Exponential Sums of Witt Towers over Affinoids
Matthew Schmidt

TL;DR
This paper develops a Dwork theory for exponential sums over affinoids in Witt towers, enabling the computation of L-function degrees and analysis of Hodge and Newton polygons.
Contribution
It introduces a novel Dwork theory framework for exponential sums over affinoids in Witt towers, advancing understanding of their L-functions and polygon relations.
Findings
Computed the degree of the L-function.
Analyzed the Hodge polygon.
Identified conditions for Hodge and Newton polygons to coincide.
Abstract
In this paper we construct a Dwork theory for general exponential sums over affinoids in Witt towers. Using this, we compute the degree of the -function, its Hodge polygon and examine when the Hodge and Newton polygons coincide.
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Taxonomy
TopicsMathematical and Theoretical Analysis · History and Theory of Mathematics · Algebraic Geometry and Number Theory
Exponential Sums of Witt Towers over Affinoids
Matthew Schmidt
Department of Mathematics, SUNY Buffalo
Abstract.
In this paper we construct a Dwork theory for general exponential sums over affinoids in Witt towers. Using this, we compute the degree of the -function, its Hodge polygon and examine when the Hodge and Newton polygons coincide.
Key words and phrases:
Exponential Sums, Dwork Theory
2010 Mathematics Subject Classification:
11T23 (primary), 11L07, 13F35
Contents
Let be a prime, a -power, and let be an integer such that . Write . Suppose we have:
[TABLE]
where are distinct elements in , is the truncated ring of Witt vectors of some ring , and is the Witt vector shifting operator. Without loss of generality, take for all and suppose that for each , the maximum is uniquely achieved. We will assume that and . If is a primitive th root of unity, the exponential sum, -function and characteristic function attached to are:
[TABLE]
In this paper we build on the methods developed in [5], where we constructed a “universal” Dwork theory that applies to exponential sums over affinoids. Using the framework from there, we build an alternative dwork cohomology using a truncated Artin-Hasse exponential which allows us to directly compute the degree of the -function. To our knowledge, the general () truncated Artin-Hasse exponential has not been used in this setting before. Furthermore, we compute the Hodge polygon via the traditional Artin-Hasse exponential and use these -adic estimates to generalize Zhu’s result from [6] about when the Newton and Hodge bounds coincide.
Our main results are the following two theorems.
Theorem 0.1**.**
The power series is a polynomial in of degree
[TABLE]
This result is not new (see Remark 4.7 in [4]), but prior results use the geometry of the Witt tower whereas we utilize -adic methods. Our second result is the computation of the Hodge polygon of , which is described as follows.
Theorem 0.2**.**
Let be the -adic Newton polygon of . Then lies above the polygon with slopes:
[TABLE]
where is such that . Furthermore, if for all , these polygons coincide if and only if .
1. Lifting
We will first lift to a -adic ring in which we can construct a -adic Dwork cohomology. Recall, that if , we denote by the image of in , that is , and for , we define the Teichmüller lift of in to be .
Lemma 1.1**.**
There is a ring isomorphism:
[TABLE]
Proof.
It’s well known that via the isomorphism
[TABLE]
and so the lemma follows from the isomorphism . ∎
Define the additive character by mapping and extend it to by composing it with the trace:
[TABLE]
The exponential sum can then be written:
[TABLE]
Using Lemma 1.1 we can lift the exponential sum from an object defined over a finite field as in (4), to one defined over a -adic ring. However, before we can choose an appropriate lifting of some technical lemmas are required.
Lemma 1.2**.**
Let such that and . Then for , .
Proof.
Clearly:
[TABLE]
When , and hence
[TABLE]
Thus and the lemma follows. ∎
Lemma 1.3**.**
Suppose is such that and , some . Then for any , .
Proof.
This is just a simple calculation:
[TABLE]
∎
We now make two observations. First, because is a Teichmüller lift, and so , some . Second, because is again a Teichmüller lift from , , any . Thus, for any , applying Lemma 1.2 yields:
[TABLE]
If we take sufficiently large so that , then
[TABLE]
and so by Lemma 1.3, if then
[TABLE]
The above discussion induces a -adic lifting of that preserves the exponential sum:
Lemma 1.4**.**
If
[TABLE]
then:
[TABLE]
Proof.
If , clearly and hence:
[TABLE]
and the lemma follows. ∎
2. -adic Banach Spaces
Define a -adic affinoid ring as follows.
Definition 2.1**.**
For , let
[TABLE]
It’s easy to see that
[TABLE]
For convenience, we will often write .
Definition 2.2**.**
Consider the affinoid space:
[TABLE]
The ring is the set of overconvergent anaytic elements on . (That is, any lies in if and only if it can be evaluated at any .)
