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