y_{p}. When p = 3 or 5 (which are the only cases we will use) X_{p} has genus zero, and explicit models are given in [7]. Over Q(μ_{p}), a choice of primitive p-th root of unity induces an isomorphism from X_{p} to one component of the usual modular curve X(p).

More generally, suppose that V is a 2-dimensional F_{p}-vector space with an action of G_{Q} and η : ∧^{2}V ⟶^{∼} μ_{p} is a non-degenerate alternating pairing. As explained in [7] Remark 2.4 or [14] §2, there is an open curve Y_{V} defined over Q whose points parametrize isomorphism classes of pairs (E, ϕ) where E is an elliptic curve and ϕ : E[5] ⟶^{∼} V is an isomorphism such that η ∘ det(ϕ) : ∧^{2}E[5] → ∧^{2}V → μ_{p} is the Weil pairing. Define X_{V} to be the compactification of Y_{V}. In particular X_{p} = X_{F_{p}×μ_{p}}.

Now suppose p = 5, E is an elliptic curve, V = E[5], and η is the Weil pairing. We will write Y_{E} and X_{E} for Y_{V} and X_{V}, and we have the following explicit result ([7] §5).

Proposition 11. Suppose E is an elliptic curve over Q and p = 5. There is an isomorphism ψ : P^{1} ⟶^{∼} X_{E} defined over Q and polynomials f_{E}(t), g_{E}(t) ∈ Q[t] of degrees 20 and 30, respectively, such that if t ∈ Q is not in the finite set ψ^{−1}(X_{E}(Q) − Y_{E}(Q)) then ψ(t) ∈ Y_{E}(Q) is represented by the elliptic curve

E_{t} : y^{2} = x^{3} + f_{E}(t)x + g_{E}(t).

In particular for every such t, ρ̄_{E_{t},5} ≅ ρ̄_{E,5}.

Similarly there are open curves Y′_{E} and Y′′_{E} whose parametrize isomorphism classes of triples (E′, ϕ, C_{3}) and (E′, ϕ, {C_{3}, C′_{3}}), respectively, where

• E′ is an elliptic curve,

• ϕ : E′[5] ⟶^{∼} E[5] is an isomorphism taking the Weil pairing on E′[5] to that on E[5],

• C_{3} and C′_{3} are (distinct) subgroups of order 3 of E′[3]. These curves come equipped with natural (forgetful) maps to Y_{E}.

Lemma 12. With notation as above, Y′_{E}(Q) and Y′′_{E}(Q) are finite.

Proof. Let X′_{E} and X′′_{E} denote the compactifications of Y′_{E} and Y′′_{E}, respectively. The classical theory of modular curves shows that X′_{E}(C) = Ĥ/(Γ(5) ∩ Γ_{0}(3)) and X′′_{E}(C) = Ĥ/(Γ(5) ∩ Γ_{split}(3)). Thus by Lemma 8, X′_{E} and X′′_{E} are both curves of genus greater than 1, and the lemma follows from Faltings’ Theorem (the Mordell Conjecture).

3. Proof of Theorem 1

Proposition 13. Suppose E is an elliptic curve over Q such that • ρ̄_{E,3} restricted to G_{Q(\sqrt{−3})} is not absolutely irreducible,

• ρ̄_{E,5} is reducible. Then j(E) ∈ {−25/2, −5^{2}241^{3}/2^{3}, −5·29^{3}/2^{5}, 5·211^{3}/2^{15}, 11^{3}/2^{3}, −29^{3}41^{3}/2^{15}}.

Proof. Suppose C_{5} is a nontrivial subgroup of E[5] stable under G_{Q}.

Case I: ρ̄_{E,3} is reducible. In this case E[3] has a proper G_{Q}-stable subgroup C_{3}, and then (E, C_{3} + C_{5}) represents a rational point of X_{0}(15). By Lemma 9(iii), j(E) ∈ {−25/2, −5^{2}241^{3}/2^{3}, −5·29^{3}/2^{5}, 5·211^{3}/2^{15}}.

