Newton Polygons of Zeta Functions and L Functions

Wan, Daqing

doi:10.2307/2946539

Cited by 68 publications

(76 citation statements)

References 7 publications

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“…Alternatively, it follows from the transfer theorem together with the π ψ -adic facial decomposition in [Wan 1993] for ψ of order p.…”

Section: And Only If the Determinant Of The Matrixmentioning

confidence: 99%

“…For n ≥ 4, it was shown in [Wan 1993; that there is an effectively computable positive integer D * ( ) depending only on such that…”

Section: Variation Of C-functions In a Familymentioning

confidence: 99%

“…Thus, it remains interesting to study exactly when f is T -adically ordinary. For this reason, in Section 6, we extend the facial decomposition theorem in [Wan 1993] to the T -adic case. Let be the convex closure in ‫ޒ‬ n of the origin and the exponents of the nonzero monomials in the Laurent polynomial f (x).…”

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confidence: 99%

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T-adic exponential sums over finite fields

Liu

Wan

2009

ANT

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We introduce T -adic exponential sums associated to a Laurent polynomial f . They interpolate all classical p m -power order exponential sums associated to f . We establish the Hodge bound for the Newton polygon of L-functions of T -adic exponential sums. This bound enables us to determine, for all m, the Newton polygons of L-functions of p m -power order exponential sums associated to an f that is ordinary for m = 1. We also study deeper properties of L-functions of T -adic exponential sums. Along the way, we discuss new open problems about the T -adic exponential sum itself.

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“…Alternatively, it follows from the transfer theorem together with the π ψ -adic facial decomposition in [Wan 1993] for ψ of order p.…”

Section: And Only If the Determinant Of The Matrixmentioning

confidence: 99%

“…For n ≥ 4, it was shown in [Wan 1993; that there is an effectively computable positive integer D * ( ) depending only on such that…”

Section: Variation Of C-functions In a Familymentioning

confidence: 99%

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T-adic exponential sums over finite fields

Liu

Wan

2009

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“…From ramification theory, one can easily see that a necessary condition for this polygon to coincide with the Hodge polygon is p ≡ 1 mod D. Adolphson and Sperber conjectured this condition is sufficient. This is true when n ≤ 3, but in higher dimensions one has to replace D by a (generally strict) multiple D * , as shown in [18], [19]. We can rewrite this result This result is known for one variable Laurent polynomials [4], [16].…”

Section: Introductionmentioning

confidence: 94%

Newton Polygons for Character Sums and Poincaré Series

Blache

2011

Int. J. Number Theory

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Abstract. 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 ∆ = ∆ 1 ⊕ ∆ 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. IntroductionIn the following, we note k a finite field with q = p a elements, and k r its degree r extension in an algebraic closure k fixed once and for all. Let x = (x 1 , . . . , x n ) be an n-tuple of indeterminates, and f (x) = i∈Z n a i x i ∈ k[x, x −1 ] be a Laurent polynomial in n variables with coefficients in k. If ψ denotes a non trivial additive character of k, let ψ r := ψ • Tr kr /k be the character induced by ψ on k r ; let χ be a multiplicative character of (k × ) n , and χ r := χ • N kr/k its extension to (k × r ) n . From f , ψ and χ, we form the character sums over each extension of kthen from these sums we define the L-functionWhen χ is trivial, we simply denote this function by L(f ; T ). From the works of Dwork and Grothendieck (cf.[8], [10]) we know that it is rational.Let us begin with a trivial χ; this is the most classical case in the literature. The first result about the L-function is due to Deligne [6, Théorème 8.4]. For a polynomial of degree d prime to p, whose higher degree form defines a nonsingular hypersurface in the projective space P n−1 , the function L ′ (f, T ) (here the sums are defined over A n , not over G n m ) has degree (d − 1) n . More generally, one can associate to the polynomial f its Newton polyhedron at infinity, which is the convex polyhedron ∆ defined in affine space R n as the convex 1991 Mathematics Subject Classification. 11M38,13A02,52B20.

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“…is a lower bound of NP(f mod P) (see [20,Propositions 2.2 and 2.3]) and that for every f ∈ A d (Q ∩Z p ) one has NP(f mod P) = HP(A d ) if and only if p ≡ 1 mod d (see [2, (3.11)]). Our Hodge polygon, which inherits that from [20,21], is defined combinatorially (so we shall refer to it as Wan's Hodge polygon in the remark below).…”

Section: Introductionmentioning

confidence: 99%

Asymptotic variation of L functions of one-variable exponential sums

Zhu¹

2004

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. Fix an integer d ≥ 3. Let A d be the dimension-d affine space over the algebraic closure Q of Q, identified with the coefficient space of degree-d monic polynomials f (x) in one variable x. For any f (x) in A d (Q), let Q(f ) be the field generated by coefficients of f in Q. For each prime p coprime to d, pick an embedding from Q to Q p , once and for all. Let P be the place in Q(f ) lying over p specified by the embedding of Q in Q p (with residue field Fq say). Suppose f ∈ A d (Q ∩ Zp), let NP(f (x) mod P) denote the q-adic Newton polygon of the L function L(f (x) mod P; T ) of exponential sums of f mod P. 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 (Q) and p large enough (depending only on f ) one has NP(f mod P) = GNP(A d ; Fp) andwhere GNP(A d ; Fp) and HP(A d ) are the generic Newton polygon and the Hodge polygon, respectively (see [23]).

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Newton Polygons of Zeta Functions and L Functions

Cited by 68 publications

References 7 publications

T-adic exponential sums over finite fields

T-adic exponential sums over finite fields

Newton Polygons for Character Sums and Poincaré Series

Asymptotic variation of L functions of one-variable exponential sums

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