| Popis: |
Let P ∈ Q p [ x , y ] P\in \Bbb Q_p[x,y] , s ∈ C s\in \Bbb C with sufficiently large real part, and consider the integral operator ( A P , s f ) ( y ) ≔ 1 1 − p − 1 ∫ Z p | P ( x , y ) | s f ( x ) | d x | \begin{equation*} (A_{P,s}f)(y)≔\frac {1}{1-p^{-1}}\int _{\Bbb Z_p}|P(x,y)|^sf(x) |dx| \end{equation*} on L 2 ( Z p ) L^2(\Bbb Z_p) . We show that if P P is homogeneous of degree d d then for each character χ \chi of Z p × \Bbb Z_p^\times the characteristic function det ( 1 − u A P , s , χ ) \det (1-uA_{P,s,\chi }) of the restriction A P , s , χ A_{P,s,\chi } of A P , s A_{P,s} to the eigenspace L 2 ( Z p ) χ L^2(\Bbb Z_p)_\chi is the q q -Wronskian of a set of solutions of a (possibly confluent) q q -hypergeometric equation, where q = p − 1 − d s q=p^{-1-ds} . In particular, the nonzero eigenvalues of A P , s , χ A_{P,s,\chi } are the reciprocals of the zeros of such q q -Wronskian. |