In this section we’ll study some fundamental properties of and look at some special quotient spaces. We start with a basic result, the Mittag-Leffler decomposition
Proposition 2.3**.**
If , there exists an isomorphism of -Banach modules
[TABLE]
such that every in can be written , with .
Proof.
See Lemma 2.1 in [6]. ∎
2.1. Quotient Spaces of
This section (along with the following Dwork cohomology) is based on the methods used by Lauder and Wan in their papers [1] and [2].
For convenience in this subsection, we will write . Fix an arbitrary polynomial in , with and and say . Define two operators on :
[TABLE]
where .
We wish to understand the quotient space . Namely, we are interested in its dimension as a module over . Our goal will be to show that is isomorphic to the following finite free -module :
Definition 2.4**.**
Consider the set
[TABLE]
and define the -module .
For the next lemma we will need some notational sugar:
[TABLE]
Lemma 2.5**.**
Fix and take . Let
[TABLE]
where if and if . (Take .) There is then a congruence
[TABLE]
Proof.
Observe that the action of on the terms is nothing but:
[TABLE]
So if :
[TABLE]
All the terms except have degree less than or equal to . When , then , and so if , by induction . The proof is similar for .
For , first compute that
[TABLE]
some . Hence:
[TABLE]
Because
[TABLE]
the terms in (2.1) except for have degree strictly less than , and so by the same induction argument with base case , the claim follows. ∎
For any , Lemma 2.5 implies that we can write uniquely
[TABLE]
and so it now remains to decompose in terms of :
Theorem 2.6**.**
.
Proof.
For a fixed and , we have by partial fraction expansion
[TABLE]
Hence for appropriate and , by induction on (with base case ), it follows that
[TABLE]
and so the Theorem follows because every can be written uniquely as
[TABLE]
∎
3. The Degree of the -function
We will now apply the prior section’s results about the space to construct our Dwork theory.
3.1. The truncated Artin-Hasse Exponential
Definition 3.1**.**
Let . Define the -truncated Artin-Hasse exponential:
[TABLE]
and take to be a solution to with .
By Theorem 4.1 in [3], the disk of convergence of is
[TABLE]
Furthermore, for any , by Theorem 4.9 also in [3], because , it is guaranteed that lies inside the disc of convergence of and it’s therefore well defined to consider the splitting functions
[TABLE]
Lemma 3.2**.**
If , then .
Proof.
By Remark 4.5 in [3], , and so
[TABLE]
∎
Lemma 3.3**.**
For and let . Then:
[TABLE]
and .
Proof.
A simple computation shows that:
[TABLE]
and so the identity follows by the defining property of . To compute the order of , observe that
[TABLE]
which has a unique minimum at . ∎
Lemma 3.4**.**
Let , and suppose such that . Then
[TABLE]
Proof.
The lemma follows from the following computation:
[TABLE]
∎
3.2. Dwork Theory via the Truncated Artin-Hasse
Definition 3.5**.**
Define:
[TABLE]
The reason for these definitions is clear. For , , applying the splitting function to yields:
[TABLE]
Remark*.*
Note that when is a Teichmüller lift, and using the same argument prior to Lemma 1.4 we see that the last equality above really does hold.
Similar to the above, define the analogous functions:
[TABLE]
so that
[TABLE]
Remark*.*
For each and , the coefficient of is equal to , and since by assumption and , the entire coefficient is nonzero. Thus .
From these functions define the corresponding Dwork maps and .
Proposition 3.6**.**
The maps and are -adically completely continuous.
Proof.
This proof is analogous to the corresponding proof in Corollary 6.10 in [5]. Fix , let be such that and define
[TABLE]
In this case, by Lemma 3.2. (Note that we are using here that because and hence ) The main observation to make is that all of the proofs following and including Lemma 6.5 in [5] can be generalized to the assumption that , where is any real number with . Here, we see that and the rest of the theory follows accordingly.
∎
Recall from subsection 2.1 that we defined the two operators and .
Lemma 3.7**.**
We have the following relations:
- (1)
. 2. (2)
. 3. (3)
. 4. (4)
.
Proof.
Let .
- (1)
Note that if , then and so . Hence,
[TABLE] 2. (2)
The identity is just the simple calculation:
[TABLE] 3. (3)
The equality is equivalent to the statement . By basic properties of and observing that ,
[TABLE] 4. (4)
This identity follows from the first three.
∎
Lemma 3.7 yields the commutative diagram:
[TABLE]
where .
First, we prove that is injective.
Lemma 3.8**.**
* is injective.*
Proof.
Suppose that so that
[TABLE]
which implies that , or
[TABLE]
some scalar . It remains to show that .
Recall that converges on the disk . So let and
[TABLE]
Hence
[TABLE]
taking and . But this implies that .
By a well known property of the gauss norm, we can therefore find such that , and so at this , is not defined, and . ∎
Proposition 3.9**.**
The degree of the -function is the degree of the first homology space, .
