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Let $f$ be a polynomial in $n$ variables over some number field and $Z$ asubscheme of affine $n$-space. The notion of motivic oscillation index of $f$at $Z$ was initiated by Cluckers (2008) and Cluckers-Musta\c{t}\v{a}-Nguyen(2019). In this paper we elaborate on this notion and raise several questions.The first one is stability under base field extension; this question is linkedto a deep understanding of the density of non-archimedean local fields overwhich Igusa's local zeta functions of $f$ has a pole with given real part. Thesecond one is around Igusa's conjecture for exponential sums with bounds interms of the motivic oscillation index. Thirdly, we wonder if the abovequestions only depend on the analytic isomorphism class of singularities. Byusing various techniques as the GAGA theorem, resolution of singularities andmodel theory, we can answer the third question up to a base field extension.Next, by using a transfer principle between non-archimedean local fields ofcharacteristic zero and positive characteristic, we can link all threequestions with a conjecture on weights of $\ell$-adic cohomology groups ofArtin-Schreier sheaves associated to jet polynomials. This way, we can answerall questions positively if $f$ is a polynomial of Thom-Sebastiani type withnon-rational singularities. As a consequence, we prove Igusa's conjecture forarbitrary polynomials in three variables and polynomials with singularities of$ADE$ type. In an appendix, we answer affirmatively a recent question ofCluckers-Musta\c{t}\v{a}-Nguyen (2019) on poles of twisted Igusa's local zetafunctions of maximal order.

On the motivic oscillation index and bound of exponential sums modulo $p^m$ via analytic isomorphisms