L-functions of symmetric powers of Kloosterman sums (unit root L-functions and p-adic estimates)
aa r X i v : . [ m a t h . N T ] M a y L -functions of symmetric powers of Kloosterman sums(unit root L -functions and p -adic estimates) C. Douglas Haessig ∗ October 8, 2018
Abstract
The L -function of symmetric powers of classical Kloosterman sums is a polynomial whose degree is now known, aswell as the complex absolute values of the roots. In this paper, we provide estimates for the p -adic absolute values ofthese roots. Our method is indirect. We first develop a Dwork-type p -adic cohomology theory for the two-variable infinitesymmetric power L -function associated to the Kloosterman family, and then study p -adic estimates of the eigenvalues ofFrobenius. A continuity argument then provides the desired p -adic estimates. Contents
Sym ∞ ,κ -cohomology 5 H κ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 H κ with κ ∈ Z p \ Z ≥ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Frobenius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 p -adic estimates 11 Sym ∞ ,κ -cohomology (again) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 H κ and H κ when κ ∈ Z p \ Z ≥ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.3 Frobenius and estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Let F q be a finite field with q = p a elements, p ≥
5. Associated to each ¯ t ∈ F ∗ q define the Kloosterman sum Kl ¯ t,m := X ¯ x ∈ F qm ¯ t ψ ◦ T r F qm ¯ t / F q (cid:18) ¯ x + ¯ t ¯ x (cid:19) m = 1 , , , . . . , where deg (¯ t ) := [ F q (¯ t ) : F q ], q ¯ t := q deg (¯ t ) , and ψ is a fixed non-trivial additive character on F q . It is well-known that theassociated L -function is quadratic: L ( Kl ¯ t , T ) := exp X m ≥ Kl ¯ t,m T m m = (1 − π (¯ t ) T )(1 − π (¯ t ) T ) , with roots satisfying π (¯ t ) π (¯ t ) = q ¯ t , | π (¯ t ) | C = | π (¯ t ) | C = √ q ¯ t , and π (¯ t ) is a p -adic 1-unit, meaning | − π (¯ t ) | p < k a positive integer, define the k -th symmetric power L -function of the Kloosterman family by L ( Sym k Kl, T ) := Y ¯ t ∈| G m / F q | k Y m =0 − π (¯ t ) k − m π (¯ t ) m T deg (¯ t ) . ∗ This work was partially supported by a grant from the Simons Foundation ( obba [16] gave the first p -adic study of this L -function, and showed among other things that it is a polynomial definedover Z , and gave a conjectural formula for its degree. Using ℓ -adic techniques, Fu-Wan [9] proved a corrected versionof this conjecture as a special case of their study of the n -variable Kloosterman sums. In that paper, motivated by aconjecture of Gouvˆea-Mazur, they asked whether there exists a uniform quadratic lower bound for the Newton polygonof L ( Sym k Kl, T ) independent of k . Our first main result of this paper affirmatively answers their question: Theorem 1.1.
Let p ≥ . For every positive integer k , writing L ( Sym k Kl, T ) = P c m T m , then ord q c m ≥ (cid:18) − p − (cid:19) m ( m − . (1)A uniform quadratic lower bound is known [18] for the L -function of the k -th symmetric product of the first relative ℓ -adic cohomology of the Legendre family of elliptic curves E t : x = x ( x − x − t ). This L -function, definedanalogously to L ( Sym k Kl, T ), equals the nontrivial part of the Hecke polynomial associated to the p -th Hecke operator T k +2 ( p ) for level 2 acting on cusp forms of weight k + 2 (see [1]). For the Kloosterman family, recent work of Yun [21],based on conjectures of Evans [3], gives an automorphic interpretation for L ( Sym k Kl, T ) when k is small. It has alsobeen shown by Fu-Wan [10] that L ( Sym k Kl, T ) is geometric (or motivic) in nature, meaning it equals the local factorat p of the zeta function of a (virtual) scheme of finite type over Z .Our motivation for the following study comes from a related but different direction. In [7], Dwork first defined theunit root L -function of the Legendre family of elliptic curves essentially as follows. Setting X := A \ { , , H ( t ) = 0 } ,where H ( t ) is the Hasse polynomial, the map f : E t t ∈ X gives a family of ordinary elliptic curves whose first relative p -adic ´etale cohomology R f ∗ Z p gives a continuous rank one representation ρ E : π arith1 ( X ) → GL ( Z p ). This has theproperty that, for a closed point t ∈ | X/ F q | , the image of the geometric Frobenius F rob t is ρ E ( F rob t ) = π ( t ), where π ( t ) is the unique p -adic unit root of the zeta function of the fiber E t . For k a positive integer, define the unit root L -function L ( ρ ⊗ kE , T ) := Y ¯ t ∈| X/ F q | − π (¯ t ) k T deg (¯ t ) . For every k , meromorphy was shown by Dwork in [7]. It also has a p -adic modular interpretation. Define the Fredholmdeterminant D ( k, T ) := det ( I − U p T | M k ), where M k is the space of overconvergent p -adic modular forms of level 2 andweight k , and U p is the Atkin operator. Then there is the relation: L ( ρ ⊗ kE , T ) = D ( k + 2 , T ) D ( k, pT ) . (2)This allows one to obtain results about modular forms from results about the unit root L -function. See [19] for a detailedexposition; see also [18]. Their special values L ( ρ ⊗ kE ,
1) are also related to geometric Iwasawa theory as discussed in [4].Unit root L -functions do not appear to have a nice cohomology theory, however a related L -function does appear tohave one, which is essentially the main result of this paper. To motivate, notice that using (2) recursively: D ( k, T ) = Y i ≥ L unit ( k − − i, q i T )= Y ¯ t ∈| X/ F q | Y i ≥ (1 − π (¯ t ) k − − i π (¯ t ) i T deg (¯ t ) ) − . The latter product is similar to the k − i ≥ ≤ i ≤ k . This motivates us to define the infinite k -symmetric power L -function of the Legendre family E by L ( Sym ∞ ,k E, T ) := Y ¯ t ∈| X/ F q | Y i ≥ (1 − π (¯ t ) k − − i π (¯ t ) i T deg (¯ t ) ) − , and thus the above relation becomes det ( I − U p T | M k ) = L ( Sym ∞ ,k − E, T ) . (3)(This relation ultimately comes from the Eichler-Selberg trace formula.) We conjecture that there is a p -adic cohomologytheory for L ( Sym ∞ ,k E, T ) that lines up with (3). Furthermore, this cohomology theory should have a natural dualtheory. One possible approach is to take a p -adic limit in k of Adolphson’s work [1]. This is likely related to p -adicmodular forms and p -adic modular symbols.Our main reason to suspect the existence of such a p -adic cohomology theory comes from the Kloosterman familystudied in this paper. Let κ ∈ Z p , where Z p denotes the p -adic integers. Define the infinite κ -symmetric power L -function(using the roots π and π from the Kloosterman sums above) L ( Sym ∞ ,κ Kl, T ) := Y ¯ t ∈| G m / F q | Y m ≥ − π (¯ t ) κ − m π (¯ t ) m T deg (¯ t ) . In this paper, we develop a p -adic cohomology theory H • κ (Section 3) for the infinite κ -symmetric power L -function whichmay be used to meromorphically describe the L -function: L ( Sym ∞ ,κ Kl, T ) = det (1 − ¯ β κ T | H κ ) det (1 − q ¯ β κ T | H κ ) , here ¯ β κ is a completely continuous operator defined on p -adic spaces H κ and H κ . We note that these types of L -functions are not expected to be rational functions in general. For the Kloosterman family, we will show H κ = 0 when κ = 0, and of dimension one when κ = 0, and when κ ∈ Z p \ Z ≥ then H κ is infinite dimensional. It is interesting thatthe case κ ∈ Z p \ Z ≥ is the easiest to handle, whereas we are unable to compute cohomology when κ is a positive integer.Studying the action of Frobenius on cohomology leads to our third main result: Theorem 1.2.
