Newton Polygons of Zeta Functions and L Functions

this paper, we give a systematic treatment of Newton polygons of exponential sums. The Newton polygon is a nice way to describe p-adic values of the zeros or poles of zeta functions and L-functions. Our main object is the Adolphson-Sperber conjecture [2] which asserts that under a simple condition the generic Newton polygon of L-functions coincides with its lower bound. We show that the conjecture is false in its full form but true in a slightly weaker form. We also show that the full form is true in various important special cases. For example, we show that for a generic projective hypersurface of degree d, the Newton polygon of the interesting part of the zeta function coincides with its lower bound (the Hodge polygon). This gives a p-adic proof of a recent theorem of Illusie, conjectured by Dwork and Mazur. For more examples, let us consider the family of ane hypersurfaces of degree d or the family of ane hypersurfaces de ned by polynomials f(x 1 ; ; x n ) of degree d i with respect to x i (1 i n), where the d i are xed positive integers. Then for all large prime number p, the generic Newton polygon for the zeta functions of each of the two families of hypersurfaces coincides with its lower bound. Our main results are several decomposition theorems obtained using certain maximizing functions from linear programming. In particular, this suggests a possible connection between Newton polygons and resolution of singularities of toric varieties