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Search: MSC category 11E12 ( Quadratic forms over global rings and fields )

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1. CMB 2009 (vol 52 pp. 63)

Dietmann, Rainer
Small Zeros of Quadratic Forms Avoiding a Finite Number of Prescribed Hyperplanes
We prove a new upper bound for the smallest zero $\mathbf{x}$ of a quadratic form over a number field with the additional restriction that $\mathbf{x}$ does not lie in a finite number of $m$ prescribed hyperplanes. Our bound is polynomial in the height of the quadratic form, with an exponent depending only on the number of variables but not on $m$.

Categories:11D09, 11E12, 11H46, 11H55

2. CMB 1997 (vol 40 pp. 402)

Carpenter, Jenna P.
On the Preservation of Root Numbers and the Behavior of Weil Characters Under Reciprocity Equivalence
This paper studies how the local root numbers and the Weil additive characters of the Witt ring of a number field behave under reciprocity equivalence. Given a reciprocity equivalence between two fields, at each place we define a local square class which vanishes if and only if the local root numbers are preserved. Thus this local square class serves as a local obstruction to the preservation of local root numbers. We establish a set of necessary and sufficient conditions for a selection of local square classes (one at each place) to represent a global square class. Then, given a reciprocity equivalence that has a finite wild set, we use these conditions to show that the local square classes combine to give a global square class which serves as a global obstruction to the preservation of all root numbers. Lastly, we use these results to study the behavior of Weil characters under reciprocity equivalence.

Categories:11E12, 11E08

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