1 Exploration of a mod pq variant of Serre’s conjecture

Consider a continuous odd Galois representation

ρ : G_{Q} → GL_{2}(Z/pqZ) ≅ GL_{2}(Z/pZ) × GL_{2}(Z/qZ)  (1)

that is irreducible when reduced modulo p and modulo q. We say that ρ is modular if there exists a newform f such that ρ ∼ ρ_{f,λ_{p}} × ρ_{f,λ_{q}}, with λ_{p} and λ_{q} primes lying over p and q, respectively. Is every mod pq representation modular? When ρ is modular, what is the minimal level N(ρ) and weight k(ρ) of a newform f giving rise to ρ? In this section we report on a first investigation, conducted jointly with B. Mazur, into this question.

1.1 Mazur’s letter

In 1998, the author received the following letter from B. Mazur.

Dear William, To put perspective on the subject of this e-mail, let me begin with an utterly untestable version of what I was suggesting to you. Let p and q be distinct prime numbers and G the Galois group of the maximal algebraic extension of Q unramified outside a finite set S of primes. Suppose that we have an odd continuous representation ρ : G → GL_{2}(Z/pqZ) which is irreducible when reduced mod p and mod q. Is ρ “modular?” To begin to give some shape to the above question, we might as well assume the “classical” Serre conjecture for single primes, so we can start out by giving ourselves two modular newforms f and g and then look for a modular newform h that is congruent mod p to f and mod q to g—under the assumption that the associated Galois representations mod p and mod q respectively are irreducible. I should be more careful about what I mean “mod p” and “mod q”: I really mean to take p and q prime ideals in the ring generated by the Fourier coefficients of f and g, and ditto with regard to h. To simplify one’s life, one can “start out” by working only with pairs f and g of newforms whose ring of Fourier coefficients is Z (i.e., the f and g in question correspond to elliptic curves). But even if one does this, one has to face the possibility (the probability, in fact) that h, if it exists at all, no longer has its ring of Fourier coefficients equal to Z. Of course, phrased this way, our question is still relatively unfalsifiable, so we have better be more specific. Ken [Ribet] and I thought a bit about how to shape a more specific testable question, and here is what we came up with—very tentatively still. Suppose that N and M are distinct prime numbers, and f and g are newforms of weight two on X_{0}(N) and X_{0}(M) respectively. Now let p be a prime number (or ideal) dividing either 1 + M − a_{M}(f) or 1 + M + a_{M}(f), and let q be a prime number (or ideal) dividing either 1+N −a_{N}(g) or 1+N +a_{N}(g). Let us suppose that the residual characteristics of the ideals p and q are distinct (i.e., if we are in a case where p and q are prime numbers, we just want them to be distinct). Suppose also that the corresponding Galois representations associated to f mod p and g mod q are irreducible. Let us say that we have maximum success if for any such choice of f, g, p, q there is a modular newform h of weight two on X_{0}(NM) which is congruent to f mod p and to g mod q. Here a_{N}(f) means the Nth Fourier coefficient of f, etc.

The reason why we call the above maximum success is that we are putting down the “evident” conditions necessary for there to be such a newform h (and looking for such an h at the minimal conceivable level, and weight). It is probably unreasonable to expect maximum success, but initial experiments might give us a sense of how far from the mark maximal success actually is, in which case we will weaken our expectations accordingly... I should also say, that the first case is N = 11 and M = 17, where the level of the sought for h is 187, i.e., not too large yet...

Barry

1.2 Congruence terminology

In order to work entirely over the integers and determine congruences exactly, we use the following modified notion of congruence. Let N be a positive integer and consider distinct weight 2 newforms f and g whose levels N_{f} and N_{g} both divide N. Denote by Ã_{f} the abelian variety generated by the images of A_{f} in J_{0}(N) under the degeneracy maps, and similarly define Ã_{g}. We say that f and g are congruent at level N modulo a rational prime p, written f ≡ g (mod p), if p divides the order of the finite group Ã_{f} ∩ Ã_{g}. Deciding whether or not f and g are congruent can be accomplished by working in the integral homology H_{1}(X_{0}(N), Z) using the algorithm of Section ??.

1.3 The first example: N = 11, M = 17

Consider the first example, in which N = 11 and M = 17. The spaces S_{2}(Γ_{0}(11)) and S_{2}(Γ_{0}(17)) are spanned by newforms f and g, respectively, where

f = q − 2q^{2} − q^{3} + 2q^{4} + q^{5} + 2q^{6} − 2q^{7} − 2q^{9} − 2q^{10} + q^{11}
−2q^{12} + 4q^{13} + 4q^{14} − q^{15} − 4q^{16} − 2q^{17} + 4q^{18}
+2q^{20} + 2q^{21} − 2q^{22} − q^{23} + ⋯,
g = q − q^{2} − q^{4} − 2q^{5} + 4q^{7} + 3q^{8} − 3q^{9} + 2q^{10} − 2q^{13} + · · · .