Let p ≥ . For every κ ∈ Z p , L ( Sym ∞ ,κ Kl, T ) is p -adic entire with T = 1 a root. Furthermore, writing L ( Sym ∞ ,κ Kl, T ) = P m ≥ c m T m ∈ T Z p [[ T ]] , then ord q c m satisfies (1). The proof of Theorem 1.1 now follows from the following identity: for k a positive integer, L ( Sym k Kl, T ) = L ( Sym ∞ ,k Kl, T ) L ( Sym ∞ , − ( k +2) Kl, q k +1 T ) . (4)Some heuristic calculations suggest that there is a chance the lower bound in Theorem 1.2 (and thus Theorem 1.1)may be improved to simply m ( m − m .We conjecture that the zeros and poles of L ( Sym ∞ ,κ Kl, T ) are all simple except for possibly finitely many, and thatadjoining the collection of zeros and poles to Q p produces a finite extension field of Q p (the so-called p -adic Riemannhypothesis).We feel strongly that a relation such as (3) exists for L ( Sym ∞ ,k Kl, T ). As the Kloosterman sums are defined overthe totally real field Q ( ζ p + ζ − p ), one could look at p -adic Hilbert modular forms for such a relation. However, sinceKloosterman sums depend on the embedding of the character ψ , a relation would involve instead the more complicated L -function L ( ⊗ σ Sym ∞ ,κ Kl σ , T ), where σ runs over the real embeddings of Q ( ζ p + ζ − p ). Further, running over differentsymmetric powers for each embedding, the L -function L ( Sym ∞ ,κ Kl σ ⊗ · · · ⊗ Sym ∞ ,κ ( p − / Kl σ ( p − / , T )is likely related to non-parallel weight ( κ , . . . , κ ( p − / ) p -adic Hilbert modular forms.The Kloosterman unit root L -function is defined by L unit ( Kl, κ, T ) := Y ¯ t ∈| G m / F q | − π (¯ t ) κ T deg (¯ t ) , (5)and note that L unit ( Kl, κ, T ) = L ( Sym ∞ ,κ Kl, T ) L ( Sym ∞ ,κ − Kl, qT ) , (6)and so we have the cohomological description of the unit root L -function: L unit ( Kl, κ, T ) = det (1 − ¯ β κ T | H κ ) det (1 − q ¯ β κ − T | H κ ) (when κ = 0 or 2) . This shows the unit root L -function has a root at T = 1 and a pole at T = 1 /q . It is unclear whether there are anycancellations among the remaining zeros and poles, however, we expect that there are few if any. We note that whileArtin’s conjecture does not carry over to geometric p -adic representations, it does for infinite symmetric powers overcurves. See [20] for more details.The p -adic cohomology theory developed here is of de Rham type, and may be seen as an extension of Dwork’s p -adiccohomology theory. We thus expect techniques from Dwork’s classical theory may be carried over to this theory. Forexample, a dual theory seems possible, perhaps giving rise to possible symmetry? Another example is studying thevariation of a family of unit root L -functions. In joint work with Steven Sperber [15], we examine how the unit roots ofa family (of unit root L -functions) vary with respect to the parameter by means of establishing a dual theory for infinitesymmetric power L -functions.While we have restricted our study to the case of the one-variable Kloosterman family, the cohomology theorydeveloped here may be used for other families, such as those studied in [13] and [14]. We hope to say more about theircohomology in a future article. Attached to the Kloosterman family are the relative cohomology spaces H t ( b ′ , b ) and H t ( b ′ , b ) defined below. Thesubscript t is meant to remind us that we will be viewing the Kloosterman family x + tx with t as a parameter, andthus H t and H t will be modules over function spaces in t . In this section, we will recall Dwork’s “Bessel cohomology”construction [6, Section 2] of H t and H t but modified according to [14]. In particular, we will see that H t = 0 and H t is a free module of rank 2 over a certain power series ring L ( b ′ ). We then study the action of Frobenius on H t . In thenext section, we will take, in an appropriate sense, the infinite symmetric power of H t and define a cohomology theoryon it.Throughout this paper we fix a prime p ≥ π ∈ Q p satisfying π p − = − p , and for convenience, set Ω := Q p ( π ). Set ˜ b := ( p − /p . Throughout the following, let b and b ′ be real numbers satisfying:˜ b ≥ b > / ( p −
1) and b ≥ b ′ , and set ε := b − p − . (7) work’s theory works best when the spaces are tailored to suit the family in hand. Observe that we may writeexp π ( x + t q x ) = P n ≥ ,u ∈ Z A ( n, u ) t n · t qm ( u ) x u , where m ( u ) := max {− u, } . This guides us in the following definition ofthe space K q ( b ′ , b ) below. Let ρ ∈ R . Define the following spaces: L ( b ′ ; ρ ) := X n ≥ A ( n ) t n | A ( n ) ∈ Ω , ord p A ( n ) ≥ b ′ n + ρ L ( b ′ ) := [ ρ ∈ R L ( b ′ ; ρ ) K q ( b ′ , b ; ρ ) := X n ≥ ,u ∈ Z A ( n, u ) t n · t qm ( u ) x u | A ( n, u ) ∈ Ω , ord p A ( n, u ) ≥ b ′ n + b | u | + ρ K q ( b ′ , b ) := [ ρ ∈ R K q ( b ′ , b ; ρ ) , where q = p a with a ≥
0. Note that K q ( b ′ /q, b ) is an L ( b ′ /q )-module. When a = 0, we will often write K ( b ′ , b ) for K ( b ′ , b ). Define the (twisted) relative boundary operator D t q : = x ∂∂x + π (cid:18) x − t q x (cid:19) = e − π ( x + t q /x ) ◦ x ∂∂x ◦ e π ( x + t q /x ) , which acts on K q ( b ′ /q, b ), and thus defines the cohomology spaces H t q ( b ′ /q, b ) := ker ( D t q | K q ( b ′ /q, b )) and H t q ( b ′ /q, b ) := K q ( b ′ /q, b ) /D t q K q ( b ′ /q, b ) . Define V q ( b ′ , b ; ρ ) := (cid:18) Ω[[ t ]] + Ω[[ t ]] · t q x (cid:19) ∩ K q ( b ′ , b ; ρ ) V q ( b ′ , b ) := [ ρ ∈ R V q ( b ′ , b ; ρ ) . When a = 0, we will write V ( b ′ , b ) for V ( b ′ , b ). Theorem 2.1 (Theorem 2.1 of [6]) . We have H t q ( b ′ /q, b ) = 0 and H t q ( b ′ /q, b ) ∼ = V q ( b ′ /q, b ) . More specifically, K q ( b ′ /q, b ; 0) = V q ( b ′ /q, b ; 0) ⊕ D t q K q ( b ′ /q, b ; ε ) . Proof.
This is [6, Theorem 2.1] but modified slightly to suit the spaces defined above. Dwork does not explicitly pointout that ker D t q = 0, however, this follows immediately from [6, Lemma 2.5].As a consequence, we from now on will identify the first cohomology group H t q ( b ′ /q, b ) with V q ( b ′ /q, b ), a free L ( b ′ /q )-module of rank two with basis { , πt q /x } . We now study the action of Frobenius on this space. Relative Frobenius.
Dwork’s Frobenius, denoted α a below, is essentially the geometric Frobenius map ψ x : x x /q but twisted in a similar manner to that of D t . It is defined as follows. First, define Dwork’s splitting function θ ( z ) := exp π ( z − z p ) = X i ≥ θ i z i , where it is well-known that ord p θ i ≥ ( p − i/p . The splitting function gives a p -adic analytic representation of anadditive character on F q : specifically, θ (1) is a primitive p -th root of unity thanks to the oddness of p -adic analysis, andfor ¯ z ∈ F q with q = p a , and ˆ z the Teichm¨uller lift of ¯ z , θ (1) Tr F q/ F p (¯ z ) = θ (ˆ z ) θ (ˆ z p ) · · · θ (ˆ z p a − ) . See [8, Prop. 6.2] for discussion of this splitting function. We now consider the p -adic analytic analogue of θ (1) Tr F q/ F p ( x + tx ) = θ (1) Tr ( x ) θ (1) Tr ( t/x ) : for each m ≥
1, define F ( t, x ) := θ ( x ) θ ( t/x ) F m ( t, x ) := m − Y i =0 F ( t p i , x p i ) . efine the operator ψ x : P A u x u P A pu x u , and observe that ψ x : K ( b ′ , b ; 0) → K ( b ′ , pb ; 0). Dwork’s Frobenius isdefined by α ( t ) : = ψ x ◦ F ( t, x )= e − π ( x + t/x ) ◦ ψ x ◦ e π ( x + t/x ) , and for m ≥ α m ( t ) : = ψ mx ◦ F m ( t, x )= α ( t p m − ) ◦ · · · α ( t p ) ◦ α ( t ) . In this definition F ( t, x ) acts via multiplication. Now, since F ( t, x ) ∈ K (˜ b/p, ˜ b/p ; 0) we see that F ( t p i , x p i ) ∈ K (˜ b/p i +1 , ˜ b/p i +1 ; 0),and thus we have the well-defined map α m ( t ) : K ( b ′ , b ; 0) → K p m ( b ′ /p m , b ; 0). Furthermore, by construction, p m D t pm ◦ α m = α m ◦ D t , and so α m induces a map on relative cohomology ¯ α m ( t ) : H t ( b ′ , b ) → H t pm ( b ′ /p m , b ).The next theorem provides details on the entries of the matrix of ¯ α m ( t ) with respect to the bases { , πt/x } and { , πt p m /x } . These entries will consist of power series in t whose specific growth conditions will help us give a well-defined Frobenius map on the infinite symmetric power spaces of H t given in Section 3.3. Theorem 2.2. [6, Theorem 2.2 and Lemma 3.2] With { , πt/x } and { , πt p m /x } as bases of H t (˜ b, ˜ b ) and H t pm (˜ b/p m , ˜ b ) ,respectively, the relative Frobenius ¯ α m satisfies ¯ α m ( t )(1) = A m, ( t ) + A m, ( t ) πt p m x and ¯ α m ( t )( πtx ) = A m, ( t ) + A m, ( t ) πt p m x , where A m, ∈ L (˜ b/p m ; 0) A m, ∈ L (˜ b/p m ; 1 p − − ˜ bp ) A m, ∈ L (˜ b/p m ; ˜ b − p − A m, ∈ L (˜ b/p m ; ˜ b − ˜ bp ) (8) Furthermore, A m, (0) = 1 A m, (0) = 0 A m, (0) = 0 A m, (0) = p m . Sym ∞ ,κ -cohomology We now consider the meaning of taking the infinite symmetric power of the space H t from the previous section and howto construct a cohomology on it. Later, in Section 3.3 we will define and study a Frobenius map on these spaces. Fornow, we continue with the same restrictions (7) on p , b , and b ′ from the beginning of Section 2.Let κ ∈ Z p . Denote by Ω[[ t, w ]] the formal power series ring in the t and w . Intuitively, we will think of Ω[[ t, w ]] asthe κ -symmetric power of the space H t with basis { , πt/x } by writing w for the basis vector πt/x , and 1 for the otherbasis vector 1. Then we should intuitively view the monomial w m ∈ Ω[[ t, w ]] as the κ -symmetric power vector 1 κ − m w m .This intuition will guide us through our definitions below.In order to avoid certain denominators in κ , we will use a normalization w ( m ) of the monomial w m . This is definedas follows. First, define the falling factorial κ m : for κ ∈ Z p \ Z ≥ and m ≥
0, the definition is conventional: κ m := κ ( κ − · · · ( κ − m + 1) , where κ := 1. For κ = k ∈ Z ≥ , define k m := m = 0 k ( k − · · · ( k − m + 1) if 0 < m ≤ kk ( k − · · · · · ˆ0 · ( − · ( − · · · ( k − m + 1) if m ≥ k + 1 , where ˆ0 means it is counted in the m terms of the product, but omitted from the actual product. For example, k k = k k +1 = k ! and k k +2 = k ! · ( − := 1, 0 := ˆ0 = 1, and 0 := −
1. We now define w ( m ) := κ m w m . A morenatural choice of basis (due to the definition of [¯ α a ] κ below) is (cid:0) κm (cid:1) w m ; however, this leads to a non-integral cohomologytheory, which in turn makes obtaining p -adic estimates difficult. As in the previous section, we need to limit our spaceto one with specific growth conditions. That is, define the subspaces of Ω[[ t, w ]]: S ( b ′ , ε ; ρ ) := X n,m ≥ A ( n, m ) t n w ( m ) | A ( n, m ) ∈ Ω , ord p A ( n, m ) ≥ b ′ n + εm + ρ S ( b ′ , ε ) := [ ρ ∈ R S ( b ′ , b ; ρ ) . e now define a boundary map on this space by taking the infinite symmetric power of the Gauss-Manin connection.Recall, the Gauss-Manin connection for the Kloosterman family is the following. With ∂ : = t ∂∂t + πtx = e − π ( x + t/x ) ◦ t ∂∂t ◦ e π ( x + t/x ) , observe that ∂ and D t commute, and thus ∂ acts on the relative cohomology space H t . Its action may be computedexplicitly: since ∂ (1) = πtx and ∂ ( πtx ) = π t, (9)we see that on H t with basis { , πt/x } the map ∂ takes the matrix form (acting on row vectors) ∂ = t ddt + H, where H := (cid:18) π t (cid:19) . (10)This is the Gauss-Manin connection of the Kloosterman family. (The solutions of this family at 0 and ∞ are studied indetail by Dwork [6].)As mentioned earlier, if we intuitively set w = πt/x then we may write (9) as ∂ (1) = w and ∂ ( w ) = π t . Viewing w m as 1 κ − m w m suggests we may define the κ -symmetric power of ∂ on Ω[[ t, w ]] by the product rule: ∂ κ ( w m ) = “ ∂ (1 κ − m w m ) ”= ( κ − m ) ∂ (1) w m + mw m − ∂ ( w )= ( κ − m ) w m +1 + mπ tw m − . Adding in t ddt finishes the definition: ∂ κ ( t n w m ) := nt n w m + ( κ − m ) t n w m +1 + mπ t n +1 w m − . Since we are using the normalized basis { w ( m ) } , which modifies the coefficients, we restate the definition. Define theboundary map ∂ κ : S ( b ′ , ε ) → S ( b ′ , ε ) as follows. Let m ≥
0. For κ ∈ Z p \ Z ≥ , define ∂ κ ( t n w ( m ) ) := nt n w ( m ) + t n w ( m +1) + m ( κ − m + 1) π t n +1 w ( m − . (11)Note that ∂ κ S ( b ′ , ε ; ε ) ⊂ S ( b ′ , ε ; 0). For κ = k ∈ Z ≥ , define ∂ k as follows. For 0 ≤ m ≤ k − m ≥ k + 2, then ∂ k ( t n w ( m ) ) is defined by (11). When m = k or m = k + 1, define ∂ k ( t n w ( k ) ) := nt n w ( k ) + kπ t n +1 w ( k − ∂ k ( t n w ( k +1) ) := nt n w ( k +1) + t n w ( k +2) + ( k + 1) π t n +1 w ( k ) . Define the cohomology spaces H κ ( S ( b ′ , ε )) := ker ( ∂ κ ) and H κ ( S ( b ′ , ε )) := S ( b ′ , ε ) /∂ κ S ( b ′ , ε ) . It is useful to have a matrix version of ∂ κ , especially when showing H κ = 0 for almost all values of κ (Theorem 3.1).To do this, we begin by writing the basis vectors 1 and πt/x in row vector form (1 ,
0) and (0 ,
1) so that (1 , H = (0 , , H = ( π t, H (1) = w and H ( πt/x ) = π t . This defines thederivation (or Leibniz rule): L κ,H ( w m ) := ( κ − m ) H (1) w m + mw m − H ( w )= ( κ − m ) w m +1 + mπ tw m − . (12)In terms of the normalized basis this means: for κ ∈ Z p \ Z ≥ , L κ,H ( w ( m ) ) = t n w ( m +1) + m ( κ − m + 1) π t n +1 w ( m − . Thus, ∂ κ on S ( b ′ , ε ) takes the form ∂ κ = t ddt + L κ,H , an infinite differential system, where the matrix of L κ,H , acting on row vectors, takes the following form. For κ ∈ Z p \ Z ≥ :matrix of L κ,H = κπ t κ − π t κ − π t . or κ = k ∈ Z > , the matrix takes the form · kπ t · ( k − π t · ( k − π t k − · π t ↓ k · · π t k + 1) · ˆ0 π t k + 2) · ( − π t . where ˆ0 means it is omitted. Note the zero in the ( k + 2) column. For κ = 0, we havematrix of L ,H = · ˆ0 π t · ( − π t H κ It follows almost immediately from this matrix description above that H κ = 0 for all κ ∈ Z p \ Z ≥ (see Theorem 3.1below). This also holds when κ is a positive integer, but it is not immediate. When κ a positive integer, then the infinitedifferential system breaks up into two systems, one finite and one infinite. The finite system will be precisely the k -thsymmetric power of the Gauss-Manin connection ∂ = t ddt + H . A study of solutions of this latter system uses Robba’swork on the symmetric powers of the Kloosterman family, and ultimately a deep result of Dwork’s. That this finitesystem exists is interesting, and is likely analogously related to how classical modular forms of weight k sit inside thespace of p -adic modular forms of weight k . We also note that in the next section, when studying H κ , we will encounterthe same issue, that things are easier when κ is not a positive integer. Theorem 3.1.
Let κ ∈ Z p . Then H κ ( S ( b ′ , ε )) = ( if κ ∈ Z p \ { } Ω if κ = 0 . Proof.
We first suppose κ ∈ Z p \ Z ≥ . Let ξ ∈ S ( b ′ , ε ) be such that ∂ κ ξ = 0. Writing ξ = P n ≥ ξ n w ( n ) , then ∂ κ ξ = 0takes the form tξ ′ + κπ tξ = 0 tξ ′ + ξ + 2( κ − π tξ = 0 (13) tξ ′ + ξ + 3( κ − π tξ = 0... ... ...We will show t m | ξ n for every m ≥ n ≥ ξ = 0. Observe that the second equation of(13) implies t | ξ , and the third equation implies t | ξ , and so forth: t | ξ n for n ≥
0. Next, suppose t m | ξ n for every n ≥
0. The first equation of (13) then implies that t m +1 | ξ . Using this, the second equation of (13) shows t m +1 | ξ .Continuing, we see that t m +1 | ξ n for every n ≥
0. Hence, ξ = 0 as desired.Suppose now that κ ∈ Z ≥ . For convenience, set k := κ . We first observe that ∂ κ ξ = 0 breaks up into two systems ofthe form tξ ′ + kπ tξ = 0 tξ ′ + ξ + 2( k − π tξ = 0 (14) tξ ′ + ξ + 3( k − π tξ = 0... ... ... tξ ′ k +1 + ξ k + ( k + 1) π tξ k +2 = 0and tξ ′ k +2 + ( k + 2) · ( − π tξ k +3 = 0 (15) tξ ′ k +3 + ξ k +2 + ( k + 3) · ( − π tξ k +4 = 0 tξ ′ k +4 + ξ k +3 + ( k + 4) · ( − π tξ k +5 = 0... ... ... s (15) is of a form essentially identical to (13), a similar argument shows ξ m = 0 for every m ≥ k + 2. Thus, (14) takesthe form of the (finite dimensional) differential system t ddt + H k , where H k is the ( k + 1) × ( k + 1) matrix H k = kπ t k − π t k − π t kπ t This is precisely the k -th symmetric power of the differential system (10). Using a deep result of Dwork’s, Robba [16,p.202] shows this system has no overconvergent solutions.Lastly, set κ = 0, then ∂ κ ξ = 0 takes the form tξ ′ + π tξ = 0 tξ ′ + 2( − π tξ = 0 (16) tξ ′ + ξ + 3( − π tξ = 0 tξ ′ + ξ + 4( − π tξ = 0... ... ...Ignoring the first equation, this system is of a similar form to (13), and so a similar argument shows ξ m = 0 for every m ≥
1. Consequently, the first equation now becomes ξ ′ = 0, and so ξ is a constant, finishing the proof. H κ with κ ∈ Z p \ Z ≥ Next we study the first cohomology group H κ , assuming throughout this section that κ ∈ Z p \ Z ≥ . We have been unableso far to handle the case when κ is a positive integer due to obstacles trying to obtain a result similar to Lemma 3.3.Our goal of this section is to prove a similar decomposition result for S ( b ′ , ε ) to that of K ( b ′ , b ) in Theorem 2.1.We start by first noticing that L ( b ′ ; ρ ) and L ( b ′ ) sit naturally in S ( b ′ , ε ) by sending ξ ξ , the series with no w ( m ) terms. We will denote by R ( b ′ ) and R ( b ′ ; ρ ) the images of these spaces in S ( b ′ , ε ). We will show that H κ ( S ( b ′ , ε )) ∼ = R ( b ′ ). Lemma 3.2.
Let κ ∈ Z p \ Z ≥ . Then R ( b ′ ) ∩ ∂ κ S ( b ′ , ε ) = { } .Proof. Let η ∈ R ( b ′ ) ∩ ∂ κ S ( b ′ , ε ). Write η = P n ≥ a n t n , and let ξ ∈ S ( b ′ , ε ; ρ ) such that ∂ κ ξ = η . Write ξ = P n,m ≥ A ( n, m ) t n w ( m ) with ord p A ( n, m ) ≥ b ′ n + εm + ρ for some ρ ∈ R . Set ξ m := P n ≥ A ( n, m ) t n . Using this, ξ = P m ≥ ξ m w ( m ) . Using the { w ( m ) } m ≥ as a basis of S ( b ′ , ε ) as an L ( b ′ )-module, ξ takes the vector form ( ξ , ξ , . . . ),and η takes the form ( P a n t n , , , . . . ). Writing ∂ κ = t ddt + L κ,H then ∂ κ ξ = η is equivalent to the system P n ≥ a n t n = tξ ′ + κπ tξ tξ ′ + ξ + 2( κ − π tξ tξ ′ + ξ + 3( κ − π tξ ...We will show by induction that t n | η for every n . Observe that the right-hand side of the first equation of the system isdivisible by t , and thus a = 0, or equivalently, t divides η . Similarly, the second equation shows t divides ξ . Continuing,we get that t divides every ξ i for all i ≥ κπ tξ in the first equation is now divisible by t , the coefficient of t in ξ must equal a . Using this, fromthe second equation and the fact that 2( κ − π tξ is divisible by t , the coefficient of t of ξ must equal − a . The sameargument using the third equation shows the coefficient of t in ξ equals a . Continuing the argument, we must have ξ m = ( − m a t + O ( t ) for every m ≥
0, and so A (1 , m ) = ( − m a . As ord p A (1 , m ) ≥ b ′ + εm + ρ for every m , itmust be that a = 0.Assume now that t n divides η and every ξ m . We proceed with an identical argument as above to show a n = 0. As κπ tξ in the first equation is now divisible by t n +1 , the coefficient of t n in ξ must equal (1 /n ) a n . Using this, from thesecond equation and the fact that 2( κ − π tξ is divisible by t n +1 , the coefficient of t n of ξ must equal − (1 /n ) a n .The same argument using the third equation shows the coefficient of t n in ξ equals (1 /n ) a n . Continuing the argument,we must have ξ m = ( − m (1 /n m +1 ) a n t n + O ( t n +1 ) for every m ≥
0. This means ord p ( n m +1 a n ) ≥ b ′ n + εm + ρ forevery m , which is impossible unless a n = 0. Thus, η = 0. Lemma 3.3.
Let κ ∈ Z p \ Z ≥ . Then S ( b ′ , ε ; 0) = R ( b ′ ; 0) ⊕ L κ,H S ( b ′ , ε ; ε ) . (17) roof. We first observe that the righthand side of (17) is contained in the left. We now show the reverse direction. First,observe that w ( m +1) = − m ( κ − m + 1) π tw ( m − + L κ,H w ( m ) . Using this recursively, it follows that:for m ≥ w (2 m +1) = L κ,H m X l =0 ζ m +1 ,l π l t l w (2( m − l )) ! for m ≥ w (2 m ) = η m, π m t m + L κ,H m − X l =0 ζ m,l π l t l w (2( m − l ) − ! , where η and ζ l are elements in Z p . The result follows easily from this. For future reference we record: η m, = 2 m ( κ/ m ( − / m ζ m,l := 2 l ( m −
12 ) l ( − κ m + 1) l ζ m +1 ,l := 2 l ( m ) l ( κ + 12 + m ) l . Theorem 3.4.
Let κ ∈ Z p \ Z ≥ . Then S ( b ′ , ε ; 0) = R ( b ′ ; 0) ⊕ ∂ κ S ( b ′ , ε ; ε ) . (18) Hence, H κ ( S ( b ′ , ε )) ∼ = R ( b ′ ) .Proof. First, observe that the righthand side of (18) is contained in the left. Let ξ ∈ S ( b ′ , ε ; 0). By Lemma 3.3, thereexists η ∈ R ( b ′ ; 0) and ζ ∈ S ( b ′ , ε ; ε ) such that ξ = η + L κ,H ζ . Setting ξ := − t ddt ζ , then ξ = η + ∂ κ ζ + ξ . As ξ ∈ S ( b ′ , ε ; ε ), which is an increase by ε in valuation, we may repeat this process to obtain ξ = P m ≥ η m + ∂ κ P m ≥ ζ m ,with η m ∈ S ( b ′ , ε ; εm ) and ζ m ∈ S ( b ′ , ε ; ε ( m + 1)). Now that we have completed our study of the cohomology spaces H κ and H κ , we move on to the study of the actionof Frobenius. First we need to make sense of taking the (infinite) κ -symmetric power of the relative Frobenius α a on S ( b ′ , ε ). As before, intuitively, if we view w m as the κ -symmetric power vector 1 κ − m w m , then the κ -symmetric powerof α a should act on this by ( α a (1)) κ − m · α a ( w ). Thus, we need to make sure ( α a (1)) κ − m is well-defined. Furthermore,we need to show such a map is well-defined on the space S ( b ′ , ε ). Unfortunately, it is not, unless we make one furtherrestriction on b . This is done now.We first show that it makes sense to write ( α a (1)) κ − m . By Theorem 2.2, ¯ α a (1) ∈ L ( b ′ /q ; 0) + L ( b ′ /q ; ε ) πt q x . Further-more, ¯ α a (1) = 1 + η + ζ πt q x with η ∈ L ( b ′ /q ; 0), t | η , and ζ ∈ L ( b ′ /q ; ε ). Define Υ q : H t q ( K q ( b ′ /q, b )) → Ω[[ t, w ]] bysending ζ + ξ πt q x ζ + ξw . Thus, for τ ∈ Z p , (Υ q ◦ ¯ α a (1)) τ is a well-defined element of Ω[[ t, w ]].We now define the κ -symmetric power of α a as follows: define [¯ α a ] κ : S ( b ′ , ε ) → S ( b ′ /q, ε ) by linearly extending overΩ[[ t ]] the action [¯ α a ] κ ( w ( m ) ) := κ m · (Υ q ◦ ¯ α a (1)) κ − m (Υ q ◦ ¯ α a πtx ) m . Due to the weighted basis { w ( m ) } , it is not immediate that this is well-defined, and in fact may not be without furtherconditions on b . We state this now. Recall, ˜ b = ( p − /p . Assume that˜ b − p − ≥ b > p − b ≥ b ′ . (19)Note, p ≥ b > / ( p −
1) and so (19) is not vacuous.
Lemma 3.5.
Assuming (19) then [¯ α a ] κ : S ( b ′ , ε ) → S ( b ′ /q, ε ) is well-defined.Proof. We now make use of the estimates from Theorem 2.2. Using the notation from that theorem, set ˜ A a, := A a, /A a, .Then [¯ α a ] κ ( w ( m ) ) = κ m ( A a, + A a, w ) κ − m ( A a, + A a, w ) m = κ m A κ − ma, (1 + ˜ A a, w ) κ − m ( A a, + A a, w ) m = κ m A κ − ma, ∞ X l =0 κ − ml ! ˜ A la, w l ! m X n =0 mn ! A m − na, A na, w n ! = X l ≥ , ≤ n ≤ m C ( l, n ) w ( l + n ) , here C ( l, n ) := A κ − ma, ˜ A la, A m − na, A na, mn ! κ − ml ! κ m κ l + n . Observe that mn ! κ − ml ! κ m κ l + n ∈ l ! Z p , and so its p -adic valuation is at least − l/ ( p − A κ − ma, ˜ A la, A m − na, A na, ∈ L ( b ′ /q ; c ), where c := l (˜ b − p − m − n )( 1 p − − ˜ bp ) + n (˜ b − ˜ bp ) . By the hypothesis on b and b ′ in (19), it follows that C ( l, n ) ∈ L ( b ′ /q ; ρ l,n ), where ρ l,n = ( l + n ) ε + n (˜ b − b ) + m ( 1 p − − ˜ bp ) . We may now define the Frobenius operator β κ . First, define the (geometric Frobenius) operator ψ t : S ( b ′ /p, ε ) → S ( b ′ , ε ) by ψ t : X n,m ≥ A ( n, m ) t n w ( m ) X n,m ≥ A ( pn, m ) t n w ( m ) , and set β κ := ψ at ◦ [¯ α a ] κ : S ( b ′ , ε ) → S ( b ′ , ε ) . This is a completely continuous operator, and so its Fredholm determinant is p -adic entire. The Dwork trace formulanow proves the following (see Section 5.1 for the proof): Theorem 3.6. L ( Sym ∞ ,κ Kl, T ) = det (1 − β κ T | S ( b ′ , ε )) δ q where δ q sends any function g ( T ) to g ( T ) /g ( qT ) . We now rewrite this using the cohomology theory introduced in Section 3. First, we note that:
Lemma 3.7. q∂ κ ◦ β κ = β κ ◦ ∂ κ . This lemma is a bit involved and so we delay its proof until Section 5.2. As a consequence of Lemma 3.7, β κ inducesmaps ¯ β κ : H κ ( S ( b ′ , ε )) → H κ ( S ( b ′ , ε )) and ¯ β κ : H κ ( S ( b ′ , ε )) → H κ ( S ( b ′ , ε )) with the property: L (Sym ∞ ,κ Kl, T ) = det (1 − ¯ β κ T | H κ ( S ( b ′ , ε ))) det (1 − q ¯ β κ T | H κ ( S ( b ′ , ε ))) . Using our knowledge of cohomology from the previous sections gives:
Theorem 3.8. If κ ∈ Z p then L ( Sym ∞ ,κ Kl, T ) is an entire function. When κ = 0 , then L ( Sym ∞ , Kl, T ) = det (1 − ¯ β T | H κ ( S ( b ′ , ε )))1 − qT , which is still an entire function by Theorem 3.9 below.Proof. If κ = 0, then the first statement follows immediately from Theorem 3.1. Suppose now that κ = 0. In this case,the constant 1 is a basis for H κ with trivial action of Frobenius: q ¯ β (1) = qψ at ◦ [¯ α a ] (1) = qψ at (Υ q ◦ ¯ α a (1)) = q. This proves the first part of the theorem. To show entireness, since the unit root L -function with κ = 0 takes the form L unit (0 , T ) = (1 − T ) / (1 − qT ), we see that L (Sym ∞ , Kl, T ) = (1 − T ) L (Sym ∞ , − Kl, qT )1 − qT . We will see in the next proposition that L (Sym ∞ , − Kl, T ) has a root at T = 1, which proves the entireness for κ = 0. Theorem 3.9.
