Characteristic functions of

Let P ∈ Q p [ x , y stretchy="false">] P\in \Bbb Q_p[x,y] , alttext="s C"> s mathvariant="double-struck">C encoding="application/x-tex">s\in C with sufficiently large real part, and consider the integral operator stretchy="false">( A f stretchy="false">) ≔ 1 − encoding="application/x-tex">\chi alttext="double-struck times"> ×<!-- × </mml:msubsup> encoding="application/x-tex">\Bbb Z_p^\times characteristic function alttext="det u chi movablelimits="true" form="prefix">det u encoding="application/x-tex">\det (1-uA_{P,s,\chi }) restriction chi"> encoding="application/x-tex">A_{P,s,\chi } s"> encoding="application/x-tex">A_{P,s} to eigenspace Z_p)_\chi alttext="q"> q encoding="application/x-tex">q -Wronskian a set solutions (possibly confluent) -hypergeometric equation, where alttext="q equals = encoding="application/x-tex">q=p^{-1-ds} In particular, nonzero eigenvalues are reciprocals zeros such -Wronskian.