Miércoles 25 de noviembre, 14:30-16:40. Sala DIR-200 IMA-PUCV
Expositor: David Grimm (USACH)
Resumen: I will first sketch and overview over the field-patching
methods developed by Harbater, Hartmann, and Krashen for homogeneous varieties over rational algebraic groups defined over the function field of an arithmetic curve over a complete discrete valuation ring.
These field-patching methods lead to a geometric local to global principle for the existence of rational points in terms of completions of local rings at special points for homogeneous varieties (by Harbater, Hartmann and Krashen in the general case, and independently by myself in the very special case of projective quadrics and when the residue characteristic is dierent from 2). I will sketch the proof of this geometric local-global principle in the special case of quadratic forms, and based on this, I present the proof of an arithmetic local-global principle for isotropy of quadratic forms by Colliot-Thelene, Parimala, and Suresh (which inspired the geometric version of local-global principle for quadratic forms in the first place). I will also point out that the arithmetic local-global principle for quadratic forms actually holds under slightly weaker hypotheses than originally formulated, and we present a consequence of this strengthened version.
Miércoles 11 de noviembre, 14:30-16:40. Sala 2-2 IMA-PUCV
Expositor: Ricardo Menares (PUCV)