Case II: ρ̄_{E,3} is irreducible. By Lemma 6, E[3] has two distinct subgroups C_{3}, C′_{3} of order 3 such that (E, C_{5}, {C_{3}, C′_{3}}) represents a rational point of X_{0,split}(5, 3). By Lemma 10(iii), j(E) ∈ {11^{3}/2^{3}, −29^{3}41^{3}/2^{15}}.

Proof of Theorem 1. Theorem 1 is now immediate from Propositions 7 and 13 and Lemma 9(iv).

4. Proof of Theorem 2

Fix an elliptic curve E over Q which is semistable at 3 and such that ρ̄_{E,5} is irreducible. We want to show that ρ̄_{E,5} is modular.

Let the notation be as in §2.3 with p = 5. Define a subset B of X_{E}(Q) by

B = (X_{E}(Q) − Y_{E}(Q)) ∪ image(Y′_{E}(Q) → Y_{E}(Q)) ∪ image(Y′′_{E}(Q) → Y_{E}(Q))

and let T = ψ^{−1}(B) ⊂ P^{1}(Q). By Lemma 12, T is a finite set.

Suppose t ∈ Q − T, and let E′ be the elliptic curve y^{2} = x^{3} + f_{E}(t)x + g_{E}(t) from Proposition 11. Since ψ(t) is not the image of a rational point of Y′_{E} or of Y′′_{E}, Proposition 6 shows that ρ̄_{E′,3} restricted to G_{Q(\sqrt{−3})} is absolutely irreducible. Further, since E is semistable at 3, Proposition 7.1 of [14] shows that E′ is semistable at 3 as well. Therefore by Theorems B and C applied with p = 3, E′ is modular, and so ρ̄_{E′,5} is modular. By Proposition 11, ρ̄_{E′,5} ≅ ρ̄_{E,5}, so ρ̄_{E,5} is modular.

5. Proof of Theorem 3

Fix a prime p > 2 and a representation ρ : G_{Q} → GL_{2}(F_{p}) such that det(ρ) is the cyclotomic character χ_{p}. Equivalently, fix a 2-dimensional F_{p}-vector space V (ρ) with a G_{Q}-action such that ∧^{2}V (ρ) is isomorphic to μ_{p} as a G_{Q}-module.

Let V_{p} = Z/pZ × μ_{p}. There is a canonical identification ∧^{2}V_{p} ≅ μ_{p}. Define

SL(V_{p}) = {ψ ∈ Aut(V_{p}) : det(ψ) = 1 on ∧^{2}V_{p}}.

Fix an F_{p}-vector space isomorphism φ : V_{p} ⟶^{∼} V (ρ).

Lemma 14. The map σ ↦ φ^{−1} ∘φ^{σ} is a one-cocycle on G_{Q} with values in SL(V_{p}). The class c ∈ H^{1}(Q, SL(V_{p})) of this cocycle depends only on the isomorphism class of ρ, and not on the choice of φ.

Proof. The fact that φ^{−1} ∘ φ^{σ} ∈ SL(V_{p}) follows from the assumption on det(ρ). The other assertions of the lemma are all easy computations. (Note that we are using non-commutative Galois cohomology [8].)

Let X_{p} be the modular curve defined in §2.3. Let E_{p} be the universal elliptic curve over X_{p} (see for example [11] or [13]), and ι : X_{p} → E_{p} the zero section. Finally, define the line bundle L_{p} to be the pull-back along ι of the cotangent bundle of E_{p} over X_{p}.

