A note on |αp - q|
A modification of a classical number-theorem on Diophantine approximations is used for generalizing H. kielhöfer's result on bifurcations of nontrivial periodic solutions to nonlinear wave equations.
Let and be holomorphic common eigenforms of all Hecke operators for the congruence subgroup of with “Nebentypus” character and and of weight and , respectively. Define the Rankin product of and bySupposing and to be ordinary at a prime , we shall construct a -adically analytic -function of three variables which interpolate the values for integers with by regarding all the ingredients , and as variables. Here is the Petersson self-inner product of .
We prove that if an n×n matrix defined over ℚ ₚ (or more generally an arbitrary complete, discretely-valued, non-Archimedean field) satisfies a certain congruence property, then it has a strictly maximal eigenvalue in ℚ ₚ, and that iteration of the (normalized) matrix converges to a projection operator onto the corresponding eigenspace. This result may be viewed as a p-adic analogue of the Perron-Frobenius theorem for positive real matrices.