We have

(1 + M − a_{M}(f))(1 + M + a_{M}(f)) = 2^{6} · 5,
(1 + N − a_{N}(g))(1 + N + a_{N}(g)) = 2^{4} · 3^{2}.

To avoid reducibility we require that p | num((11 − 1)/12) = 5 and q | num((17 − 1)/12) = 4, so only the pair p = 2, q = 3 remains. The representation

ρ = ρ_{f,2} × ρ_{g,3} : G_{Q} → GL(2, Z/6Z)

is irreducible mod 2 and mod 3. Choose newforms h_{1}, h_{2}, …, h_{6} from each Galois conjugacy class in S_{2}(Γ_{0}(11 · 17)), ordered as in Section ??. We find that f ≡ h_{2}, h_{3}, h_{4}, h_{5}, h_{6} (mod 2) and g ≡ h_{1}, h_{4} (mod 3). Because h_{4} occurs in both of these lists, ρ is modular; ρ arises from

h_{4} = q + aq^{2} + (−a − 1)q^{3} − 2aq^{4} + (a − 1)q^{5} + (a − 2)q^{6} − 2q^{7} + ⋯

where a^{2} + 2a − 2 = 0. We have maximal success.

1.4 Two more examples

1.4.1 N = 11, M = 19:

A basis for S_{2}(Γ_{0}(19)) is

g = q − 2q^{3} − 2q^{4} + 3q^{5} − q^{7} + q^{9} + 3q^{11} + 4q^{12} − 4q^{13} + · · · .

We find that

(1 + M − a_{M}(f))(1 + M + a_{M}(f)) = 2^{4} · 5^{2},
(1 + N − a_{N}(g))(1 + N + a_{N}(g)) = 2^{5} · 5.

The only pair of primes for which ρ is irreducible is p = 2, q = 5. The four classes of newforms at level 11 · 19 are h_{1}, …, h_{4}. The relevant congruences are f ≡ h_{1}, h_{2}, h_{3}, h_{4} (mod 2) and g ≡ h_{4} (mod 5). Therefore ρ_{f,2} × ρ_{g,5} arises from the eigenform h_{4}; the coefficient a_{2} of h_{4} satisfies the irreducible polynomial x^{7} + x^{6} − 14x^{5} − 10x^{4} + 59x^{3} + 27x^{2} − 66x − 30. Again, we have maximal success.

1.4.2 N = 11, M = 23:

We have

g = q + aq^{2} + (−2a − 1)q^{3} + (−a − 1)q^{4} + 2aq^{5} + (a − 2)q^{6} + (2a + 2)q^{7} +

(−2a − 1)q^{8} + 2q^{9} + (−2a + 2)q^{10} + (−2a − 4)q^{11} + (a + 3)q^{12} + 3q^{13} + ⋯,

with a^{2} + a − 1 = 0; and

(1 + M − a_{M}(f))(1 + M + a_{M}(f)) = 5^{2} · 23,
Norm((1 + N − a_{N}(g))(1 + N + a_{N}(g))) = 2^{4} · 5 · 11 · 19

We must take p = 23; for q take each of 2, 5, 19. The relevant congruences are f ≡ h_{4} (mod 23), and g ≡ h_{3}, h_{4} (mod 2), g ≡ h_{2} (mod 5), g ≡ h_{1} (mod 19). We do not have maximal success; only ρ_{f,23} × ρ_{g,2} arises from S_{2}(Γ_{0}(11 · 23)).

1.5 Is every ρ modular?

As in Mazur’s letter (Section 1.1), consider f, g, p, q with f, g of level N, M, respectively. Suppose that there is no form h at level NM such that h ≡ f (mod p) and h ≡ g (mod q) (congruence as in Section 1.2). Our choice of p and q implies that there is h_{1} and h_{2} at level NM such that h_{1} ≡ f (mod p) and h_{2} ≡ g (mod q). Using the Cebotarev Density Theorem (see e.g., [Lan94, VIII.4]) and Ribet’s level raising theorem [Rib90] we see that there are infinitely many primes ℓ for which there exists newforms b_{1} and b_{2} at level NMℓ such that b_{1} ≡ h_{1} (mod p) and b_{2} ≡ h_{2} (mod q). Does b_{1} = b_{2} in one of these infinitely many cases? More generally, begin with two newforms f and g both of some level N and consider the infinitely many primes ℓ such that there exists h_{1} and h_{2} of level Nℓ with h_{1} ≡ f (mod p) and h_{2} ≡ g (mod q). Must there always be an ℓ so that we can take h_{1} = h_{2}? If yes, then every mod pq representation is modular; at least, if we admit the classical Serre conjectures.

Answers to the above questions may be viewed as a generalization of Ribet’s level raising machinery [Rib90]. The introduction of two primes at once obscures the algebraic interpretation in terms of ideals of the Hecke algebra; making it appear necessary to find new arguments. For this reason, the author suspects that there are non-modular mod pq representations. Even if not all mod pq representations are modular, the question remains of finding the minimal weight and level of those which are.

