Diagonal forms of higher degree over function fields of p-adic curves

Abstract

We investigate diagonal forms of degree [Formula: see text] over the function field [Formula: see text] of a smooth projective [Formula: see text]-adic curve: if a form is isotropic over the completion of [Formula: see text] with respect to each discrete valuation of [Formula: see text], then it is isotropic over certain fields [Formula: see text], [Formula: see text] and [Formula: see text]. These fields appear naturally when applying the methodology of patching; [Formula: see text] is the inverse limit of the finite inverse system of fields [Formula: see text]. Our observations complement some known bounds on the higher [Formula: see text]-invariant of diagonal forms of degree [Formula: see text]. We only consider diagonal forms of degree [Formula: see text] over fields of characteristic not dividing [Formula: see text].