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# Asian Journal of Mathematics

## Volume 8 (2004)

### Number 3

### Variation of *p*-adic Newton polygons for L-functions of exponential sums

Pages: 427 – 472

DOI: http://dx.doi.org/10.4310/AJM.2004.v8.n3.a4

#### Author

#### Abstract

In this paper, we continue to develop the systematic decomposition theory [18] for the generic Newton polygon attached to a family of zeta functions over finite fields and more generally a family of L-functions of *n*-dimensional exponential sums over finite fields. Our aim is to establish a new collapsing decomposition theorem (Theorem 3.7) for the generic Newton polygon. A number of applications to zeta functions and L-functions are given, including the full form of the remaining 3 and 4-dimensional cases of the Adolphson-Sperber conjecture [2], which were left unresolved in [18]. To make the paper more readable and useful, we have included an expanded introductory section as well as detailed examples to illustrate how to use the main theorems.