For every κ ∈ Z p , T = 1 is a root of L ( Sym ∞ ,κ Kl, T ) .Proof. While one may use the cohomology above to prove this, we will use a different argument. From [16, TheoremB] T = 1 is a root of the k -th symmetric power L -function L ( Sym k Kl, T ) for every positive integer k . The result nowfollows by continuity: let { k m } be any sequence of positive integers which tend to infinity and k m → κ p -adically, thenlim m →∞ L ( Sym k m Kl, T ) = L ( Sym ∞ ,κ Kl, T ) . Now that L (Sym ∞ ,κ Kl, T ) is an entire function, we next investigate the zeros using the q -adic Newton polygon. Wecould use the above theory to give a lower bound for the Newton polygon, however, since we are using the “first” splittingfunction, the estimate is weaker than it could be. In Section 4 we switch to using the infinite splitting function which willprovide a stronger estimate. This switch comes at the cost of reworking much of the theory we just established becausethe map β κ and the spaces involved are altered. p -adic estimates In order to obtain the best possible p -adic estimates, we modify the splitting function used earlier. Unfortunately, thisaffects nearly all the spaces and maps defined earlier as well, which means we need to redefine them accordingly, as wellas reprove certain results. We do this now. Let γ ∈ Q p be a root of P ∞ i =0 t pi p i with ord p ( γ ) = p − , and set Ω := Q p ( γ ). Let E ( t ) := exp (cid:18)P ∞ i =0 t pi p i (cid:19) be theArtin-Hasse exponential. Define γ l := P li =0 γ pi p i and note that ord p ( γ l ) ≥ p l +1 p − − l −
1. Dwork’s infinite splitting functionis defined as θ ∞ ( t ) := E ( γt ) = P ∞ i =0 λ i t i , and it is well-known that the coefficients satisfies ord p ( λ i ) ≥ ip − since theArtin-Hasse exponential has p -adic integral coefficients. It also satisfies the same properties as θ ( z ) defined in Section 2in terms of being a p -adic analytic lift of an additive character on F q . Set ˆ b := p/ ( p − b and b ′ be real numbers satisfying:ˆ b ≥ b > / ( p −
1) and b ≥ b ′ , and set ε := b − p − . (20)With q = p a for a ≥
0, define the spaces L ( b ′ ; ρ ) := ( ∞ X n =0 A ( n ) t n | A ( n ) ∈ Ω , ord p A ( n ) ≥ b ′ n + ρ ) L ( b ′ ) := [ ρ ∈ R L ( b ; ρ ) K q ( b ′ , b ; ρ ) := X n ≥ ,u ∈ Z A ( n, u ) t n · t qm ( u ) x u | A ( n, u ) ∈ Ω , ord p A ( n, u ) ≥ b ′ n + b | u | + ρ K q ( b ′ , b ) := [ ρ ∈ R K ( b ′ , b ; ρ ) . We will denote K ( b ′ , b ) by K ( b ′ , b ). Note that these spaces are precisely the same as in Section 2 but with π replaced by γ . Relative cohomology.
Our first step is to obtain relative cohomology, which requires a boundary map b D t defined asfollows. As we did before, consider the p -adic analogue of θ (1) Tr F q/ F p ( x + tx ) = θ (1) Tr ( x ) θ (1) Tr ( t/x ) : define F ∞ ( t, x ) := θ ∞ ( x ) θ ∞ ( tx ) F ∞ ,a ( t, x ) := a − Y i =0 F ∞ ( t p i , x p i ) , Next define a function G ( t, x ) such that F ∞ ( t, x ) = G ( t,x ) G ( t p ,x p ) . Using this equation recursively, we see that G ( t, x ) mustbe defined by G ( t, x ) := ∞ Y j =0 F ∞ ( t p j , x p j ) ∈ Ω [[ t, x ]] . Define the (twisted) boundary operator b D t on K ( b ′ , b ) by b D t : = 1 G ( t, x ) ◦ x ∂∂x ◦ G ( t, x )= x ∂∂x + W ( t, x )where, setting f x := x ∂∂x f ( t, x ), W ( t, x ) := ∞ X j =0 γ j p j f x ( t p j , x p j ) . (21)As W ∈ K (ˆ b, ˆ b ; −
1) and acts via multiplication, b D t is a well-defined endomorphism of K ( b ′ , b ). This allows us to definethe relative cohomology spaces H t ( K ( b ′ , b )) := ker ( b D t | K ( b ′ , b )) and H t ( K ( b ′ , b )) := K ( b ′ , b ) / b D t K ( b ′ , b ) . We will now identify H t ( K ( b ′ , b )) with a simpler space and show H t = 0. Define the spaces (with ρ ∈ R ) V q ( b ′ , b ; ρ ) := (cid:18) Ω [[ t ]] + Ω [[ t ]] t q x (cid:19) ∩ K q ( b ′ , b ; ρ ) V q ( b ′ , b ) := [ ρ ∈ R V q ( b ′ , b ; ρ ) . The paper could have been written using only the results from this section, however, I feel there is something to be gained from the simplicityusing the first splitting function as well as potential future work. heorem 4.1. We have H t q ( K ( b ′ , b )) = 0 and H t q ( K ( b ′ /q, b )) ∼ = V q ( b ′ /q, b ) . Furthermore, K q ( b ′ /q, b ; 0) = V q ( b ′ /q, b ; 0) ⊕ b D t q K q ( b ′ /q, b ; ε ) . Proof.
The result follows from [6, Lemma 2.1], and a similar argument to [12, Section 3.3].
Relative Frobenius.
Define the relative Frobenius α ∞ , ( t ) := ψ x ◦ F ∞ ( t, x ) , and, for m ≥ α ∞ ,m ( t ) : = ψ mx ◦ F ∞ ,m ( t, x )= α ∞ , ( t p m − ) ◦ · · · ◦ α ∞ , ( t p ) ◦ α ∞ , ( t ) . It follows from F ∞ ( t, x ) ∈ K (ˆ b/p, ˆ b/p ; 0) that F ∞ ( t p i , x p i ) ∈ K (ˆ b/p i +1 , ˆ b/p i +1 ; 0), and thus α ∞ ,m ( t ) : K ( b ′ , b ; 0) →K p m ( b ′ /p m , b ; 0). As p m b D t pm ◦ α ∞ ,m = α ∞ ,m ◦ b D t , we see that α ∞ ,m induces a map on relative cohomology ¯ α ∞ ,m ( t ) : H t ( K ( b ′ , b )) → H t pm ( K ( b ′ /p m , b )). Again, we needto understand the power series entries of the matrix of ¯ α ∞ ,m ( t ): Theorem 4.2.
With { , γt/x } and { , γt p m /x } as bases of H t ( K (ˆ b, ˆ b )) and H t pm ( K (ˆ b/p m , ˆ b )) , respectively, the matrixof the relative Frobenius ¯ α ∞ ,m satisfies ¯ α ∞ ,m (1) = A m, ( t ) + A m, ( t ) γt p m x and ¯ α ∞ ,m ( πtx ) = A m, ( t ) + A m, ( t ) γt p m x , where A m, ∈ L (ˆ b/p m ; 0) A m, ∈ L (ˆ b/p m ; 1 p − − ˆ bp ) A m, ∈ L (ˆ b/p m ; ˆ b − p − A m, ∈ L (ˆ b/p m ; ˆ b − ˆ bp ) (22) and A m, (0) = 1 .Proof. This follows from an analogous argument to [2, Section 3] or [12, Section 3].We now move on to the infinite symmetric power theory.
