Serre's conjecture mod pq (new idea!?)
From:
William Stein
To:
mazur@math.harvard.edu
Date:
Tue, 28 May 2002 05:34:02 +0000
Dear Barry,
I had an idea that is relevant to our recurring investigations into an
analogue of Serre's conjecture for two-dimensional odd Galois
representations modulo pq. A few years ago, we used newforms f and g
of primes levels N and M to construct Galois representations rho
modulo p*q. You asked me if these have "Serre level NM" in the sense
that they arise from a single newform h in S_2(Gamma_0(NM)).
Frequently, the answer was no. It just occured to me that restricting
to weight 2 is very unnatural. We should have asked: is there a
newform h in S_k(Gamma_0(NM)) for some k such that h = f (mod p) and
h = g (mod q)?
Still starting with f and g of weight 2, I think the reasonable
possibilities for k are of the form
2 + a*LCM(p-1, q-1)
for integers a. In our examples, NM is already three digit size and
LCM(p-1,q-1) is often large, so it is very difficult to search for h
for any value of a except a=0. Instead, I just tried replacing f and
g by forms of the same weight k, but with k > 2. Here's an and
illustrative example of the sort of behaviour I've found in examples.
Let f be the unique newform in S_4(Gamma_0(5)) and g be the unique
newform in S_4(Gamma_0(7)). Let p = 31 and q = 7, so by Theorem A of
Diamond-Taylor there exists newforms h1 and h2 in S_4(Gamma_0(35))
such that h1 = f (mod 31) and h2 = g (mod 7). In fact, there are
exactly 3 newforms in S_4(Gamma_0(35)); the second is congruent to
f and the third to g and that's it. In particular, we DON'T find
a newform h in S_4(Gamma_0(35)) that gives rise to the Galois representation
attached to <(f,31),(g,7)>. However, if we consider S_{4+30}(Gamma_0(35)),
we find four newforms, and the first one h is congruent to both f modulo 31
and to g modulo 7.
CONJECTURE: Let f and g be newforms of prime levels Gamma_0(N) and
Gamma_0(M) of weight k. Suppose p, q > k+1 are primes such that
a_M(f)^2 = (M+1)^2 M^(k-2) (mod p), and
a_N(g)^2 = (N+1)^2 N^(k-2) (mod q).
Then there exists a newform h of level Gamma_0(MN) and weight
k + a*LCM(p-1,q-1), for some a, such that h = f (mod p) and h = g (mod q).
What is a?
Regards,
William