We deduce Katz's theorems for $(A,B)$-exponential sums over finite fields using $\ell$-adic cohomology and a theorem of Denef-Loeser, removing the hypothesis that $A+B$ is relatively prime to the characteristic $p$. In some degenerate cases, the Betti number estimate is improved using toric decomposition and Adolphson-Sperber's bound for the degree of $L$-functions. Applying the facial decomposition theorem in \cite{W1}, we prove that the universal family of $(A,B)$-polynomials is generically ordinary for its $L$-function when $p$ is in certain arithmetic progression.
Many exponential sums over finite fields, including Gauss sums and Kloosterman sums, arise as the Fourier transform with respect to a character of the trace function of an $\ell$-adic sheaf on a commutative algebraic group. We study the equidistribution of these sums when the sheaf is fixed but the character varies over larger and larger extensions of the finite field. For the additive group, monodromy governs equidistribution by a theorem of Deligne. A few years ago, Katz solved the multiplicative variant of the question in a work where Tannakian ideas play an essential role.
The classical Kloosterman sums give rise to a Galois representation of the function field unramfied outside 0 and $\infty$. We study the local monodromy of this representation at $\infty$ using $l$-adic method based on the work of Deligne and Katz. As an application, we determine the degrees and the bad factors of the $L$-functions of the symmetric products of the above representation. Our results generalize some results of Robba obtained through $p$-adic method.
$T$-adic exponential sums associated to a Laurent polynomial $f$ are introduced. They interpolate all classical $p^m$-power order exponential sums associated to $f$. The Hodge bound for the Newton polygon of $L$-functions of $T$-adic exponential sums is established. 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 $f$ which is ordinary for $m=1$. Deeper properties of $L$-functions of $T$-adic exponential sums are also studied. Along the way, new open problems about the $T$-adic exponential sum itself ar…
The $C$-function of $T$-adic exponential sums is studeid. An explicit arithmetic bound is established for the Newton polygon of the $C$-function. This polygon lies above the Hodge polygon. It gives a sup-Hodge bound of the $C$-function of $p$-power order exponential sums.
Generic Newton polygons for L-functions of exponential sums associated to Laurent polynomials in one variable are determined. The corresponding Hasse polynomials are also determined.
We prove a vanishing theorem for the p-adic cohomology of exponential sums on affine space. In particular, we obtain new classes of exponential sums on affine space that have a single nonvanishing p-adic cohomology group. The dimension of this cohomology group equals a sum of Milnor numbers.
We give two applications of our earlier work "Exponential sums on A^n, II" (math.AG/9909009). We compute the p-adic cohomology of certain exponential sums on A^n involving a polynomial whose homogeneous component of highest degree defines a projective hypersurface with at worst weighted homogeneous isolated singularities. This study was motivated by recent work of Garcia (Exponential sums and singular hypersurfaces, Manuscripta Math., v. 97 (1998), pp. 45-58). We also compute the p-adic cohomology of certain exponential sums on A^n whose degree is divisible by the characteristic.
We prove the intersection of Igusa's Conjecture of [Igusa, J., "Lectures on forms of higher degree", Lect. math. phys., Springer-Verlag, 59 (1978)] and the Denef - Sperber Conjecture of [Denef, J. and Sperber, S., "Exponential sums mod p^n and Newton polyhedra", Bull. Belg. Math. Soc., suppl. (2001) 55-63] on nondegenerate exponential sums modulo p^m.
The twisted $T$-adic exponential sum associated to $x^{d}+λx$ is studied. If $λ\neq0,$ then an explicit arithmetic polygon is proved to be the Newton polygon of the twisted $C$-function of the T-adic exponential sum. It gives the Newton polygons of the $L$-functions of twisted $p$-power order exponential sums.
The twisted $T$-adic exponential sum associated to a polynomial in one variable is studied. An explicit arithmetic polygon is proved to be the generic Newton polygon of the twisted $C$-function of the T-adic exponential sum. It gives the generic Newton polygon of the twisted $L$-functions of $p^m$-power order exponential sums, for all $m$.
The $T$-adic exponential sum associated to a Laurent polynomial in one variable is studied. An explicit arithmetic polygon is proved to be the generic Newton polygon of the $C$-function of the T-adic exponential sum. It gives the generic Newton polygon of $L$-functions of $p$-power order exponential sums.
We give bounds for exponential sums over curves defined over Galois rings. We first define summation subsets as the images of lifts of points from affine opens of the reduced curve, and we give bounds for the degrees of their coordinate functions. Then we get bounds for exponential sums, extending results of Kumar et al., Winnie Li over the projective line, and Voloch Walker over elliptic and $C_{ab}$ curves.