Sym ∞ ,κ -cohomology (again) Analogously to what we did in Section 3, within Ω [[ t, w ]] and setting w ( m ) := κ m w m , define the spaces S ( b ′ , ε ; ρ ) := X n,m ≥ A ( n, m ) t n w ( m ) | A ( n, m ) ∈ Ω , ord p A ( n, m ) ≥ b ′ n + εm + ρ S ( b ′ , ε ) := [ ρ ∈ R S ( b ′ , b ; ρ ) . We now use the Gauss-Manin connection to define a boundary operator for this space. As our cohomology has changedsince Section 3, so has the connection. Define b ∂ : = 1 G ( t, x ) ◦ t ∂∂t ◦ G ( t, x )= t ∂∂t + W ( t, x )where, setting f t ( t, x ) := t ∂∂t f ( t, x ), W ( t, x ) := ∞ X j =0 γ j p j f t ( t p j , x p j ) = ∞ X j =0 γ j p j (cid:18) tx (cid:19) p j . Note that b ∂ is an endomorphism of K ( b ′ , b ) since W ( t, x ) ∈ K (ˆ b, ˆ b ; − b ∂ commutes with b D ( t ) as endomorphismsof K ( b ′ , b ), and thus it induces an operator on relative cohomology b ∂ : H t ( K ( b ′ , b )) → H t ( K ( b ′ , b )). With respect to thebasis { , γtx } on H t ( K ( b ′ , b )), we may write the boundary map in matrix form b ∂ = t ddt + b H , where b H is a two-by-twomatrix with entries in L ( b ′ ). We define the κ -symmetric power of the boundary operator b ∂ on S ( b ′ , ε ) by b ∂ κ := t ddt + L κ, b H ,where L κ, b H is defined similarly to (12) but using the matrix b H . Define the cohomology spaces H κ ( S ( b ′ , ε )) := ker ( b ∂ κ | S ( b ′ , ε )) and H κ ( S ( b ′ , ε )) := S ( b ′ , ε ) / b ∂ κ S ( b ′ , ε ) . .2 H κ and H κ when κ ∈ Z p \ Z ≥ Throughout this section we will assume κ ∈ Z p \ Z ≥ . As we did before, we will show H κ = 0 and identify H κ with asimpler space R ( b ′ ) defined as follows. The spaces L ( b ′ ; ρ ) and L ( b ′ ) sit naturally in S ( b ′ , ε ) by sending ξ ξ , the serieswith no w ( m ) terms. We will denote by R ( b ′ ) and R ( b ′ ; ρ ) the images of these spaces in S ( b ′ , ε ). We will show that H κ ( S ( b ′ , ε )) ∼ = R ( b ′ ).Abusing notation slightly, define the matrix H := (cid:18) γ t (cid:19) . (Note, this is the same matrix as in Section 3 with π replaced by γ .) In the following lemma we relate b H with the matrix H . This lemma is a technical key which allows us to prove results using the simpler H matrix and then lift the result tothat of the more complicated b H . Lemma 4.3.
There exists η a 1-unit in L ( b ′ ; 0) and a matrix b R such that b H = η H + b R, where b R = b R b R b R b R ! satisfies b R ∈ L ( b ′ ; 0) b R ∈ L ( b ′ ; ε ) b R ∈ L ( b ′ ; − ε ) b R ∈ L ( b ′ ; 0) . Proof.
Write f x ( t p j , x p j ) = f x ( t, x ) p j + ph ( t, x ), where h has p -adic integral coefficients. Using this, define Q and R such that W = γf x Q + R . It follows from (21) that R , Q ∈ K (ˆ b, ˆ b ; 0) and Q is a 1-unit. Similarly, define Q (no R is required) such that W = γf t Q , and note that Q ∈ K (ˆ b, ˆ b ; 0) is a 1-unit.Consider now b ∂ (1) = W . From [6, Lemma 2.1], there exists a 1-unit η ∈ L ( b ′ ; 0) and h ∈ K ( b ′ , b ; ε ) such that Q = η + γf x h . Then W = γf t Q = γf t η + γf x ( γf t h )= γf t η + ( W − R ) Q − ( γf t h )= γf t η + b D t ( ζ ) − ξ where ζ ∈ K ( b ′ , b ; 0) and ξ ∈ K ( b ′ , b ; 0). By Theorem 4.1, write ξ = ξ (0)1 + ξ (1)1 γtx + b D t ( ζ ) where ξ (0)1 ∈ L ( b ′ ; 0), ξ (1)1 ∈ L ( b ′ ; ε ), and ζ ∈ K ( b ′ , b ; ε ). Then b ∂ (1) = η γtx + ξ (0)1 + ξ (1)1 γtx + b D t ( ζ + ζ ) . This shows, b R = ξ (0)1 ∈ L ( b ′ ; 0) and b R = ξ (1)1 ∈ L ( b ′ ; ε ).We now compute b ∂ ( γtx ). Write b ∂ (cid:18) γtx (cid:19) = γtx + W γtx = γtx + γf t Q γtx = γtx + γf t ( η + γf x h ) γtx = γtx + η (cid:18) γtx (cid:19) + γf x ( γf t h ) γtx . Now, b D t (cid:18) γtx (cid:19) = − γtx + W γtx = − γtx + ( γf x Q + R ) γtx = − γtx + γ t − (cid:18) γtx (cid:19) + γf x ξ + ξ here ξ := ( Q − γtx and ξ := R γtx are elements in K ( b ′ , b ; − ε ). By Theorem 4.1, γf x ξ + ξ = ˜ η + ˜ η γtx + b D t (˜ ζ )for some ˜ η ∈ L ( b ′ ; − ε ), ˜ η ∈ L ( b ′ ; 0), and ˜ ζ ∈ K ( b ′ , b ; 0). Next, setting ˜ ξ := γf t h γtx ∈ K ( b ′ , b ; − ε ), then γf x ( πf t h γtx ) = γf x ˜ ξ = b D t ( ζ ) + ζ + ζ γtx where ζ ∈ K ( b ′ , b ; − ε ), ζ ∈ L ( b ′ ; − ε ), and ζ ∈ L ( b ′ ; 0). Consequently, b ∂ ( γtx ) = γtx + η ( γtx ) + γf x ( γf t h γtx )= γtx + η (cid:18) − γtx + π t + ˜ η + ˜ η γtx + b D t (˜ ζ ) (cid:19) + b D t ( ζ ) + ζ + ζ ( γtx )= η γ t + b R + b R γtx + b D t ( ζ + η ˜ ζ ) , where b R := η ˜ η + ζ ∈ L ( b ′ ; − ε ) b R := 1 − η + η ˜ η + ζ ∈ L ( b ′ ; 0) . This finished the proof.It follows immediately from the estimates on the matrix of b R that: Corollary 4.4.
We have1. L κ, b H = η L κ,H + L κ, b R , where L κ,H and L κ, b R is defined similarly to (12).2. L κ, b R S ( b ′ , ε ; 0) ⊂ S ( b ′ , ε ; 0) . Lemma 4.5. S ( b ′ , ε ; 0) = R ( b ′ , ε ; 0) + L κ, b H S ( b ′ , ε ; ε ) .Proof. Let ξ ∈ S ( b ′ , ε ; 0). We will show ξ is contained in the righthand side. By the proof of Lemma 3.3 but with π replaced by γ in the matrix of H , there exists η ∈ R ( b ′ , ε ; 0) and ζ ∈ S ( b ′ , ε ; ε ) such that ξ = η + L κ,H ( ζ ). By Corollary4.4, we may write L κ,H = ( L κ, b H − L κ, b R ) η − , and so ξ = η + ( L κ, b H − L κ, b R ) η − ζ = η + L κ, b H ( η − ζ ) − L κ, b R ( η − ζ ) . By Corollary 4.4, since η − ζ ∈ S ( b ′ , ε ; ε ), L κ, b R ( η − ζ ) ∈ S ( b ′ , ε ; ε ). We may now repeat this procedure with L κ, b R ( η − ζ )and so forth, thus showing ξ ∈ R ( b ′ , ε ; 0) + L κ, b H S ( b ′ , ε ; ε ).To prove the other direction, let ζ ∈ S ( b ′ , ε ; ε ). Again by Corollary 4.4, L κ, b H ( ζ ) = η L κ,H ( ζ ) + L κ, b R ( ζ ). Now L κ,H ( ζ ) ∈ S ( b ′ , ε ; 0), and Corollary 4.4 gives L κ, b R ( ζ ) ∈ S ( b ′ , ε ; ε ). This proves the result. Lemma 4.6. R ( b ′ , ε ) ∩ L κ,H S ( b ′ , ε ) = { } .Proof. Let ξ = P m ≥ ξ m w ( m ) ∈ S ( b ′ , ε ) and η ∈ R ( b ′ , ε ) be such that L κ,H ξ = η . This is equivalent to the system ofequations η = + κγ tξ ξ + 2( κ − γ tξ ξ + 3( κ − γ tξ ξ + 4( κ − γ tξ ...Observe that the second equation shows t | ξ , and the fourth equation shows t | ξ , and hence t | ξ . Repeating thisshows t m | ξ for any m ≥
