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C. Douglas Haessig

Publications and source records attributed to C. Douglas Haessig.

17 recordsLinked to original sources
arXiv2026

On partial T-adic exponential sums and partial exponential sums with p-power conductor

Liu and Wan introduced T-adic exponential sums as a way to interpolate all character sums with character having p-power conductor. In this paper, we generalize their T-adic theory to partial T-adic exponential sums. We prove that the associated L-functions are T-adic meromorphic, and as a consequence, give a p-adic proof of rationality for all partial L-functions of characters with p-power conductor. We also give Newton-over-Hodge estimates.

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arXiv2026

$p$-adic Theory for Partial Toric Exponential Sums

Wan proved the rationality of partial toric $L$-functions using $\ell$-adic techniques. In this paper, we present a $p$-adic proof in the spirit of Dwork. We demonstrate that partial $L$-functions can be expressed as an alternating product of twisted Fredholm determinants. These twisted determinants appear to be intrinsic to the analytic structure of partial $L$-functions, and unlike their classical counterparts, twisted Fredholm determinants of completely continuous operators are not automatically $p$-adic entire functions. However, for partial $L$-functions they will be $p$-adic meromorphic. After proving rationality, we construct a $p$-adic cohomology theory and give a $p$-adic cohomological formula for partial toric $L$-functions. Last, we show they have a unique $p$-adic unit root which may be explicitly written in terms of $A$-hypergeometric series.

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arXiv2024

Symmetric Power L-functions of the hyper-Kloosterman Family

The symmetric power L-function of the hyper-Kloosterman family is a rational function over the integers. Its degree and complex absolute values of its zeros and poles are now known through the work of Fu and Wan. The purpose of this paper is to study the p-adic absolute value of these zeros and poles. In particular, we give a uniform lower bound, independent of the symmetric power, of the q-adic Newton polygon of this $L$-function under suitable conditions. We also give similar results for any other linear algebra operation of the hyper-Kloosterman family, such as tensor, exterior, symmetric powers, or combinations thereof.

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arXiv2022

Partial zeta functions, partial exponential sums, and p-adic estimates

Partial zeta functions of algebraic varieties over finite fields generalize the classical zeta function by allowing each variable to be defined over a possibly different extension field of a fixed finite field. Due to this extra variation their rationality is surprising, and even simple examples are delicate to compute. For instance, we give a detailed description of the partial zeta function of an affine curve where the number of unit poles varies, a property different from classical zeta functions. On the other hand, they do retain some properties similar to the classical case. To this end, we give Chevalley-Warning type bounds for partial zeta functions and L-functions associated to partial exponential sums.

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arXiv2020

Kloosterman sums and Hecke polynomials in characteristics 2 and 3

In this paper we give a modular interpretation of the $k$-th symmetric power $L$-function of the Kloosterman family of exponential sums in characteristics 2 and 3, and in the case of $p=2$ and $k$ odd give the precise 2-adic Newton polygon. We also give a $p$-adic modular interpretation of Dwork's unit root $L$-function of the Kloosterman family, and give the precise 2-adic Newton polygon when $k$ is odd. In a previous paper, we gave an estimate for the $q$-adic Newton polygon of the symmetric power $L$-function of the Kloosterman family when $p \geq 5$. We discuss how this restriction on primes was not needed, and so the results of that paper hold for all $p \geq 2$.

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arXiv2016

L-functions of symmetric powers of Kloosterman sums (unit root L-functions and p-adic estimates)

The L-function of symmetric powers of classical Kloosterman sums is a polynomial whose degree is now known, as well as the complex absolute values of the roots. In this paper, we provide estimates for the p-adic absolute values of these roots. Our method is indirect. We first develop a Dwork-type p-adic cohomology theory for the two-variable infinite symmetric power L-function associated to the Kloosterman family, and then study p-adic estimates of the eigenvalues of Frobenius. A continuity argument then provides the desired p-adic estimates.

