quadratic lattices
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2019 ◽  
Vol 374 (1-2) ◽  
pp. 323-329
Author(s):  
Rainer Schulze-Pillot
Keyword(s):  

2019 ◽  
Vol 15 (02) ◽  
pp. 309-325 ◽  
Author(s):  
Markus Kirschmer ◽  
Gabriele Nebe

We relate proper isometry classes of maximal lattices in a totally definite quaternary quadratic space [Formula: see text] with trivial discriminant to certain equivalence classes of ideals in the quaternion algebra representing the Clifford invariant of [Formula: see text]. This yields a good algorithm to enumerate a system of representatives of proper isometry classes of lattices in genera of maximal lattices in [Formula: see text].


2017 ◽  
Vol 23 (6) ◽  
pp. 983-1002 ◽  
Author(s):  
Marlyse Njinkeu Sandjon ◽  
Amílcar Branquinho ◽  
Mama Foupouagnigni ◽  
Iván Area

2016 ◽  
Vol 13 (06) ◽  
pp. 1611-1616
Author(s):  
Ruiqing Wang

In this paper, we prove that there exist indecomposable lattices of ranks 5 and 6 over a Hasse domain of any global function field in which [Formula: see text] is not a square, which solves a problem proposed by Gerstein.


2014 ◽  
Vol 151 (5) ◽  
pp. 793-827 ◽  
Author(s):  
Sungmun Cho

The celebrated Smith–Minkowski–Siegel mass formula expresses the mass of a quadratic lattice $(L,Q)$ as a product of local factors, called the local densities of $(L,Q)$. This mass formula is an essential tool for the classification of integral quadratic lattices. In this paper, we will describe the local density formula explicitly by observing the existence of a smooth affine group scheme $\underline{G}$ over $\mathbb{Z}_{2}$ with generic fiber $\text{Aut}_{\mathbb{Q}_{2}}(L,Q)$, which satisfies $\underline{G}(\mathbb{Z}_{2})=\text{Aut}_{\mathbb{Z}_{2}}(L,Q)$. Our method works for any unramified finite extension of $\mathbb{Q}_{2}$. Therefore, we give a long awaited proof for the local density formula of Conway and Sloane and discover its generalization to unramified finite extensions of $\mathbb{Q}_{2}$. As an example, we give the mass formula for the integral quadratic form $Q_{n}(x_{1},\dots ,x_{n})=x_{1}^{2}+\cdots +x_{n}^{2}$ associated to a number field $k$ which is totally real and such that the ideal $(2)$ is unramified over $k$.


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