1. Hence, ξ = 0, and thus ξ m = 0 for m ≥
1. A similar argument using the odd rows shows ξ m +1 = 0 for m ≥ Theorem 4.7.
Let κ ∈ Z p \ Z ≥ . Then H κ ( S ( b ′ , ε )) = 0 , and H κ ( S ( b ′ , ε )) ∼ = R ( b ′ ) . Furthermore, S ( b ′ , ε ; 0) = R ( b ′ , ε ; 0) ⊕ b ∂ κ S ( b ′ , ε ; ε ) . (23) roof. First note that the righthand side of (23) is contained in the left. Next, let ξ ∈ S ( b ′ , ε ; 0) and set E := t ddt . ByLemma 4.5, there exists η ∈ R ( b ′ , ε ; 0) and ζ ∈ S ( b ′ , ε ; ε ) such that ξ = η + L κ, b H ( ζ )= η + b ∂ κ ( ζ ) − E ( ζ ) . As E ( ζ ) ∈ S ( b ′ , ε ; ε ), we may repeat this procedure to obtain S ( b ′ , ε ; 0) = R ( b ′ , ε ; 0) + b ∂ κ S ( b ′ , ε ; ε ).We now show directness. Let η ∈ R ( b ′ , ε ) ∩ b ∂ κ S ( b ′ , ε ). Let ζ ∈ S ( b ′ , ε ) be such that b ∂ κ ζ = η . If ζ = 0, then thereexists c ∈ R such that ζ ∈ S ( b ′ , ε ; c ) but ζ
6∈ S ( b ′ , ε ; c + ε ). Now, by Corollary 4.4, η = b ∂ κ ζ = Eζ + η L κ,H ζ + L κ, b R ζ. Set ξ := Eζ + L κ, b R ζ ∈ S ( b ′ , ε ; c ), By the first part of this proof, there exists η ∈ R ( b ′ , ε ; c ) and ζ ∈ S ( b ′ , ε ; c + ε ) suchthat ξ = η + b ∂ κ ζ = η + E ζ + η L κ,H ( ζ ) + L κ, b R ( ζ ) . Set ξ := E ( ζ ) + L κ, b R ζ ∈ S ( b ′ , ε ; c + ε ). Iterating this procedure, we obtain η = ∞ X i =1 η i + η L κ,H ( ζ + ∞ X i =1 ζ i ) , which we rewrite as L κ,H ( ζ + X i ≥ ζ i ) = ( η − X i ≥ η i ) η − . By Lemma 4.6, we must have ζ = − P i ≥ ζ i ∈ S ( b ′ , ε ; c + ε ), which contradicts our choice of c .Lastly, observe that setting η = 0 shows ker b ∂ κ = 0. Now that we have finished the study of cohomology, we move on to the Frobenius. Most of the arguments are the sameas in Section 3.3. Define the κ -symmetric power of ¯ α ∞ ,a as follows: define [¯ α ∞ ,a ] κ : S ( b ′ , ε ) → Ω [[ t, w ]] by linearlyextending over L ( b ′ ) the action[¯ α ∞ ,a ] κ ( w ( m ) ) := κ m · (Υ q ◦ ¯ α ∞ ,a (1)) κ − m (Υ q ◦ ¯ α ∞ ,a γtx ) m , where Υ q : H t q ( K q ( b ′ /q, b )) → Ω [[ t, w ]] by sending ζ + ξ γt q x ζ + ξw . Just as in Lemma 3.5, [¯ α ∞ ,a ] κ is an endomorphismof S ( b ′ , ε ) when ˆ b − p − ≥ b > p − and b ≥ b ′ . (Note that p ≥ b > / ( p −
1) so that such a b exists.)Define the Frobenius map β ∞ ,κ := ψ at ◦ [ α ∞ ,a ] κ : S ( b ′ , ε ) → S ( b ′ , ε ) . An analogous result to Lemma 3.7 shows q b ∂ κ ◦ β ∞ ,κ = β ∞ ,κ ◦ b ∂ κ , and so β ∞ ,κ induces maps on cohomology ¯ β ∞ ,κ : H κ ( S ( b ′ , ε )) → H κ ( S ( b ′ , ε )) and ¯ β ∞ ,κ : H κ ( S ( b ′ , ε )) → H κ ( S ( b ′ , ε )). Combining this with the Dwork trace formula(analogous to Theorem 3.6), and Theorem 4.7 gives: Theorem 4.8.
Set ˆ b − p − ≥ b > p − and b ≥ b ′ . For κ ∈ Z p \ Z ≥ , L ( Sym ∞ ,κ Kl, T ) = det (1 − ¯ β ∞ ,κ T | H κ ( S ( b ′ , ε ))) . We are now able to finish this section by providing an estimate for the q -adic Newton polygon of L ( Sym ∞ ,κ Kl, T )for every κ . While we have concentrated in this section on the case when κ is not a positive integer, the function L ( Sym ∞ ,κ Kl, T ) is continuous in the variable κ . Thus, we need only prove the result for κ ∈ Z p \ Z ≥ for the estimateto hold. Theorem 4.9.
Let κ ∈ Z p . Writing L ( Sym ∞ ,κ Kl, T ) = P ∞ m =0 c m T m , then for every m ≥ , ord q c m ≥ (cid:18) − p − (cid:19) m ( m − . Proof.
We will prove this assuming κ ∈ Z p \ Z ≥ . As this set is dense in Z p , the result will follow by continuity in κ of L ( Sym ∞ ,κ Kl, T ).Defining the operator β ∞ ,κ, := ψ t ◦ [¯ α ∞ , ] κ : S ( b ′ , ε ) → S ( b ′ , ε ), then we see that β a ∞ ,κ, = ψ t ◦ [¯ α ∞ , ( t )] κ ◦ · · · ◦ ψ t ◦ [¯ α ∞ , ( t )] κ = ψ at ◦ [¯ α ∞ , ( t p a − )] κ ◦ · · · ◦ [¯ α ∞ , ( t p )] κ ◦ [¯ α ∞ , ( t )] κ = ψ at ◦ [¯ α ∞ , ( t p a − ) ◦ · · · ◦ ¯ α ∞ , ( t p ) ◦ ¯ α ∞ , ( t )] κ = ψ at ◦ [¯ α ∞ ,a ( t )] κ = β ∞ ,κ,a , here we have used an argument similar to [11, Corollary 2.4] for the third equality. Hence, on cohomology, ¯ β a ∞ ,κ, =¯ β ∞ ,κ,a .Now, det (1 − ¯ β ∞ ,κ,a T a | H κ ( S ( b ′ , ε ))) = det (1 − ¯ β a ∞ ,κ, T a | H κ ( S ( b ′ , ε )))= Y ζ a =1 det (1 − ζ ¯ β ∞ ,κ, T | H κ ( S ( b ′ , ε ))) . (24)Counting multiplicities, let m i denote the number of reciprocal roots of det (1 − ¯ β ∞ ,κ, T | H κ ( S ( b ′ , ε ))) which have slope s i ; note, we say λ ∈ C p has slope s i if ord p ( λ ) = s i . Then, from (24), det (1 − ¯ β ∞ ,κ,a T | H κ ( S ( b ′ , ε ))) has m i reciprocalroots of slope s i /a , or alternatively, it has m i reciprocal roots of q -adic slope s i .In order to have the map β ∞ ,κ, well-defined, we require ˆ b − p − ≥ b > p − . Thus, set b to be the maximum value b = ˆ b − p − . By definition, ε := b − p − = ˆ b − p − . Note that γ n t n ∈ R (ˆ b/p ; 0) ⊂ S (ˆ b/p, ˆ b − p − ; 0). As [¯ α ∞ , ] κ is well-defined on this space, we have β ∞ ,κ, ( γ n t n ) ∈ S (ˆ b, ε ; 0). Unfortunately, we are unable to control the reductionof this space in cohomology using Theorem 4.7 since we need “ ε = b − p − ” and “ b ≥ b ′ ”; in our case of S (ˆ b, ε ; 0), wehave b ′ = ˆ b > b = ˆ b − p − . We may fix this by viewing β ∞ ,κ, ( γ n t n ) ∈ S (ˆ b, ε ; 0) ⊂ S (ˆ b − p − , ε ; 0). Using Theorem4.7, ¯ β ∞ ,κ, ( γ n t n ) = P m ≥ B ( m, n ) t m ∈ R (ˆ b − p − ; 0). Writing P m ≥ B ( m, n ) t m = P m ≥ B ( m, n ) γ − m · γ m t m , wesee that ord p B ( m, n ) γ − m ≥ b − / ( p − m − m/ ( p −
1) = 2 m (1 − p − ). The result now follows from the previousparagraph and the argument given by Dwork in [5, Section 7] . In order to prove this result, we need to first recall the Dwork trace formula on the fibers. Fix ¯ t ∈ F ∗ q and let ˆ t be itsTeichm¨uller lift. Set d (¯ t ) := [ F q (¯ t ) : F q ] and define q ¯ t := q d (¯ t ) . Define α ˆ t := ψ ad (¯ t ) x ◦ F ad (¯ t ) (ˆ t, x ) . Observe that α ˆ t is an endomorphism of K ˆ t ( b ), where K ˆ t ( b ) denotes the space obtained from K ( b ′ , b ) by specializing t = ˆ t .Dwork’s trace formula states ( q m ¯ t − n T r ( α m ˆ t | K ˆ t ( b )) = X ¯ x ∈ F ∗ qm ¯ t Ψ ◦ T r F qm ¯ t / F q ( x + ¯ tx ) , or equivalently L ( Kl ¯ t , T ) = det (1 − α ˆ t T | K ˆ t ( b )) det (1 − q ¯ t α ˆ t T | K ˆ t ( b )) . By Theorem 2.2, the operator D ˆ t := x ddx + π (cid:16) x − ˆ tx (cid:17) acts on the space K ˆ t ( b ) such that the associated cohomologysatisfies H ( K ˆ t ( b )) := ker ( D ˆ t ) = 0 and H ( K ˆ t ( b )) := K ˆ t ( b ) /D ˆ t K ˆ t ( b ) ∼ = Ω(ˆ t ) + Ω(ˆ t ) π ˆ tx . Further, the Frobenius α ˆ t inducesa map ¯ α ˆ t on cohomology satisfying L ( Kl ¯ t , T ) = det (1 − ¯ α ˆ t T | H ( K ˆ t ( b )))= (1 − π (¯ t ) T )(1 − π (¯ t ) T ) . We now move on to the infinite symmetric powers of the fibers, defined analogously to S ( b ′ , ε ). Define S ˆ t ( ε ) as thespace obtained from S ( b ′ , ε ) by specializing t = ˆ t . As a consequence of Theorem 2.2, observe that ¯ α ˆ t (1) = 1 + η ˆ t + ζ ˆ t x for some elements η ˆ t , ζ ˆ t ∈ Ω(ˆ t ) satisfying | η ˆ t | p < | ζ ˆ t | p <
