## Variations of $$p$$-adic Newton polygons for $$L$$-functions of exponential sums.(English)Zbl 1084.11067

Let $$F_q$$ be the finite field of $$q$$ elements and for $$k\geq 1$$ let $$F_{q^k}$$ be the extension of $$F_q$$ of degree $$k$$. For any Laurent polynomial $$f\in F_q[X_1,X_1^{-1},\ldots,X_n,X_n^{-1}]$$ form the additive character sum
$S_k(f)=\sum_{(x_1,\ldots,x_n)\in (F_{q^k}\setminus \{0\})^n} \psi_k(f(x_1,\ldots,x_n)),$
where $$\psi_k$$ is the additive canonical character of $$F_{q^k}$$. With the Laurent polynomial is associated an $$L$$-function
$L(f,T)=\exp\left(\sum_{k=1}^\infty S_k(f)\frac{T^k}{k}\right).$
If $$f$$ is not degenerated, this $$L$$-function is a polynomial. A. Adolphson and S. Sperber [Ann. Math. (2) 130, 367–406 (1989; Zbl 0723.14017)] calculated the degree of $$P$$ from the Newton polyhedron of $$f$$ at infinity and gave a lower bound for the Newton polygon of $$P$$. Moreover, they conjectured that the Newton polygon of the polynomial $$P$$ associated with the generic Laurent polynomial $$f$$ with given Newton polyhedron at infinity coincides with this lower bound. Combined with his work in [Newton polygons of zeta functions and $$L$$ functions. Ann. Math. (2) 137, No. 2, 249–293 (1993; Zbl 0799.11058)] the author proves that this conjecture is true in every dimension $$n\leq 3$$ and false for every $$n\geq 4$$. Moreover, he shows that a weaker form of the Adolphson-Sperber conjecture is always true. These results are applications of a flexible collapsing decomposition theorem introduced in this paper.

### MSC:

 11T23 Exponential sums 11G25 Varieties over finite and local fields 11M41 Other Dirichlet series and zeta functions 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)

### Citations:

Zbl 0723.14017; Zbl 0799.11058
Full Text: