Hodge Polygon Research Articles

The integral Hodge polygon for p-divisible groups with endomorphism structure

The integral Hodge polygon for p-divisible groups with endomorphism structure

Newton polygons of L-functions for two-variable generalized Kloosterman sums

In this paper, we study the Newton polygon of the [Formula: see text]-function of a generalized Kloosterman polynomial with two variables over finite fields. We give the explicit form of the monomial basis of the top dimensional cohomology space of the [Formula: see text]-adic complex associated to the [Formula: see text]-function. One consequence of this result is a concrete method for computing the Hodge polygon. Using decomposition theorems of Wan, we determine when a generalized Kloosterman polynomial is ordinary and when its Newton polytope is generically ordinary.

Newton polygon of exponential sums in two variables with triangular base

Let f(x1,x2) be a two-variable polynomial over a finite field of characteristic p, and assume that its convex hull is a triangle Δ. The goal of this paper is to study the Newton polygon NP(f,χ) of the L-function associated to f and a finite character χ of Zp. It is a partial generalization of [DWX], in which the authors studied polynomials of one variable.In this paper, we prove an improved lower bound IHP(Δ) for NP(f,χ), which, when f is non-ordinary, is strictly higher than the classical Hodge polygon. Moreover, we prove that if NP(f,χ) and IHP(Δ) coincide at one certain point, then they coincide at infinitely many points. This recovers the key property of the proof from [DWX].

Frobenius and the Hodge Filtration of the Generic Fiber

Abstract We use the Nygaard filtration on Ainf cohomology to give a new proof of the following result: for a smooth proper (formal) scheme over a mixed characteristic valuation ring, the Newton polygon of its special fiber lies on or above the Hodge polygon of its generic fiber.

T-adic exponential sums over affinoids

We introduce and develop (π1/c,p)-adic Dwork theory for L-functions of exponential sums associated to one-variable rational functions, interpolating pk-order exponential sums over affinoids. Namely, we prove a generalization of the Dwork-Monsky-Reich trace formula and apply it to establish an analytic continuation of the C-function Cf(s,π). We compute the lower (π1/c,p)-adic bound, the Hodge polygon, for this C-function. Along the way, we also show why a strictly π-adic theory will not work in this case.

Newton polygons for L-functions of generalized Kloosterman sums

Abstract In the present paper, we study the Newton polygons for the L-functions of n-variable generalized Kloosterman sums. Generally, the Newton polygon has a topological lower bound, called the Hodge polygon. In order to determine the Hodge polygon, we explicitly construct a basis of the top-dimensional Dwork cohomology. Using Wan’s decomposition theorem and diagonal local theory, we obtain when the Newton polygon coincides with the Hodge polygon. In particular, we concretely get the slope sequence for the L-function of F ¯ ⁢ ( λ ¯ , x ) := ∑ i = 1 n x i a i + λ ¯ ⁢ ∏ i = 1 n x i - 1 , \bar{F}(\bar{\lambda},x):=\sum_{i=1}^{n}x_{i}^{a_{i}}+\bar{\lambda}\prod_{i=1}% ^{n}x_{i}^{-1}, with a 1 , … , a n {a_{1},\ldots,a_{n}} being pairwise coprime for n ≥ 2 {n\geq 2} .

L-functions of exponential sums and Hasse polynomials

Let f be a non-degenerate Laurent polynomial over a finite field and L⁎(f,T) the associated L-function of the toric exponential sums of f. The Newton polygon can be used to study the p-adic valuations of roots or poles of L⁎(f,T), which has a lower bound called the Hodge polygon that depends only on the Newton polytope of f. These two polygons coincide when the coefficients of f are not zeros of the certain Hasse polynomial. Using the Dwork trace formula, we find a formula for the Hasse polynomial of the slope one side for a class of Laurent polynomials, which generalizes the results of Zhang, Feng and Chen.

Ordinary primes in Hilbert modular varieties

A well-known conjecture, often attributed to Serre, asserts that any motive over any number field has infinitely many ordinary reductions (in the sense that the Newton polygon coincides with the Hodge polygon). In the case of Hilbert modular cuspforms $f$ of parallel weight $(2,\ldots ,2)$, we show how to produce more ordinary primes by using the Sato–Tate equidistribution and combining it with the Galois theory of the Hecke field. Under the assumption of stronger forms of Sato–Tate equidistribution, we get stronger (but conditional) results. In the case of higher weights, we formulate the ordinariness conjecture for submotives of the intersection cohomology of proper algebraic varieties with motivic coefficients, and verify it for the motives whose $\ell$-adic Galois realisations are abelian on a finite-index subgroup. We get some results for Hilbert cuspforms of weight $(3,\ldots ,3)$, weaker than those for $(2,\ldots ,2)$.

