The \(u\)-invariant of \(p\)-adic function fields (Q2838629)

This beautifully written paper represents a significant advance in the study of the \(u\)-invariant, an important field invariant pertaining to quadratic forms. Classically, this invariant is defined to be the maximum of the dimensions of finite-dimensional anisotropic quadratic forms over \(K\), or \(\infty\) if such a maximum does not exist. In the case of formally real fields, the \(u\)-invariant would therefore always be \(\infty\) and one restricts the definition in that case to anisotropic forms that are torsion in the Witt ring of \(K\). The main results of the present paper only deal with nonformally real fields. There is an extensive amount of literature about the \(u\)-invariant regarding possible values and the determination for explicitly given fields, but many problems remain open, one of the most striking perhaps the following: Given a field \(K\) with \(u(K)<\infty\). Does it follow that \(u(K(x))<\infty\) where \(K(x)\) denotes the rational function field in one variable\,? If \(K\) is a \(C_i\)-field, then by definition of this property one gets \(u(K)\leq 2^{i}\) and thus \(2u(K)\leq u(K(x))\leq 2^{i+1}\) since \(K(x)\) will be \(C_{i+1}\) and the first inequality follows from an easy valuation argument. But even in the case where \(K\) is a \(p\)-adic number field and therefore \(u(K)=4\), the above question remained open for a long time until Merkurjev and independently \textit{D. W. Hoffmann} and \textit{J. Van Geel} [J.\ Ramanujan Math.\ Soc.\ 13, No. 2, 85--110 (1998; Zbl 0922.11032)] showed that \(u(L)\leq 26\) resp. \(u(L)\leq 22\) for \(L\) a finitely generated field extension of transcendence degree \(1\) over a \(p\)-adic nondyadic number field. Eventually, \textit{R. Parimala} and \textit{V. Suresh} [Ann.\ Math.\ (2) 172, No.\ 2, 1391--1405 (2010; Zbl 1208.11053)] proved that then one in fact has \(u(L)=8\) using methods from cohomology. Another proof of this result using so-called patching techniques has been given by \textit{D. Harbater, J. Hartmann} and \textit{D. Krashen} [Invent. Math. 178, No. 2, 231--263 (2009; Zbl 1259.12003)].NEWLINENEWLINEIn the present paper, the author far exceeds anything that was previously known about the \(u\)-invariant of function fields over \(p\)-adic number fields including the dyadic case. In fact, he shows that if \(K\) is any field that is finitely generated over \(\mathbb{Q}_p\) of transcendence degree \(m\), then \(u(K)=2^{m+2}\). For the proof, the author introduces the property \(\mathcal{A}_i(d)\) for a field \(K\) which is satisfied if any system of \(r\) homogeneous forms of degree \(d\) over \(K\) in \(n>rd^i\) variables has a common nontrivial zero in some finite extension of \(K\) of degree prime to \(d\).NEWLINENEWLINEOne can then show that \(K\) having property \(\mathcal{A}_i(2)\) implies \(u(K)\leq 2^{i}\). Furthermore, consider the following types of field extensions of a field \(K\): rational function field \(L=K(x)\) (type (a)), or Laurent series field \(L=K(\!(x)\!)\) (type (b)), or \(L/K\) finite algebraic (type (c)). It is then shown that if \(K\) has property \(\mathcal{A}_i(d)\), then \(L\) has \(\mathcal{A}_{i+1}(d)\) for type (a) and (b) extensions, and \(L\) has \(\mathcal{A}_i(d)\) for type (c) extensions provided \(d\) is a prime power. The main result now follows by showing that a \(p\)-adic number field has property \(\mathcal{A}_2(2)\). This is done by invoking an important result due to \textit{D. R. Heath-Brown} [Compos.\ Math.\ 146, No.\ 2, 271--287 (2010; Zbl 1194.11047)] stating that if \(K\) is a \(p\)-adic field with residue field \(F\), \(\{ Q_1,\ldots,Q_r\}\) a system of \(r\) quadratic forms over \(K\) in \(n>4r\) Variables and if \(|F|\geq (2r)^r\), then this system has a common nontrivial zero over \(K\).NEWLINENEWLINEThe author generalizes his result on the \(u\)-invariant even further. A field extension \(L/K\) is called basic of type \(m\) if it can be written as a tower \(K=K_0\subset K_1\subset K_2\subset\ldots\subset K_t=L\) such that each \(K_{i+1}/K_i\) is of type (a), (b) or (c), and exactly \(m\) of the \(K_{i+1}/K_i\) are of type (a) or (b). It is then shown that if \(K\) is basic of type \(m\) over \(\mathbb{Q}_p\), then \(u(K)=2^{m+2}\) and also \(u(K(\!(x,y)\!))=2^{m+4}\).