The u-invariant of the function fields of p-adic curves

The u-invariant of a field is the maximum dimension of ansiotropic quadratic forms over the field. It is an open question whether the u-invariant of function fields of p-aidc curves is 8. In this paper, we answer this question in the affirmative for function fields of non-dyadic p-adic curves.Comment: 20 page

Failure of the local-global principle for isotropy of quadratic forms over rational function fields

We prove the failure of the local-global principle, with respect to all discrete valuations, for isotropy of quadratic forms over a rational function field of transcendence degree at least 2 over the complex numbers. Our construction involves the generalized Kummer varieties considered by Borcea and Cynk--Hulek.Comment: 7 pages, comments welcome

Entropy spectrum of (1+1) dimensional stringy black holes

We explore the entropy spectrum of $(1+1)$ dimensional dilatonic stringy black holes via the adiabatic invariant integral method and the Bohr-Sommerfeld quantization rule. It is found that the corresponding spectrum depends on black hole parameters like charge, ADM mass and more interestingly on the dilatonic field. We calculate the entropy of the present black hole system via the Euclidean treatment of quantum gravity and study the thermodynamics of the black hole and find that the system does not undergo any phase transition.Comment: 10 pages, 2 figure