Generic Newton polygon for exponential sums in n variables with parallelotope base

Let $p$ be a prime number. Every $n$-variable polynomial $f(\underline{x})$ over a finite field of characteristic $p$ defines an Artin-Schreier-Witt tower of varieties whose Galois group is isomorphic to ${\Bbb Z}_p$. Our goal of this paper is to study the Newton polygon of the $L$-function associated to a nontrivial finite character of ${\Bbb Z}_p$ and a generic polynomial whose convex hull is an $n$-dimensional paralleltope $\Delta$. We denote this polygon by ${\rm GNP}(\Delta)$. We prove a lower bound of ${\rm GNP}(\Delta)$, which is called the improved Hodge polygon ${\rm IHP}(\Delta)$. We show that ${\rm IHP}(\Delta)$ lies above the usual Hodge polygon ${\rm HP}(\Delta)$ at certain infinitely many points, and when $p$ is larger than a fixed number determined by $\Delta$, it coincides with ${\rm GNP}(\Delta)$ at these points. As a corollary, we roughly determine the distribution of the slopes of ${\rm GNP}(\Delta)$.

On the Hodge–Newton filtration for p-divisible groups of Hodge type

A p-divisible group, or more generally an F-crystal, is said to be Hodge-Newton reducible if its Hodge polygon passes through a break point of its Newton polygon. Katz proved that Hodge-Newton reducible F-crystals admit a canonical filtration called the Hodge-Newton filtration. The notion of Hodge-Newton reducibility plays an important role in the deformation theory of p-divisible groups; the key property is that the Hodge-Newton filtration of a p-divisible group over a field of characteristic p can be uniquely lifted to a filtration of its deformation. We generalize Katz's result to F-crystals that arise from an unramified local Shimura datum of Hodge type. As an application, we give a generalization of Serre-Tate deformation theory for local Shimura data of Hodge type. We also apply our deformation theory to study some congruence relations on Shimura varieties of Hodge type.

Groupesp-divisibles avec condition de Pappas-Rapoport et invariants de Hasse

We study $p$-divisible groups $G$ endowed with an action of the ring of integers of a finite (possibly ramified) extension of $\mathbb{Q}_p$ over a scheme of characteristic $p$. We suppose moreover that the $p$-divisible group $G$ satisfies the Pappas-Rapoport condition for a certain datum $\mu$ ; this condition consists in a filtration on the sheaf of differentials $\omega_G$ satisfying certain properties. Over a perfect field, we define the Hodge and Newton polygons for such $p$-divisible groups, normalized with the action. We show that the Newton polygon lies above the Hodge polygon, itself lying above a certain polygon depending on the datum $\mu$. We then construct Hasse invariants for such $p$-divisible groups over an arbitrary base scheme of characteristic $p$. We prove that the total Hasse invariant is non-zero if and only if the $p$-divisible group is $\mu$-ordinary, i.e. if its Newton polygon is minimal. Finally, we study the properties of $\mu$-ordinary $p$-divisible groups. The construction of the Hasse invariants can in particular be applied to special fibers of PEL Shimura varieties models as constructed by Pappas and Rapoport.

Symmetric powers do not stabilize

We discuss the stabilization of symmetric products Sym^n(X) of a smooth projective variety X in the Grothendieck ring of varieties. For smooth projective surfaces X with non-zero h^0(X, \omega_X), these products do not stabilize; we conditionally show that they do not stabilize in another related sense, in response to a question of R. Vakil and M. Wood. There are analogies between such stabilization, the Dold-Thom theorem, and the analytic class number formula. Finally, we discuss Hodge-theoretic obstructions to the stabilization of symmetric products, and provide evidence for these obstructions in terms of a relationship between the Newton polygon of a certain motivic zeta function associated to a curve, and its Hodge polygon.