1. Define Υ q ¯ t : H ( K ˆ t ( b )) → S ˆ t ( ε ) by ζ + ξ π ˆ tx ζ + ξw .For any τ ∈ Z p , (Υ q ¯ t ◦ α ˆ t (1)) τ is a well-defined element of Ω(ˆ t )[[ w ]]. Define [¯ α ˆ t ] κ acting on S ˆ t ( ε ) by linearly extending[¯ α ˆ t ] κ ( w ( m ) ) := κ m (Υ q ¯ t ◦ ¯ α ˆ t (1)) κ − m (Υ q ¯ t ◦ ¯ α ˆ t π ˆ tx ) m . By Lemma 3.5, [¯ α ˆ t ] κ is a well-defined endomorphism of S ˆ t ( ε ) when ˜ b − p − ≥ b . The main purpose for working on thefibers is that, by an argument similar to [11, Corollary 2.4, part 2], we have det (1 − [¯ α ˆ t ] κ T | S ˆ t ( ε )) = ∞ Y m =0 (cid:0) − π (¯ t ) κ − m π (¯ t ) m T (cid:1) , (25)which is the local factor in the Euler product of L ( Sym κ, ∞ Kl, T ). We may now prove the theorem. roof of Theorem 3.6. Let B κ ( t ) be the infinite dimensional matrix of [¯ α a ] κ with respect to the basis B := { w ( m ) : m ≥ } . Write B κ ( t ) = P n ≥ b n t n , where b n is an infinite matrix with entries in C p . Define F B κ := ( b qn − m ) ( n,m ) where n, m ≥
0, and we set b qn − m := 0 if qn − m <
0, the zero matrix. As described prior to [13, Lemma 2.3], the matrix of β κ with respect to B is F B κ . By [17, Lemma 4.1], the Dwork trace formula gives( q m − T r ( β mκ ) = ( q m − T r ( F mB κ )= X ¯ t ∈ F ∗ qm ˆ t =Teich(¯ t ) T r ( B κ (ˆ t q m − ) · · · B κ (ˆ t q ) B κ (ˆ t ))= X ¯ t ∈ F ∗ qm ˆ t =Teich(¯ t ) T r ([¯ α ˆ t ] mκ | S ˆ t ( ε ))It now follows from (25) that L ( Sym ∞ ,κ Kl, T ) = det (1 − β κ T | S ( b ′ , ε )) δ q . (See the argument succeeding [11, Equation 8] for details.) The result follows from a limit using finite symmetric powers. Let k be a positive integer. Define the map Sym k ¯ α m : Sym kL ( b ′ ) H t ( b ′ , b ) → Sym kL ( b ′ /p m ) H t pm ( b ′ /p m , b )Define the length( w ( m ) ) := m . For ξ ∈ S ( b ′ , ε ), define length( ξ ) as the supremum of the lengths of the individualterms in the series defining ξ . In most cases, the length of ξ will be infinite. Set w := 1 and w := πt/x , so that { w , w } is a basis of H t ( b ′ , b ), and { w k − m w m : 0 ≤ m ≤ k } is a basis of Sym kL ( b ′ ) H t ( b ′ , b ) over L ( b ′ ). As the basisis finite, { w k − m w ( m )1 : 0 ≤ m ≤ k } is also a basis of Sym kL ( b ′ ) H t ( b ′ , b ) over L ( b ′ ), where w ( m )1 := k m w m . Define S ( k ) ( b ′ , ε ) := { ξ ∈ S ( b ′ , ε ) | length( ξ ) ≤ k } . We may identify S ( k ) ( b ′ , ε ) with Sym k H t ( b ′ , b ) using the map w ( m ) w k − m w ( m )1 . Let k n be a sequence of positive integers tending to infinity such that p -adically k n → κ . For each n ∈ Z ≥ , define theapproximation map [¯ α a ] ( κ ; n ) : S ( b ′ , ε ) → S ( b ′ /q, ε ) by[¯ α a ] ( κ ; n ) ( w ( m ) ) := ( [¯ α a ] k n ( w ( m ) ) if m ≤ k n k n -symmetric power, we have[¯ α a ] ( κ ; n ) ( w ( m ) ) = (Υ q ◦ ¯ α a (1)) k n − m (Υ q ◦ ¯ α a πtx ) m ∼ = (cid:16) Sym k n ¯ α a (cid:17) ( w k n − m w ( m )1 ) , and thus [¯ α a ] ( κ ; n ) ∼ = Sym k n ¯ α a .Next, an analogous argument to that in [11, Lemma 2.2] demonstrates that lim n →∞ [¯ α a ] ( κ ; n ) = [¯ α a ] κ as maps from S ( b ′ , ε ) → S ( b ′ /q, ε ). Consequently, if we define β ( κ ; n ) := ψ at ◦ [¯ α a ] ( κ ; n ) then as operators on S ( b ′ , ε ),lim n →∞ β ( κ ; n ) = β κ . (26)Lastly, define ∂ ( κ ; n ) on S ( b ′ , ε ) as follows. For 0 ≤ m ≤ k n − ≤ m ≤ k n − ∂ ( κ ; n ) ( t r w ( m ) ) := rt r w ( m ) + t r w ( m +1) + m ( k n − m + 1) π t r +1 w ( m − ,m = k n : ∂ ( κ ; n ) ( t r w ( k ) ) := rt r w ( k ) + k n π t r +1 w ( k − ,m > k n : ∂ ( κ ; n ) ( t r w ( k ) ) := 0 . Similarly, define ˜ ∂ ( κ ; n ) on Sym k n L ( b ′ ) H t ( b ′ , b ) as follows. For ( ξ , . . . , ξ k n ) ∈ H t ( b ′ , b ) ⊕ k n , define˜ ∂ ( κ ; n ) ( ξ · · · ξ k n ) := k n X i =1 ξ · · · ˆ ξ i · · · ξ k n ∂ ( ξ i ) , where ∂ was defined by (10). Then ∂ ( κ ; n ) ∼ = ˜ ∂ ( κ ; n ) through the identification of the spaces S ( k n ) ( b ′ , ε ) and Sym k n L ( b ′ ) H t ( b ′ , b ).Further, again using the identification, q∂ ( κ ; n ) ◦ β ( κ ; n ) = β ( κ ; n ) ◦ ∂ ( κ ; n ) . (27)Lastly, observe that with the topology of coefficient-wise convergence on S ( b ′ , ε ), ∂ ( κ ; n ) → ∂ κ , which follows by consideringthe case 0 ≤ m ≤ k n − | ( ∂ ( κ ; n ) − ∂ κ )( t r w ( m ) ) | = | m ( k n − κ ) π t r +1 w ( m − | , which tends to zero as n → ∞ . Lemma 3.7 now follows by taking the limit of (27). eferences [1] Alan Adolphson, A p -adic theory of Hecke polynomials , Duke Math. J. (1976), no. 1, 115–145.[2] Alan Adolphson and Steven Sperber, Exponential Sums and Newton Polyhedra: Cohomolgy and Estimates , Annalsof Math. (1989), no. 2, 367–406.[3] H. Timothy Choi and Ronald Evans,
Congruences for sums of powers of Kloosterman sums , Int. J. Number Theory (2007), no. 1, 105–117. MR 2310495 (2008d:11090)[4] Richard Crew, L -functions of p -adic characters and geometric Iwasawa theory , Invent. Math. (1987), no. 2,395–403. MR 880957 (89g:11049)[5] B. Dwork, On the zeta function of a hypersurface: II , Annals of Math (1964), no. 2, 227–299.[6] B. Dwork, Bessel functions as p -adic functions of the argument , Duke Math. J. (1974), 711–738. MR 0387281(52 On Hecke polynomials , Inventiones math. (1971), 249–256.[8] Bernard Dwork, Giovanni Gerotto, and Francis J. Sullivan, An introduction to G -functions , Annals of MathematicsStudies, vol. 133, Princeton University Press, Princeton, NJ, 1994. MR 1274045[9] Lei Fu and Daqing Wan, L -functions for symmetric products of Kloosterman sums , J. Reine Angew. Math. (2005), 79 – 103.[10] , L -functions of symmetric products of the Kloosterman sheaf over Z , Math. Ann. (2008), no. 2, 387–404.MR 2425148 (2009i:14022)[11] C. Douglas Haessig, Meromorphy of the rank one unit root L -function revisited , Finite Fields Appl. (2014),191–202. MR 3249829[12] C. Douglas Haessig and Antonio Rojas-Le´on, L -functions of symmetric powers of the generalized Airy family ofexponential sums , Int. J. Number Theory (2011), no. 8, 2019–2064. MR 2873140 (2012k:11114)[13] C. Douglas Haessig and Steven Sperber, L -functions associated with families of toric exponential sums , J. NumberTheory (2014), 422–473. MR 3239170[14] , Families of generalized kloosterman sums , Trans. Amer. Math. Soc. DOI: 10.1090/tran/6720 (2015).[15] C. Douglas Haessig and Steven Sperber, p-adic variation of unit root L-functions , (2015).[16] Philippe Robba,
Symmetric powers of the p -adic Bessel equation , J. Reine Angew. Math. (1986), 194 – 220.[17] Daqing Wan, Meromorphic continuation of L -functions of p -adic representations , Ann. of Math. (2) (1996),no. 3, 469–498.[18] , Dimension variation of classical and p -adic modular forms , Invet. Math. (1998), 469–498.[19] , A quick introduction to Dwork’s conjecture , Contemporary mathematics (1999), 147–163.[20] , L -functions of function fields , Number theory, Ser. Number Theory Appl., vol. 2, World Sci. Publ., Hack-ensack, NJ, 2007, pp. 237–241. MR 2364844 (2009a:11186)[21] Zhiwei Yun, Galois representations attached to moments of Kloosterman sums and conjectures of Evans , Compos.Math. (2015), no. 1, 68–120, Appendix B by Christelle Vincent. MR 3305309(2015), no. 1, 68–120, Appendix B by Christelle Vincent. MR 3305309