Unimodular lattices over real quadratic fields

Scharlau, Rudolf

doi:10.1007/bf02572333

Cited by 9 publications

(8 citation statements)

References 18 publications

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“…Further A and D stand for the root lattices of the corresponding type. The four one-class genera of unimodular lattices over totally real quadratic fields have been found by Scharlau in [Sch94]. In this paper, he gives explicit constructions for these lattices and the corresponding automorphism groups Aut(H 4 ) = (SL 2 (5) • SL 2 (5)) : 2 in the notation of [NP95], Aut(D 4 ( √ 2)) = Aut(D 4 ).2, Aut(D ∼ 4 ) and Aut((2A 2 ) ∼ ).…”

Section: Resultsmentioning

confidence: 89%

One-class genera of maximal integral quadratic forms

Kirschmer

2014

Journal of Number Theory

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Suppose Q is a definite quadratic form on a vector space V over some totally real field K = Q. Then the maximal integral ZK -lattices in (V, Q) are locally isometric everywhere and hence form a single genus. We enumerate all orthogonal spaces (V, Q) of dimension at least 3, where the corresponding genus of maximal integral lattices consists of a single isometry class. It turns out, there are 471 such genera. Moreover, the dimension of V and the degree of K are bounded by 6 and 5 respectively. This classification also yields all maximal quaternion orders of type number one.

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Section: Resultsmentioning

confidence: 89%

One-class genera of maximal integral quadratic forms

Kirschmer

2014

Journal of Number Theory

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“…In fact, extending A 4 and A * 4 to lattices over Z[τ ], compare [19], one sees that I is intermediate between them, but is still integral for tr(xȳ). When viewed as a Z[τ ]-module, I has class number 1, see [19,Thm. 3.4].…”

Section: Lemmamentioning

confidence: 97%

“…An alternative way to prove Proposition 1 would be to use Remark 2 and to show that all potential x ∈ I with x = x are elements of the dual of L (taken with respect to tr(xȳ)), but that the coset representative of A * 4 /A 4 cannot be chosen in I. In fact, extending A 4 and A * 4 to lattices over Z[τ ], compare [19], one sees that I is intermediate between them, but is still integral for tr(xȳ). When viewed as a Z[τ ]-module, I has class number 1, see [19,Thm.…”

Section: Lemmamentioning

confidence: 99%

Similar sublattices of the root lattice A4

2008

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Similar sublattices of the root lattice A 4 are possible [J.H. Conway, E.M. Rains, N.J.A. Sloane, On the existence of similar sublattices, Can. J. Math. 51 (1999) 1300-1306] for each index that is the square of a non-zero integer of the form m 2 +mn−n 2 .Here, we add a constructive approach, based on the arithmetic of the quaternion algebra H(Q( √ 5)) and the existence of a particular involution of the second kind, which also provides the actual sublattices and the number of different solutions for a given index. The corresponding Dirichlet series generating function is closely related to the zeta function of the icosian ring.

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“…[40, §16.6, pp133-134], [35, §5.16, pp33-34]) or small class number, however almost all actual computational results have been limited to either the case F = Q or to unimodular lattices over real quadratic fields. When [F : Q] > 1, the author is only aware of the papers [11,22,21,23,31].…”

Section: Neighbors and Generamentioning

confidence: 99%

Algorithms for computing maximal lattices in bilinear (and quadratic) spaces over number fields

Hanke¹

2013

Diophantine Methods, Lattices, and Arithmetic Theory of Quadratic Forms

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In this paper we describe an algorithm that quickly computes a maximal a-valued lattice in an F -vector space equipped with a non-degenerate bilinear form, where a is a fractional ideal in a number field F . We then apply this construction to give an algorithm to compute an a-maximal lattice in a quadratic space over any number field F where the prime p = 2 is unramified. We also develop the theory of p-neighbors for a-valued quadratic lattices at an arbitrary prime p of O F (including when p | 2) and prove its close connection to the residual geometry of certain quadrics mod p. Finally we give a well-known application of p-neighboring lattices and exact mass formulas to compute a complete set of representatives for the classes in a given genus of (totally definite) quadratic O F -lattices.

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Unimodular lattices over real quadratic fields

Cited by 9 publications

References 18 publications

One-class genera of maximal integral quadratic forms

One-class genera of maximal integral quadratic forms

Similar sublattices of the root lattice A4

Algorithms for computing maximal lattices in bilinear (and quadratic) spaces over number fields

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