Families of Artin–Schreier curves with Cartier–Manin matrix of constant rank

Let k be an algebraically closed field of characteristic p>0. Every Artin–Schreier k-curve X has an equation of the form yp−y=f(x) for some f(x)∈k(x) such that p does not divide the least common multiple L of the orders of the poles of f(x). Under the condition that p≡1modL, Zhu proved that the Newton polygon of the L-function of X is determined by the Hodge polygon of f(x). In particular, the Newton polygon depends only on the orders of the poles of f(x) and not on the location of the poles or otherwise on the coefficients of f(x). In this paper, we prove an analogous result about the a-number of the p-torsion group scheme of the Jacobian of X, providing the first non-trivial examples of families of Jacobians with constant a-number. Equivalently, we consider the semi-linear Cartier operator on the sheaf of regular 1-forms of X and provide the first non-trivial examples of families of curves whose Cartier–Manin matrix has constant rank.

On the Hodge–Newton Filtration for p-Divisible Groups with Additional Structures

We prove that, for a $p$-divisible group with additional structures over a complete valuation ring of rank one $O_K$ with mixed characteristic $(0,p)$, if the Newton polygon and the Hodge polygon of its special fiber possess a non trivial contact point, which is a break point for the Newton polygon, then it admits a Hodge-Newton over $O_K$. The proof is based on the theories of Harder-Narasimhan filtration of finite flat group schemes and admissible filtered isocrystals. We then apply this result to the study of some larger class of Rapoport-Zink spaces and Shimura varieties than those studied previously by Mantovan, and confirm some new cases of Harris's conjecture.

On the zeta function of a projective complete intersection

We compute a basis for the $p$-adic Dwork cohomology of a smooth complete intersection in projective space over a finite field and use it to give $p$-adic estimates for the action of Frobenius on this cohomology. In particular, we prove that the Newton polygon of the characteristic polynomial of Frobenius lies on or above the associated Hodge polygon. This result was first proved by B. Mazur using crystalline cohomology.

Hodge-Stickelberger polygons for $L$-functions of exponential sums of $P(x^s)$

Let $\F_q$ be a finite field of cardinality $q$ and characteristic $p$. Let $\bar{P}(x)$ be any one-variable Laurent polynomial over $\F_q$ of degree $(d_1,d_2)$ respectively and $p\nmid d_1d_2$. For any fixed $s\geq 1$ coprime to $p$, we prove that the $q$-adic Newton polygon of the $L$-functions of exponential sums of $\bar{P}(x^s)$ has a tight lower bound which we call Hodge-Stickelberger polygon, depending only on the $d_1,d_2,s$ and the residue class of $(p\bmod s)$. This Hodge-Stickelberger polygon is a certain weighted convolution of the Hodge polygon for $L$-function of exponential sums of $\bar{P}(x)$ and the Newton polygon for the $L$-function of exponential sums of $x^s$ (which is precisely given by the classical Stickelberger theory). We have an analogous Hodge-Stickelberger lower bound for multivariable Laurent polynomials as well. For any $\nu\in(\Z/s\Z)^\times$, we show that there exists a Zariski dense open subset $ {\cU} _\nu$ defined over $\Q$ such that for every Laurent polynomial $P$ in $\cU_\nu(\bar{\Q})$ the $q$-adic Newton polygon of $L(\bar{P}(x^s)/\F_q;T)$ converges to the Hodge-Stickelberger polygon as $p$ approaches infinity and $p\equiv \nu\bmod s$. As a corollary, we obtain a tight lower bound for the $q$-adic Newton polygon of the numerator of the zeta function of an Artin-Schreier curve given by affine equation $y^p-y=\bar{P}(x^s)$. This estimates the $q$-adic valuations of reciprocal roots of the numerator of the zeta function of the Artin-Schreier curve.

The L-functions of twisted Witt extensions

The L-function of a non-degenerate twisted Witt extension is proved to be a polynomial. Its Newton polygon is proved to lie above the Hodge polygon of that extension. And the Newton polygons of the Gauss–Heilbronn sums are explicitly determined, generalizing the Stickelberger theorem.

