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| On these pages, we present most recent results of our joint research, directly from the lab. |
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Basic bibliography:
K. Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), 1213-1237 A. Derhem, Capitulation dans les extensions quadratiques de corps de nombres cubiques cycliques, Thèse de doctorat, Université Laval, Quebec, 1988 D. C. Mayer, Multiplicities of dihedral discriminants, Math. Comp. 58 (1992), 831-847 and S55-S58 |
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Web master's e-mail address:
contact@algebra.at |
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| Cubic discriminants with irregular conductors (2002/02/08) |
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Dan (02/02/03): On January 6th, 2002, I suggested to study systematically
series of imaginary quadratic discriminants d congruent -3(mod 9) with 3-rank 2.
The target of the investigation should be evidence for the
actual existence of the 4 cases of 3-defect configurations
for irregular prime conductors f = 3^2:
(i1) delta_3(3^2) = 2 and delta_3(3) = 1 (i2) delta_3(3^2) = 1 and delta_3(3) = 1 (i3) delta_3(3^2) = 1 and delta_3(3) = 0 (i4) delta_3(3^2) = 0 and delta_3(3) = 0 This sounds rather abstract, doesn't it. Therefore, I now characterize the cases with multiplicities, in a manner as it would appear in your CCF algorithm: we want to find quadratic discriminants d = -3(mod 9) for which there exist (i1) 4 cubics with D=d, 0 cubics with D=9d, 0 cubics with D=81d (i2) 4 cubics with D=d, 0 cubics with D=9d, 9 cubics with D=81d (i3) 4 cubics with D=d, 9 cubics with D=9d, 0 cubics with D=81d (i4) 4 cubics with D=d, 9 cubics with D=9d, 27 cubics with D=81d |
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Karim (02/02/05): All occur quite soon
[found via CCF the list of d < 10^9 with (r_3 = 2) + used Hasse lemma to check m(9d) and m(81d)], I'm listing minimal occurrences: (i1) 4 cubics with D=d, 0 cubics with D=9d, 0 cubics with D=81d d = -8751 (i2) 4 cubics with D=d, 0 cubics with D=9d, 9 cubics with D=81d d = -42591 (i3) 4 cubics with D=d, 9 cubics with D=9d, 0 cubics with D=81d d = -128451 (i4) 4 cubics with D=d, 9 cubics with D=9d, 27 cubics with D=81d d = -2069688 [this was only number 867 in list of discriminants with r_3 = 2] |
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Dan (02/02/08): Thank you very much for determining the minimal occurrences
of the four 3-defect configurations for d = -3(mod 9), d < 0, r_3 = 2. Two of these discriminants were already known to me since 1991: 8751 as the least |d| with delta_3(3) = 1 and therefore m_3(d,3) = 0, and also simply as the smallest d = -3(mod 9) with r_3 = 2 128451 as the least |d| with delta_3(3) = 0 and thus m_3(d,3) > 0 (exactly: m_3(d,3) = 9) I did neither know that delta_3(9) = 2 for d = -8751 nor that delta_3(9) = 1 for d = -128451. Completely new for me are: 42591 as the least |d| such that delta_3(3) = delta_3(9) = 1, whence m_3(d,9) = 9. This also implies that the 10 discriminants d = -20568,...,-41583 between -8751 and -42591 must be of the same type as d = -8751. 2069688 as the least |d| with delta_3(3) = delta_3(9) = 0 and hence m_3(d,3) = 9, m_3(d,9) = 27. A consequence is that all the discriminants with free prime conductor 3, d = -283908,-344667,...>-2069688, must be of the same type as d = -128451. Well, here we have the 4 basis discriminants (bd) ! Welcome, there you go ;-) However, now an arduous task arises: we must map the 3-admissible prime conductors q for the bd's to the lattice of subspaces of my vector space V of non-trivial generators of ideal cubes, i. e., determine the association q --> V_q to the 3-ring spaces mod q. (Compare "Positions of 3-ring spaces" in the numerical examples of my Vienna Congress presentation) This includes more information than simply to calculate, if delta_3(q) = 1 or = 0 ! Yesterday I have restored a small program which forms the low level layer of my DIFFQI algorithm and which enabled me to prepare the first few 3-admissible q's for the bd's: d = -8751 ... q = 3,3^2,43,47,59,67,71,73,79,83,89,103 d = -42591 ... q = 3,7,3^2,13,17,19,23,29,37,47,53,61,67 d = -128451 ... q = 2,3,3^2,11,29,53,61,67,71,89,103 d = -2069688 ... q = 3,5,7,3^2,11,13,17,19,43,59,61,67,71 The question is, assuming that CCF/CRF alone will not be able to solve these kinds of problems, if PARI extended with your implementation of Hasse's lemma can, in some way, compare the subgroups of index 3 associated with the individual q's. For example, taking d = -42591, which has nice small q's, q = 3, 7, 9, 13, 17, 19, ..., we know that delta_3(3) = delta_3(9) = 1, which certainly implies V_3 = V_9 < V_1 (using that 3 | 9 implies V_9 <= V_3). Further, I found in my data of 1991 that delta_3(7) = 1 and thus V_7 < V_1. The following question arises: is V_7 = V_3 or not? I could give an answer only with the aid of the high level layer of my DIFFQI algorithm. |
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