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arXiv2015

p-adic variation of unit root L-functions

Dwork's conjecture, now proven by Wan, states that unit root L-functions "coming from geometry" are p-adic meromorphic. In this paper we study the p-adic variation of a family of unit root L-functions coming from a suitable family of toric exponential sums. In this setting, we find that the unit root L-functions each have a unique p-adic unit root. We then study the variation of this unit root over the family of unit root L-functions. Surprisingly, we find that this unit root behaves similarly to the classical case of families of exponential sums. That is, the unit root is essentially a ratio of A-hypergeometric functions.

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arXiv2015

p-adic unit roots of L-functions over finite fields

In this brief note, we consider p-adic unit roots or poles of L-functions of exponential sums defined over finite fields. In particular, we look at the number of unit roots or poles, and a congruence relation on the units. This raises a question in arithmetic mirror symmetry.

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arXiv2013

Meromorphy of the rank one unit root L-function revisited

We demonstrate that Wan's alternate description of Dwork's unit root L-function in the rank one case may be modified to give a proof of meromorphy that is classical, eliminating the need to study sequences of uniform meromorphic functions.

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arXiv2013

Families of generalized Kloosterman sums

We construct p-adic relative cohomology for a family of toric exponential sums which generalize the classical Kloosterman sums. Under natural hypotheses such as quasi-homogeneity and nondegeneracy, this cohomology is acyclic except in the top dimension. Our construction enables sufficiently sharp estimates for the action of Frobenius on cohomology so that our earlier work may be applied to the L-functions coming from linear algebra operations on these families to deduce a number of basic properties.

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arXiv2013

L-functions associated with families of toric exponential sums

We consider arbitrary algebraic families of lower order deformations of nondegenerate toric exponential sums over a finite field. We construct a relative polytope with the aid of which we define a ring of coefficients consisting of p-adic analytic functions with polyhedral growth prescribed by the relative polytope. Using this we compute relative cohomology for such families and calculate sharp estimates for the relative Frobenius map. In applications one is interested in L-functions associated with linear algebra operations (symmetric powers, tensor powers, exterior powers and combinations thereof) applied to the relative Frobenius. Using methods pioneered by Ax, Katz and Bombieri we prove estimates for the degree and total degree of the associated L-function and p-divisibility of the reciprocal zeros and poles. Similar estimates are then established for affine families and pure Archimedean weight families (in the simplicial case).

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arXiv2010

L-functions of symmetric powers of the generalized Airy family of exponential sums: ell-adic and p-adic methods

For ψa nontrivial additive character on the finite field F_q, the map t \mapsto \sum_{x \in F_q} ψ(f(x)+tx) is the Fourier transform of the map t \mapsto ψ(f(t))$. As is well-known, this has a cohomological interpretation, producing a continuous ell-adic Galois representation. This paper studies the L-function attached to the k-th symmetric power of this representation using both ell-adic and p-adic methods. Using ell-adic techniques, we give an explicit formula for the degree of this L-function and determine the complex absolute values of its roots. Using p-adic techniques, we study the p-adic absolute values of the roots.

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arXiv2008

On the zeta function of divisors for projective varieties with higher rank divisor class group

Given a projective variety X defined over a finite field, the zeta function of divisors attempts to count all irreducible, codimension one subvarieties of X, each measured by their projective degree. When the dimension of X is greater than one, this is a purely p-adic function, convergent on the open unit disk. Four conjectures are expected to hold, the first of which is p-adic meromorphic continuation to all of C_p. When the divisor class group (divisors modulo linear equivalence) of X has rank one, then all four conjectures are known to be true. In this paper, we discuss the higher rank case. In particular, we prove a p-adic meromorphic continuation theorem which applies to a large class of varieties. Examples of such varieties are projective nonsingular surfaces defined over a finite field (whose effective monoid is finitely generated) and all projective toric varieties (smooth or singular).

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arXiv2008

L-functions of symmetric powers of cubic exponential sums

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. 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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arXiv2007

On the $p$-adic meromorphy of the function field height zeta function

In this brief note, we will investigate the number of points of bounded (twisted) height in a projective variety defined over a function field, where the function field comes from a projective variety of dimension greater than or equal to 2. A first step in this investigation is to understand the $p$-adic analytic properties of the height zeta function. In particular, we will show that for a large class of projective varieties this function is $p$-adic meromorphic.

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