Colliot-Th\'el\`ene's conjecture and finiteness of u-invariants

We show that Colliot-Th'el`ene’s conjecture on 0-cycles of degree 1 implies finiteness for the u-invariant of the function field of a curve over a totally imaginary number field and period-index bounds for the Brauer groups of arbitrary fields of transcendence degree 1 over the rational numbers.