Euclidean quadratic forms and ADC forms: I
On elementary equivalence, isomorphism and isogeny
Motivated by recent work of Florian Pop, we study the connections between three notions of equivalence of function fields: isomorphism, elementary equivalence, and the condition that each of a pair of fields can be embedded in the other, which we call isogeny. Some of our results are purely geometric: we give an isogeny classification of Severi-Brauer varieties and quadric surfaces. These results are applied to deduce new instances of “elementary equivalence implies isomorphism”: for all genus zero...
The period-index problem in WC-groups IV: a local transition theorem
Let be a complete discretely valued field with perfect residue field . Assuming upper bounds on the relation between period and index for WC-groups over , we deduce corresponding upper bounds on the relation between period and index for WC-groups over . Up to a constant depending only on the dimension of the torsor, we recover theorems of Lichtenbaum and Milne in a “duality free” context. Our techniques include the use of of torsors under abelian varieties with good reduction and a generalization...
Euclidean quadratic forms and ADC forms II: integral forms
We study ADC quadratic forms and Euclidean quadratic forms over the integers, obtaining complete classification results in the positive case.
On the number of representations of an integer by a linear form.
Page 1