Newton polygons for twisted exponential sums and polynomials $P(x^d)$

Details Newton polygons for twisted exponential sums and polynomials $P(x^d)$ Newton polygons for twisted exponential sums and polynomials $P(x^d)$

2007-02-16

arxiv.org/abs/math/0702502v2

Mathematics

Number Theory

The results in this preprint have been strenghened in arXiv:0706.2340; please look at this new preprint

Scientific paper

We study the $p$-adic absolute value of the roots of the $L$-functions associated to certain twisted character sums, and additive character sums associated to polynomials $P(x^d)$, when $P$ varies among the space of polynomial of fixed degree $e$ over a finite field of characteristic $p$. For sufficiently large $p$, we determine in both cases generic Newton polygons for these $L$-functions, which is a lower bound for the Newton polygons, and the set of polynomials of degree $e$ for which this generic polygon is attained. In the case of twisted sums, we show that the lower polygon defined in \cite{as1} is tight when $p\equiv 1 [de]$, and that it is the actual Newton polygon for any degree $e$ polynomial.

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