Define Aut_{0}(E_{p}) to be the isomorphisms from E_{p} to itself which map the image of the zero section ι(X_{p}) to itself. There are natural G_{Q}-equivariant maps

SL(V_{p}) → Aut_{0}(E_{p}) → Aut(L_{p}) → Aut(X_{p})

and we apply these maps to the class c of Lemma 14

H^{1}(Q, SL(V_{p})) → H^{1}(Q, AutQ̄(L_{p})) → H^{1}(Q, AutQ̄(X_{p}))
c  ↦  c_{L}  ↦  c_{X}. (1)

Define η_{ρ} = det(φ^{−1}) : ∧^{2}V (ρ) → μ_{p}. Let X(ρ) denote the modular curve X_{V (ρ)} of §2.3. Then X(ρ) contains an open curve Y (ρ) whose points correspond to isomorphism classes of pairs (E, ϕ) where E is an elliptic curve and ϕ : E[p] ⟶^{∼} V (ρ) is an isomorphism such that η_{ρ}∘det(ϕ) : ∧^{2}E[p] → ∧^{2}V (ρ) → μ_{p} is the Weil pairing. In particular (see for example Proposition 2.1 of [7]) a point of Y (ρ)(Q) gives an elliptic curve E over Q such that ρ̄_{E,p} ≅ ρ.

As explained in [14] §2 and [7] Remark 2.4, X(ρ) is the twist of X_{p} by c_{X} in the sense of [8]. Now suppose p = 3 or 5, so X_{p} ≅ P^{1} over Q. To prove Theorem 4, we need only show that Y (ρ)(Q) is nonempty. We will do this by showing that c_{X} = 0, so X(ρ) ≅ X_{p} ≅ P^{1} over Q. For a related discussion see [10].

Lemma 15. Suppose p = 3 or 5, and fix an isomorphism X_{p} ≅ P^{1} over Q.

(i) AutQ̄(X_{p}) ≅ PGL_{2}(Q̄) ≅ PSL_{2}(Q̄). (ii) AutQ̄(L_{p}) ≅ GL_{2}(Q̄)/μ_{m}, where

m = \begin{cases}1 & if p = 3 \\ 5 & if p = 5 \end{cases}

and μ_{m} lies inside GL_{2}(Q̄) as scalars.

Proof. Assertion (i) is well-known.

By [4] §12.1, the degree of the line bundle L_{p} is #(PSL_{2}(F_{p}))/12 = m. Since X_{p} ≅ P^{1}, L_{p} is determined by its degree so we can identify H^{0}(X_{p}, L_{p}), the space of global sections of L_{p}, with the space of homogeneous polynomials of degree m in two variables. Further L_{p} is generated by its global sections, so this identification gives

Aut(L_{p}) ≅ Aut(H^{0}(X_{p}, L_{p})) ≅ GL_{2}/μ_{m}

where ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2} sends a polynomial f(x, y) to f(ax + by, cx + dy).

Proof of Theorem 3. To simplify notation we give the proof in the case that F = Q. The case of an arbitrary field of characteristic different from p is identical.

Let p = 3 or 5 and m = 1 or 5, respectively, as in Lemma 15(ii). Let ρ : G_{Q} → GL_{2}(F_{p}) be a representation whose determinant is χ_{p} and let c_{X} and c_{L} be as above. By Lemma 15, there are exact sequences

0 → ±1 → SL_{2}(Q̄) → AutQ̄(X_{p}) → 0
0 → μ_{m} → GL_{2}(Q̄) → AutQ̄(L_{p}) → 0.

Since H^{1}(Q, GL_{2}(Q̄)) = H^{1}(Q, SL_{2}(Q̄)) = 0 (see [8] III Lemme 1), the induced long exact sequences from Galois cohomology show that c_{X} ∈ H^{1}(Q, AutQ̄(X_{p})) is killed by 2 and c_{L} ∈ H^{1}(Q, AutQ̄(L_{p})) is killed by m. But m is odd, and c_{X} is the image of c_{L} under (1), so c_{X} = 0. Therefore X(ρ) is isomorphic over Q to P^{1}, so Y (ρ)(Q) is infinite and there are (infinitely many) elliptic curves E over Q with ρ̄_{E,p} isomorphic to ρ.