1.6 Serre mod pq table

In Table 1.6 we tabulate computations that address Mazur’s letter (see Section 1.1). We analyzed each pair f, g of newforms of levels N, M, with NM ≤ 671. Each pair of rows in the table corresponds to a pair of forms f, g. Let h_{1}, …, h_{n} be representatives for the Galois conjugacy classes of newforms at level NM. The “Congruences” column is a block of columns that contain, in order, the primes p, q such that f ≡ h_{i} (mod p) and g ≡ h_{i} (mod q). The “Irreducible nonsuccess” column lists the pairs p, q, with p and q both non-Eisenstein primes listed in column two, such that ρ_{f,p} × ρ_{g,q} does not arise from a modular form of level NM. The third column is empty exactly when we have maximal success; this occurs for only 4 of the 23 pairs f, g. The table gives only a small part of the full computation, which covers all pairs f, g such that NM < 2000. [[A summary of computation up to level 2000 will go here.]]

Table 1: Serre mod pq data for NM ≤ 671

f, g  Congruences  Irreducible nonsuccess

11A 1 2 2 2 2 2 17A 3 1 1 3 1 2

\begin{matrix}11A & 2 & 2 & 2·5 & 2 \\ 19A & 3 & 1 & 1 & 3·5 \end{matrix}

11A 1 5 1 23 23A 19 5 2 2  23 23 5 19
\begin{matrix}11A & 2 & 3 & 2 & 3·5 & 1 \\ 29A & 1 & 17 & 1 & 1 & 167 \end{matrix}  2 2  3 17 167 167
\begin{matrix}11A & 5 & 1 & 1 & 3·13 \\ 31A & 5 & 2 & 7 & 2 \end{matrix}  3 13 7 7
11A 1 7 1 41 37A 7 1 1 17  7 7 41 7 17 7
\begin{matrix}11A & 1 & 7 & 1 & 41 \\ 37B & 1 & 3 & 3·5 & 1 \end{matrix}  7 41 5 5
\begin{matrix}11A & 2 & 1 & 5 & 17 & 2 \\ 41A & 1 & 1 & 5 & 1 & 2·863 \end{matrix}  {17 \atop 863}
11A 2 1 5 1 1 19 2 43A 3 5 1 1 1 3 1  2 19 5 5
11A 2 1 5 1 1 19 2 43B 1 1 1 1 23 1 113  2 19 19 23 23 113
11A 2 2 2 2 1 7 2 1 1 2 2 47A 3 1 3 31 7 1 3 1 3 2 2  2 7 7 7 7 7 2 3 7 31
\begin{matrix}11A & 2 & 3 & 2 & 2 & 2 & 1 & 2·3 & 2 & 2 \\ 53A & 1 & 2 & 3 & 1 & 3 & 2 & 1 & 1 & 2 \end{matrix}  3 3
\begin{matrix}11A & 2 & 3 & 2 & 2 & 2 & 1 & 2·3 & 2 & 2 \\ 53B & 1 & 2 & 1 & 1 & 1 & 2 & 43 & 281 & 2 \end{matrix}  {3 \atop 281}
17A 1 2 3 2 2 2 19A 5 1 7 1 3 1  3 5
\begin{matrix}17A & 1 & 1 & 5 & 1 & 2·7 \\ 23A & 2 & 1 & 5 & 109 & 2 \end{matrix}  5 5  7 7 2 109 5 109
17A 3 3 1 1 1 2 2 2 29A 1 2 2 2 7 1 31 2  {3 \atop 31}
\begin{matrix}17A & 1 & 7 & 1 & 1 & 2·3 \\ 31A & 1 & 11 & 2 & 109 & 2 \end{matrix} 3 3 7 7 11 109 2 109

\begin{matrix}19A & 2 & 2 & 1 & 1 & 2 & 2 & 2 & 2·3 \\ 23A & 1 & 1 & 1 & 11 & 1 & 2 & 3 & 2 \end{matrix}

\begin{matrix}19A & 1 & 1 & 2 & 2 & 1 & 2 & 2·3 & 2 \\ 29A & 2 & 2 & 7 & 1 & 2 & 1 & 2 & 2·13 \end{matrix}

\begin{matrix}19A & 2 & 7 & 2·3 & 2 & 2 \\ 31A & 1 & 1 & 79 & 1 & 5·79 \end{matrix}  {7 \atop 79}
\begin{matrix}11A & 1 & 1 & 11 & 1 & 1 & 5·13 \\ 59A & 2 & 7 & 19 & 2·2671 & 2 & 2 \end{matrix}  11 11 11  13 13 13 2 7 2671 7 19 2671
11A 5 2 37 2 61A 1 7 17 1  2 37 17 7
\begin{matrix}11A & 5 & 2 & 37 & 2 \\ 61B & 5·17 & 1 & 1 & 13·331 \end{matrix}  2 37 37 37 17 13 17 331

References

[Lan94] S. Lang, Algebraic number theory, second ed., Springer-Verlag, New York, 1994.

[Rib90] K. A. Ribet, Raising the levels of modular representations, Séminaire de Théorie des Nombres, Paris 1987–88, Birkhäuser Boston, Boston, MA, 1990, pp. 259–271.