Milnor's conjecture for p-adic curves

The first cases of Milnor's conjecture on the fundamental filtration of the Witt ring of a field are settled by Kummer theory and by Merkurjev's theorem. Merkurjev's theorem---that every 2-torsion Brauer class is represented by the Clifford algebra of a quadratic form---is in general false when the base is no longer a field. Parimala, Scharlau, and Sridharan found smooth complete p-adic curves for which Merkurjev's theorem is equivalent to the existence of a rational theta characteristic. We'll discuss how replacing quadratic forms with their natural reductive counterparts, the so-called line bundle-valued quadratic forms, a generalized version of Milnor's conjecture for p-adic curves is obtained.