Hodge Polygon Research Articles

A positivity property of the Satake isomorphism

Let G be an unramified reductive group over a local field. We consider the matrix describing the Satake isomorphism in terms of the natural bases of the source and the target. We prove that all coefficients of this matrix which are not obviously zero are in fact positive numbers. The result is then applied to an existence problem of F-crystals which is a partial converse to Mazur's theorem relating the Hodge polygon and the Newton polygon.

Tensor products in p-adic Hodge theory

There is a classical relation between the p-adic absolute value of the eigenvalues of Frobenius on crystalline cohomology and Hodge numbers, for a variety in characteristic p: “the Newton polygon lies on or above the Hodge polygon” [14], [1]. For a variety in characteristic p with a lift to characteristic 0, Fontaine conjectured and Faltings proved a more precise statement: there is an inequality which relates the slope of Frobenius on any Frobenius-invariant subspace of the crystalline cohomology to the Hodge filtration, restricted to that subspace [7], [4]. A vector space over a p-adic field together with a σ-linear endomorphism and a filtration which satisfies this inequality is called a weakly admissible filtered isocrystal (see section 1 for the precise definition). The category of such objects is one possible p-adic analogue of the category of Hodge structures: in particular, it is an abelian category. We give a new proof of Faltings’s theorem that the tensor product of weakly admissible filtered isocrystals over a p-adic field is weakly admissible [5]. By a similar argument, we also prove a characterization of weakly admissible filtered isocrystals with G-structure in terms of geometric invariant theory, which was conjectured by Rapoport and Zink [19]. Before Faltings, Laffaille [12] had proved the tensor product theorem in the case of filtered isocrystals over an unramified extension of Qp. Faltings’s proof works by reducing this problem of σ-linear algebra to a different problem of pure linear algebra, the problem of showing that the tensor product of two vector spaces, each equipped with a finite “semistable” set of filtrations, is semistable. The latter problem is solved by constructing suitable integral lattices (in [5]) or hermitian metrics (in [20]) on vector spaces with a semistable set of filtrations, just as one can prove that the tensor product of semistable bundles on an algebraic curve is semistable using Narasimhan-Seshadri’s hermitian metrics ([6], [16]). In this paper, we can avoid the reduction from filtered isocrystals to filtered vector spaces. The point is that Ramanan and Ramanathan’s algebraic proof [17] that the tensor product of semistable vector bundles is semistable can be modified to apply directly to filtered isocrystals. We have an inequality to prove for a class of linear subspaces S of a tensor product V ⊗W . The inequality is obvious for sufficiently general subspaces S and also if S is a very special subspace, say if S is a decomposable subspace S1 ⊗ S2 ⊂ V ⊗W . But it is not clear how to prove the inequality we want if S is somewhere in the middle. The solution, following Ramanan and Ramanathan, is to use geometric invariant theory to give a sharp dichotomy between “general” subspaces and “special” subspaces of V ⊗W , in such a way that we get useful information in either case.

Newton Polygons of Zeta Functions and L Functions

In this article 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 zeroes or poles of zeta functions and L functions. Our main objective is to show that the Adolphson-Sperber conjecture 12], which asserts that under a simple condition the generic Newton polygon of L functions coincides with its lower bound, 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 affine hypersurfaces of degree d or the family of affine hypersurfaces defined by polynomials f(xi, . . ., x7n) of degree di with respect to xi (1 < i < n), where the di are fixed positive integers. Then, for all large prime numbers p, the generic Newton polygon for the zeta functions of each of the two families of hypersurfaces coincides with its lower bound. We obtain our main results, namely several decomposition theorems, using certain maximizing functions from linear programming. Our work suggests a possible connection between Newton polygons and the resolution of singularities of toric varieties. Let p be a prime, q = pa, and let Fq be the finite field of q elements and Fqm its extension of degree m. Fix a nontrivial additive character qP of Fp. For any Laurent polynomial f(xi, . . ., xn) E Fq[xl, xj1,.. . ., x, xi1] we form

On the p-adic Theory of Exponential Sums

where V is an arbitrary affine variety defined over Fq, V(Fq) denotes the Fq-rational points of V, f is a regular function on V, and Tq is an arbitrary additive character on Fq. The estimate (1.2) is given in terms of the dimension of the ambient space, and the degree of polynomials which definef and V. The work generalizes the result of Katz [4] which gives a best-possible estimate for the sum(*) in the case Tq is the trivial character (equivalently, Tq is non-trivial but f is the zero function). Thus Katz's estimate is a best-possible estimate for the p-divisibility of N(V) the number of Fq rational points of V. These questions on p-divisibility trace their origins to a problem posed by Artin and solved by Chevalley and Warning [10]. The results may be interpreted as specifying a sharp lower bound for the slope of the Newton polygon of the L-function associated with the sum(*). A finer measure of the p-adic behavior of this L-function is given by the shape of its Newton-polygon. Dwork, in [3], showed that for nonsingular projective hypersurfaces the Newton polygon of the zeta function lies over its Hodge polygon. Katz's result in [4] showed that the same relationship holds for the first slopes of the two polygons in the case of a nonsingular projective complete intersection; at the same time, he formulated a general conjecture which was subsequently proved by Mazur [6] using crystalline cohomology. In section 2, we prove the analogue for exponen-