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By Kubota T.

ISBN-10: 0470509201

ISBN-13: 9780470509203

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N: © 2012 by Taylor & Francis Group, LLC Introduction to Lattices 13 (1) There are vectors ym+1 , . . , yn ∈ L such that the following n vectors form a basis of L: x1 , x2 , . . , xm−1 , z, ym+1 , . . , yn . (2) The greatest common divisor of the integers am+1 , . . , an is 1. Proof. 17). Up to this point we have been considering “full-rank” sublattices; the dimension of the sublattice M is equal to the dimension of the lattice L. For the next definition and theorem we consider a more general situation.

1 for these points in R3 : A = (77, 9, 31), B = (20, −61, −48), C = (24, 65, 86). 5. 1 for these points in R4 : A = (4, −9, −2, −5), B = (1, 7, 8, −1), C = (−6, −8, −2, −2). 6. 1 for these points in R4 : A = (−62, −33, −68, −67), B = (42, 18, −59, 12), C = (52, −13, 82, 72). 7. Let C be an n × n matrix with integer entries and determinant ±1. Prove that C −1 also has integer entries. 8. Show that these three bases of R2 generate the same lattice. For each ordered pair of bases, find a sequence of unimodular row operations which converts from the first basis to the second: { x1 , x2 } = { y1 , y2 } = { z1 , z2 } = −41 −82 −79 −461 1 −99 , , 26080957 43756088 , −198 −1103 , , 3875510 6501953 .

1 The Euclidean algorithm Euclid(a, b) for the greatest common divisor Repeating this process, adding 1 to the subscripts at each step, we obtain 1419 = 1 · 726 + 693 726 = 1 · 693 + 33 693 = 21 · 33 + 0 r2 = q4 r3 + r4 r3 = q5 r4 + r5 q4 = 1 q5 = 1 r4 = 693 r5 = 33 r4 = q6 r5 + r6 q6 = 21 r6 = 0 We now have a remainder of 0, so the algorithm terminates. The last nonzero remainder, r5 = 33, is the greatest common divisor of a = 7854 and b = 2145. 2. Let n be the number of iterations of step (3) of the Euclidean algorithm.

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An elementary theory of Eisenstein series by Kubota T.


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