Asymptotic variation of L functions of one-variable exponential sums

Let d>2 and let p be a prime coprime to d. Let Z_pbar be the ring of integers of Q_pbar. Suppose f(x) is a degree-d polynomial over Qbar and Z_pbar. Let P be a prime ideal over p in the ring of integers of Q(f), where Q(f) is the number field generated by coefficients of f in Qbar. Let A^d be the dimension-d affine space over Qbar, identified with the space of coefficients of degree-d monic polynomials. Let NP(f mod P) denote the p-adic Newton polygon of L(f mod P;T). Let HP(A^d) denote the p-adic Hodge polygon of A^d. We prove that there is a Zariski dense open subset U defined over Q in A^d such that for every geometric point f(x) in U(Qbar) we have lim_{p-->oo} NP(f mod P) = HP(A^d), where P is any prime ideal in the ring of integers of Q(f) lying over p. This proves a conjecture of Daqing Wan.

L-functions of exponential sums over one-dimensional affinoids: Newton over Hodge

This paper proves a sharp lower bound for Newton polygons of L-functions of exponential sums of one-variable rational functions. Let p be a prime and let F¯p be the algebraic closure of the finite field of p elements. Let f¯(x) be any one-variable rational function over F¯p with ℓ poles of orders d1,…,dℓ. Suppose p is coprime to d1,…,dℓ. We prove that there exists a tight lower bound which we call Hodge polygon, depending only on the dj's, to the Newton polygon of L-function of exponential sums of f¯(x)⁠. Moreover, we show that for any f¯(x) these two polygons coincide if and only if p ≡ 1 mod dj for every 1 ≤ j ≤ ℓ. As a corollary, we obtain a tight lower bound for the p-adic Newton polygon of zeta-function of an Artin-Schreier curve given by affine equations yp−y=f¯(x)⁠.

The First Slope Case of Wan's Conjecture

Let d ≥ 2 and p a prime coprime to d. For f(x) ∈ (Zp ∩Q)[x], let NP1(f mod p) denote the first slope of the Newton polygon of the L-function of the exponential sums ∑ x∈F p` ζ TrF p` /Fp (f(x)) p . We prove that there is a Zariski dense open subset U in the space Ad of degree-d monic polynomials over Q such that for all f(x) ∈ U we have limp→∞ NP1(f mod p) = 1 d . This is a “first slope case” of a conjecture of Wan. Let d ≥ 2 be an integer and p a prime coprime to d. Let A be the set of all degreed monic polynomials over Q. For any f(x) = x+ad−1x+ . . .+a0 ∈ (Zp∩Q)[x] and for any integer ` ≥ 1 let S`(f) := ∑ x∈F p` ζ TrF p` /Fp (f(x)) p . The L function of f(x) mod p is defined by L(f mod p;T ) = exp (∑∞ `=1 S`(f) T ` ` ) . It is a theorem of Dwork-Bombieri-Grothendieck that L(f mod p;T ) = 1 + b1T + . . . + bd−1T d−1 ∈ Z[ζp][T ] for some p-th root of unity ζp in Q. Define the Newton polygon of f mod p, denoted by NP(f mod p), as the lower convex hull of the points (`, ordpb`) in R for 0 ≤ ` ≤ d − 1 where we set b0 = 1. It is exactly the p-adic Newton polygon of the polynomial L(f mod p;T ). Let NP1(f mod p) denote its first slope. Define the Hodge polygon HP(f) as the convex hull in R of the points (`, `(`+1) 2d ) for 0 ≤ ` ≤ d − 1. It is proved that the Newton polygon is always lying above the Hodge polygon ( see [3] [6] and [2]). The following conjecture was proposed by Wan in the Berkeley number theory seminar in the fall of 2000, a general form of which will appear in [7, Section 2.5]. Conjecture 1 (Wan). There is a Zariski dense open subset U in A such that for all f(x) ∈ U we have limp→∞NP(f mod p) = HP(f). The cases d = 3 and 4 are proved in [6] and [4], respectively. It is also known that if p ≡ 1 mod d then NP(f mod p) = HP(f) for all f ∈ A (see [1]). In this paper we use an elementary method to prove the “first slope case” of this conjecture. For any real number r let dre denote the least integer greater than or equal to r. For any integer N and for any Laurent polynomial g(x) in one variable, we use [g(x)]xN to denote the x -coefficient of g(x). Theorem 2. Let d ≥ 2 and p a prime coprime to d. Let f(x) be a degree-d monic polynomial in (Zp ∩Q)[x]. Suppose [ f(x)d p−1 d e ] xp−1 6≡ 0 mod p. If p > d2 + 1 then NP1(f mod p) = ⌈ p−1 d ⌉ /(p− 1). Date: February 1, 2002. 1991 Mathematics Subject Classification. 11L, 14H, 14M.