p -adic variation of L functions of one variable exponential sums, I

“…The q-adic Newton polygon NP q (P (x); F q ) of this L-function is defined as the lower convex hull of the points (i, ord q (b i )) i≥0 on the (x, y)-plane. Results about this Newton polygon can be found in [16,20,21]. This polygon is independent of the choice of base field F q in F p (even though the reciprocal roots of the L-function do depend on F q ).…”

Section: Introductionmentioning

confidence: 95%

“…In fact, one can carry out calculations in the spirit of [4] to get explicitly the generic Newton polygons, and Hasse polynomials that describe exactly which polynomials attain this polygon. [20,21] and was generalized to Laurent polynomials in [12] that there is a Zariski dense open subset U in A d1,d2 defined over Q such that for every f ∈ U(Q) we have its limit of Newton polygon approaching the Hodge polygon as p → ∞. It has been fascinating researchers to know what (Laurent) polynomials f over Q that would fail the asymptotic property lim p→∞ NP(f mod P) = HP(A d1,d2 ).…”

Section: Newton Polygons For Laurent Polynomials P (X S )mentioning

confidence: 99%

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

Blache

Férard

Zhu

2008

Mathematical Research Letters

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Abstract. Let Fq be a finite field of cardinality q and characteristic p. Let P (x) be any one-variable Laurent polynomial over Fq of degree (d 1 , d 2 ) respectively and p d 1 d 2 . For any fixed s ≥ 1 coprime to p, we prove that the q-adic Newton polygon of the L-functions of exponential sums of P (x s ) has a tight lower bound which we call HodgeStickelberger polygon, depending only on the d 1 , d 2 , s and the residue class of (p mod s). This Hodge-Stickelberger polygon is a certain weighted convolution of the Hodge polygon for L-function of exponential sums of 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 ν ∈ (Z/sZ) × , we show that there exists a Zariski dense open subset Uν defined over Q such that for every Laurent polynomial P in Uν (Q) the q-adic Newton polygon of L(P (x s )/Fq; T ) converges to the Hodge-Stickelberger polygon as p approaches infinity and p ≡ ν mod 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 = P (x s ). This estimates the q-adic valuations of reciprocal roots of the numerator of the zeta function of the Artin-Schreier curve.

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“…(Note, the following theorem was first proven by Zhu [19] for the family x d + ax where a ∈ Q and assuming p is sufficiently large. )…”

Section: Frobenius Estimatesmentioning

confidence: 99%

L-functions of symmetric powers of cubic exponential sums

Haessig¹

2009

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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For each positive integer k, we investigate the L-function attached to the k-th symmetric power of the F -crystal associated to the family of cubic exponential sums of x 3 + λx where λ runs over Fp. We explore its rationality, field of definition, degree, trivial factors, functional equation, and Newton polygon. The paper is essentially self-contained, due to the remarkable and attractive nature of Dwork's p-adic theory.A novel feature of this paper is an extension of Dwork's effective decomposition theory when k < p. This allows for explicit computations in the associated p-adic cohomology. In particular, the action of Frobenius on the (primitive) cohomology spaces may be explicitly studied.

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p -adic variation of L functions of one variable exponential sums, I

Cited by 36 publications

References 16 publications

Generic T-adic exponential sums in one variable

Generic T-adic exponential sums in one variable

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

L-functions of symmetric powers of cubic exponential sums

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