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 $\rho: G \rightarrow \mbox{\rm GL}_2(\mathbf{Z}/pq\mathbf{Z})$ which is irreducible when reduced mod p and mod q. Is $\rho$ ``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 pto 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 fand g are newforms of weight two on X₀(N) and X₀(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 qare 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 gmod 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 X₀(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