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 
