CoCalc Public Fileswww / tables / serremodpq / higher_weight_idea.txt
Author: William A. Stein
1Serre's conjecture mod pq (new idea!?)
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3
4
5From:
6William Stein <[email protected]>
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8
9To:
10[email protected]
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12
13Date:
14Tue, 28 May 2002 05:34:02 +0000
15Dear Barry,
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17
18I had an idea that is relevant to our recurring investigations into an
19analogue of Serre's conjecture for two-dimensional odd Galois
20representations modulo pq.  A few years ago, we used newforms f and g
21of primes levels N and M to construct Galois representations rho
22modulo p*q.  You asked me if these have "Serre level NM" in the sense
23that they arise from a single newform h in S_2(Gamma_0(NM)).
24Frequently, the answer was no.  It just occured to me that restricting
25to weight 2 is very unnatural.  We should have asked: is there a
26newform h in S_k(Gamma_0(NM)) for some k such that h = f (mod p) and
27h = g (mod q)?
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30Still starting with f and g of weight 2, I think the reasonable
31possibilities for k are of the form
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34                2  + a*LCM(p-1, q-1)
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37for integers a.  In our examples, NM is already three digit size and
38LCM(p-1,q-1) is often large, so it is very difficult to search for h
39for any value of a except a=0.  Instead, I just tried replacing f and
40g by forms of the same weight k, but with k > 2.  Here's an and
41illustrative example of the sort of behaviour I've found in examples.
42Let f be the unique newform in S_4(Gamma_0(5)) and g be the unique
43newform in S_4(Gamma_0(7)).  Let p = 31 and q = 7, so by Theorem A of
44Diamond-Taylor there exists newforms h1 and h2 in S_4(Gamma_0(35))
45such that h1 = f (mod 31) and h2 = g (mod 7).  In fact, there are
46exactly 3 newforms in S_4(Gamma_0(35)); the second is congruent to
47f and the third to g and that's it.  In particular, we DON'T find
48a newform h in S_4(Gamma_0(35)) that gives rise to the Galois representation
49attached to <(f,31),(g,7)>.  However, if we consider S_{4+30}(Gamma_0(35)),
50we find four newforms, and the first one h is congruent to both f modulo 31
51and to g modulo 7.
52
53
54CONJECTURE:  Let f and g be newforms of prime levels Gamma_0(N) and
55Gamma_0(M) of weight k.  Suppose p, q > k+1 are primes such that
56         a_M(f)^2 = (M+1)^2 M^(k-2) (mod p), and
57         a_N(g)^2 = (N+1)^2 N^(k-2) (mod q).
58Then there exists a newform h of level Gamma_0(MN) and weight
59k + a*LCM(p-1,q-1), for some a, such that h = f (mod p) and h = g (mod q).
60
61
62What is a?
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65Regards,
66   William
67