The $T$-adic exponential sum of a polynomial in one variable is studied. An explicit arithmetic polygon in terms of the highest two exponents of the polynomial is proved to be a lower bound of the Newton polygon of the $C$-function of the T-adic exponential sum. This bound gives lower bounds for the Newton polygon of the $L$-function of exponential sums of $p$-power order.
The twisted $T$-adic exponential sum associated to a polynomial in one variable is studied. An explicit arithmetic polygon in terms of the highest two exponents of the polynomial is proved to be a lower bound of the Newton polygon of the $C$-function of the twisted T-adic exponential sum. This bound gives lower bounds for the Newton polygon of the $L$-function of twisted $p$-power order exponential sums.
Given a prime $p$ and an integer $d>1$, we give a numerical criterion to decide whether the $\ell$-adic sheaf associated to the one-parameter exponential sums $t\mapsto \sum_xψ(x^d+tx)$ over ${\mathbb F}_p$ has finite monodromy or not, and work out some explicit cases where this is computable.
Our previous theorems on exponential sums often did not apply or did not give sharp results when certain powers of a variable appearing in the polynomial were divisible by p. We remedy that defect in this paper by systematically applying "p-power reduction," making it possible to strengthen and extend our earlier results.
In this article, we use Robba's method to give an estimate of the Newton polygon for the L function on torus and we can draw the Newton polygon in some special cases.
We prove an upper bound on the log canonical threshold of a hypersurface that satisfies a certain power condition and use it to prove several generalizations of Igusa's conjecture on exponential sums, with the log-canonical threshold in the exponent of the estimates. We show that this covers optimally all situations of the conjectures for non-rational singularities, by comparing the log canonical threshold with a local notion of the motivic oscillation index.
In this paper we define the $p$-density of a finite subset $D\subset\ma{N}^r$, and show that it gives a good lower bound for the $p$-adic valuation of exponential sums over finite fields of characteristic $p$. We also give an application: when $r=1$, the $p$-density is the first slope of the generic Newton polygon of the family of Artin-Schreier curves associated to polynomials with their exponents in $D$.
We propose a conjecture for exponential sums which generalizes both a conjecture by Igusa and a local variant by Denef and Sperber, in particular, it is without the homogeneity condition on the polynomial in the phase, and with new predicted uniform behavior. The exponential sums have summation sets consisting of integers modulo $p^m$ lying $p$-adically close to $y$, and the proposed bounds are uniform in $p$, $y$, and $m$. We give evidence for the conjecture, by showing uniform bounds in $p$, $y$, and in some values for $m$. On the way, we prove new bounds for log-canonical thresholds which a…
In this article, we consider the estimation of exponential sums along the points of the reduction mod $p^{m}$ of a $p$-adic analytic submanifold of $ \mathbb{Z}_{p}^{n}$. More precisely, we extend Igusa's stationary phase method to this type of exponential sums. We also study the number of solutions of a polynomial congruence along the points of the reduction mod $% p^{m}$ of a $p$-adic analytic submanifold of $\mathbb{Z}_{p}^{n}$. In addition, we attach a Poincare series to these numbers, and establish its rationality. In this way, we obtain geometric bounds for the number of solutions of the…
This thesis contains some new results on the uniform rationality of Poincaré series of p-adic equivalence relations and Igusa's conjecture on exponential sums
In this article, we prove a comparison theorem between the Dwork cohomology introduced by Adolphson and Sperber and the rigid cohomology. As a corollary, we can calculate the rigid cohomology of Dwork isocrystal on torus.
Let f be a polinomial with coefficients in a finite field F. Let $Ψ: F \to C^{\ast}$ be a non-trivial additive character. In this paper we give bounds for the exponential sums $\sum_{x\in F^n} Ψ(Tr_{F/F_p} (f(x)))$ in some cases where the highest degree form of f defines a singular projective hypersurface X (e.g. when X is an arrangement of lines in P^2). The bound involves the Milnor numbers of the singularities of X. The proof goes via the classical cohomological interpretation of this exponential sums through Grothendieck's trace formula.
Gelfond and Khovanskii found a formula for the sum of the values of a Laurent polynomial at the zeros of a system of n Laurent polynomials in the complex n-torus whose Newton polyhedra have generic mutual positions. An exponential change of variables gives a similar formula for exponential sums with rational frequencies. We conjecture that this formula holds for exponential sums with real frequencies. We give an integral formula which proves the existence-part of the conjectured formula not only in the complex situation but also in a very general real setting. We also prove the conjectured for…
We prove the multivariate Fujiwara bound for exponential sums: for a $d$-variate exponential sum $f$ with scaling parameter $μ$, if $x$ is contained in the amoeba $\mathscr{A}(f)$, then the distance from $x$ to the Archimedean tropical variety associated to $f$ is at most $d \sqrt{d}\, 2\log(2 + \sqrt{3})/ μ$. If $f$ is polynomial, then the bound can be improved to $d \log(2 + \sqrt{3})$.
