Newton polygons for character sums and Poincaré series

Régis Blache (1)

1 LAMIA - Laboratoire de Mathématiques Informatique et Applications

In this paper, we precise the asymptotic behaviour of Newton polygons of $L$-functions associated to character sums, coming from certain $n$ variable Laurent polynomials. In order to do this, we use the free sum on convex polytopes. This operation allows the determination of the limit of generic Newton polygons for the sum $\Delta=\Delta_1\oplus \Delta_2$ when we know the limit of generic Newton polygons for each factor. To our knowledge, these are the first results concerning the asymptotic behaviour of Newton polygons for multivariable polynomials when the generic Newton polygon differs from the combinatorial (Hodge) polygon associated to the polyhedron.

Mots clés: Character sums; $L$-functions; Newton polygons and polytopes

Domaines: Théorie des nombres [math.NT]; Géométrie algébrique [math.AG]

Régis Blache. Newton polygons for character sums and Poincaré series. 2011. ⟨hal-00551461⟩