Remark . The above proof of Theorem 3 relies heavily on the fact that X_{p} has genus zero, which is true for p = 3 and 5 but false for p > 5. However, one can replace p by 4 and define X_{4} corresponding to V_{4} = Z/4Z × μ_{4} (see [14]). Then X_{4} has genus zero, and one can define E_{4} and L_{4} and apply the method of the proof above. But the corresponding integer m of Lemma 15(ii) is #(PSL_{2}(Z/4Z))/12 = 2, so we cannot conclude that c_{X} = 0.

In fact, it is not true that every representation ρ : G_{Q} → GL_{2}(Z/4Z) with determiant equal to the mod 4 cyclotomic character is isomorphic to the representation of G_{Q} on E[4] for some elliptic curve E over Q. For example, let ρ be the nontrivial representation of G_{Q} into \{1, ( \begin{matrix}1 & 2 \\ 2 & 3 \end{matrix} )\} ⊂ GL_{2}(Z/4Z) factoring through Gal(Q(i)/Q). Then det(ρ) is the mod 4 cyclotomic character, but if ρ came from an elliptic curve E we would have E(R)[4] ≅ Z/2Z × Z/2Z which is impossible.

References

[1] B. Birch, W. Kuyk, eds., Modular functions of one variable IV. Lecture Notes in Math. 476 Springer-Verlag, Berlin-New York (1975) 74–144.

[2] J. Cremona, Algorithms for modular elliptic curves, Cambridge Univ. Press, Cambridge (1992).

[3] F. Diamond, On deformation rings and Hecke rings. To appear in Annals of Math.

[4] F. Diamond, J. Im, Modular forms and modular curves. In: Seminar on Fermat’s Last Theorem. Canadian Math. Soc. Conf. Proc. 17 (1995) 39-133.

[5] P. Deligne, M. Rapoport, Les schémas de modules de courbes elliptiques. In: Modular functions of one variable II. Lecture Notes in Math. 349 (1973) 143–316.

[6] R. Langlands, Base change for GL(2). Annals of Math. Studies 96, Princeton University Press, Princeton (1980).

[7] K. Rubin, A. Silverberg, Families of elliptic curves with constant mod p representations. In: Elliptic curves, modular forms, and Fermat’s Last Theorem (Hong Kong, December 1994), Coates and Yau, eds. International Press, Cambridge (1995) 148–161.

[8] J-P. Serre, Cohomologie galoisienne. Lecture Notes in Math. 5, Springer-Verlag, Berlin-New York (1964).

[9] , Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent math. 15 (1972) 259–331.

[10] , Extensions icosaédriques (Letter to J.D. Gray). In: Collected Papers Vol. III, Springer-Verlag, Berlin-New York (1986) 550–554.

[11] G. Shimura, Moduli and fibre systems of abelian varieties. Annals of Math 83 (1966) 294–338.

[12] , Introduction to the arithmetic theory of automorphic functions. Princeton Univ. Press, Princeton (1971).

[13] T. Shioda, On elliptic modular surfaces. J. Math. Soc. Japan 24 (1972) 20–59.

[14] A. Silverberg, Explicit families of elliptic curves with prescribed mod N representations. This volume.

[15] J. Silverman, The arithmetic of elliptic curves, Graduate Texts in Math. 106, Springer-Verlag, Berlin-New York (1986).

[16] R. Taylor, A. Wiles, Ring theoretic properties of certain Hecke algebras. Annals of Math. 141 (1995) 553– 572.

[17] J. Tunnell, Artin’s conjecture for representations of octahedral type. Bull. A.M.S. 5 (1981) 173–175.

[18] A. Wiles, Modular elliptic curves and Fermat’s Last Theorem. Annals of Math. 141 (1995) 443–551.

Department of Mathematics, Ohio State University, Columbus, Ohio 43210, USA

E-mail address: rubin@math.ohio-state.edu