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The First Slope Case of Wan's Conjecture

✍ Scribed by Jasper Scholten; Hui June Zhu

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
93 KB
Volume
8
Category
Article
ISSN
1071-5797

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✦ Synopsis

Let d ! 2 and p a prime coprime to d. For f ðxÞ 2 ðZ p \ QÞ½x, let NP 1 ðf mod pÞ denote the first slope of the Newton polygon of the L-function of the exponential sums X x2F p ' z Tr F p ' =Fp ðf ðxÞÞ p :

We prove that there is a Zariski dense open subset U in the space A d of degree-d monic polynomials over Q such that for all f ðxÞ 2 U we have lim p!1 NP 1 ðf mod pÞ ¼ 1 d . This is a ''first slope case'' of a conjecture of Wan.

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