Primitive ternary quadratic lattices (local cases)
Let L be a primitive ternary quadratic lattice over Zp. If D=prD0 is the discriminant of L with r=ordp(D), then we may assume that D0=1 by scaling of L with D0. Thus we assume that D=pr in the following.
1. Non-dyadic case.
w : a non-square unit, u,v = 1 or w, H = [1] + [-1], K = [1] + [-w] (orthogonal splittings)
(1) Unimodular case (r = 0)
| L | Remark |
| [1] + [1] + [1] | - |
(2) Non-unimodular case (r > 0)
| L | Remark |
| H + [-pr] | r > 0 |
| K + [-pr·w] | r > 0 |
| [-1] + psH | r = 2s > 0 |
| [-w] + psK | r = 2s > 0 |
| [u] + [ps·v] + [pt·uv] | 0 < s < t, r = s + t |
2. Dyadic case.
u,v,w = 1,3,5,7, H = hyperbolic (2xy), K = non-hyperbolic (2x^2+2xy+2y^2)
(1) Even Case (r > 0)
| L | Remark |
| H + [-2r] | r > 1 |
| K + [-2r·5] | r > 1 |
| H + [-2] = K + [-2·5] | r = 1 |
(2) Odd (non-diagonal) Case (r > 0)
| L | Remark |
| [-1] + 2sH | r = 2s > 2 |
| [-5] + 2sK | r = 2s > 2 |
| [-1] + 2H = [-5] + 2K | r = 2 |
(3) Odd (mixed) Case (r = 0)
| L |
| [1] + [1] + [1] = [1] + [5] + [5] = [3] + K |
| [1] + [3] + [3] = [1] + [7] + [7] = [3] + [5] + [7] = [7] + H |
(4) Odd (diagonal) Case (r > 0)
-1- s = 0, t = 1 : L = [u] + [v] + [2·w], L# = [w] + [2·v] + [2·u] (uvw = 1)
| uvw |
| 111 = 551 = 133 = 313 = 573 = 753 = 375 = 735 |
| 177 = 717 = 155 = 515 = 357 = 537 = 331 = 771 |
-2- s = 0, t = 2 : L = [u] + [v] + [22·w], L#=[w] + [22·v] + [22·u] (uvw = 1)
| uvw |
| 111 = 155 = 515 = 551 |
| 331 = 771 = 375 = 735 |
| 133 = 313 = 177 = 717 = 357 = 537 = 573 = 753 |
-3- s =0, t > 2 : L = [u] + [v] + [2t·w], L# = [w] + [2t·v] + [2t·u] (uvw = 1)
| uvw |
| 111 = 551 |
| 155 = 515 |
| 331 = 771 |
| 375 = 735 |
| 133 = 313 = 573 = 753 |
| 177 = 717 = 357 = 537 |
-4- s = 1, t = 1 : L = [u] + [2·v] + [22·w], L# = [w] + [2·v] + [22·u] (uvw = 1)
| uvw |
| 111 = 331 = 515 = 735 = 133 = 357 = 537 = 753 |
| 177 = 717 = 573 = 313 = 155 = 375 = 551 = 771 |
-5- s = 1, t = 2 : L = [u] + [2·v] + [23·w], L# = [w] + [22·v] + [23·u] (uvw = 1)
| uvw |
| 111 = 155 = 331 = 375 |
| 313 = 357 = 537 = 573 |
| 177 = 717 = 133 = 753 |
| 771 = 735 = 551 = 515 |
-6- s = 1, t > 2 : L = [u] + [2·v] + [2t+1·w], L# = [w] + [2t·v] + [2t+1·u] (uvw = 1)
| uvw |
| 111 = 331 |
| 155 = 375 |
| 133 = 753 |
| 177 = 717 |
| 357 = 537 |
| 313 = 573 |
| 515 = 735 |
| 551 = 771 |
-7- s = 2, t = 2 : L = [u] + [22·v] + [24·w], L# = [w] + [22·v] + [24·u] (uvw = 1)
| uvw |
| 111 = 155 = 515 = 551 |
| 133 = 177 = 537 = 573 |
| 313 = 357 = 717 = 753 |
| 331 = 375 = 735 = 771 |
-8- s = 2, t > 2 : L = [u] + [22·v] + [2t+2·w], L# = [w] + [2t·v] + [2t+2·u] (uvw = 1)
| uvw |
| 111 = 551 |
| 155 = 515 |
| 133 = 573 |
| 177 = 537 |
| 313 = 753 |
| 331 = 771 |
| 375 = 735 |
| 357 = 717 |
-9- s > 2, t > 2 : L = [u] + [2s·v] + [2s+t·w], L# = [w] + [2t·v] + [2s+t·u]
u,v = 1,3,5,7 (uvw = 1)