Proof.
Just as in Theorem 6.13 in [5], because is completely continuous by Proposition 3.6, we see that . A simple computation then shows that
[TABLE]
Since ,
[TABLE]
But by the relation from Lemma 3.7, (since is bijective on ). Thus:
[TABLE]
And so (7) becomes:
[TABLE]
that is and the same formula holds when is replaced with since . Thus by the well known formula
[TABLE]
we have the identity
[TABLE]
and the claim follows. ∎
Theorem 3.10**.**
.
Proof.
We apply the results of subsection 2.1 to the polynomial in definition 3.5. By Lemma 2.5, it’s clear that:
[TABLE]
where . With this computation, the degree of follows from Theorem 2.6. ∎
4. The Hodge Polygon
In this second section we utilize a more classical style Dwork theory from the Artin-Hasse exponential and compute the Hodge polygon of the -function and generalize the main result of [6]. The theory in this section is analogous to the work from [5], but we provide a detailed sketch with a focus on -adic estimates that we will use to compute the Newton polygon.
4.1. The Artin-Hasse Exponential
Definition 4.1**.**
Let . Define the Artin-Hasse exponential:
[TABLE]
and take to be a solution to with .
Define the splitting function:
[TABLE]
When there is no confusion, we will write and .
4.2. -adic Estimates
Definition 4.2**.**
Let , :
[TABLE]
Lemma 4.3**.**
For and ,
[TABLE]
with equality if and only if and (which is satisfied if ).
Proof.
We compute that:
[TABLE]
Hence,
[TABLE]
and so the lemma is clear. ∎
Lemma 4.4**.**
For and ,
[TABLE]
with equality if and only if and (which is satisfied if ).
Proof.
Observe that
[TABLE]
Then by Lemma 4.3,
[TABLE]
and this minimum is achieved when and otherwise. Furthermore, this minimum is unique, under our assumption that is the unique maximum in and obtained if and only if and . ∎
Lemma 4.5**.**
For and :
[TABLE]
and equality holds if and only if , and .
Proof.
We follow Lemma 4.4.7 in [5]. First note that:
[TABLE]
where such that . ∎
Proposition 4.6**.**
For and :
[TABLE]
with equality if and only if , and .
Proof.
Following Proposition 4.4.9 in [5] and using Lemma 4.5,
[TABLE]
where we write to ease notation. The minimum is then uniquely achieved when since . ∎
Corollary 4.7**.**
For and :
[TABLE]
with equality if and only if , and .
4.3. Computing the Hodge
Let represent the matrix for with respect to the weighted basis , with the entries of lying in . Write:
[TABLE]
so that
[TABLE]
Following the proof of Theorem 7.2 in [5], line (9) along with Corollary 4.7 yields the Hodge bound with slopes:
[TABLE]
Moreover, noting when equality holds in Corollary 4.7 implies that if this Hodge bound is obtained, then for all . Oppositely, if for all , then the Hodge bound is necessarily achieved under the assumption that , because and .
Remark* (The “Truncated” Hodge Polygon).*
If we use the Dwork theory developed in Section 3 utilizing the truncated Artin-Hasse exponential, we can derive lower bounds for the Newton polygon of , just like we would to compute the Hodge polygon in the traditional case. However, we see that the the resulting lower bound is actually lower than the Hodge bound.
Recall from Proposition 3.6 that
[TABLE]
Applying the same procedure as above yields a lower polygon consisting of slopes
[TABLE]
Interestingly, 10 implies that as , this “truncated” Hodge polygon converges upward towards the regular Hodge bound. (This is expected since the coefficients of the truncated Artin-Hasse functions converge upon the coefficients of the classical Artin-Hasse as , i.e. as .)
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] Alan Lauder and Daqing Wan: Computing zeta functions of Artin-Schreier curves over finite fields, Journal of Complexity 5 , 35–55 (2004).
- 2[2] Alan Lauder and Daqing Wan: Computing zeta functions of Artin-Schreier curves over finite fields II, Journal of Complexity 20 , 331–349 (2004).
- 3[3] Keith Conrad: Artin-Hasse-Type Series and Roots of Unity, http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/A Hrootofunity.pdf (1984).
- 4[4] Michiel Kosters and Daqing Wan: Genus growth in ℤ p subscript ℤ 𝑝 \mathbb{Z}_{p} towers of function fields, Proc. Amer. Math. Soc 146 , 1481-1494 (2018).
- 5[5] Matthew Schmidt: T 𝑇 T -adic Exponential Sums over Affinoids, https://arxiv.org/abs/1901.05516 (2018).
- 6[6] Hui June Zhu: L-Functions of Exponential Sums over One-Dimensional Affinoids, International Mathematics Research Notices 30 (2004), 1529-1550.
