On slopes of $L$-functions of $\mathbb{Z}_p$-covers over the projective line

0.45 MB·

by  Michiel Kosters

ON SLOPES OF L-FUNCTIONS OF Z -COVERS OVER THE
p
PROJECTIVE LINE
MICHIELKOSTERSANDHUIJUNEZHU
7
1
0
2

n Abstract. LetP :···→C2→C1→P1 beaZp-coveroftheprojectiveline
a overafinitefieldofcardinalityqandcharacteristicpwhichramifiesatexactly
J onerationalpoint,andisunramifiedatotherpoints. Inthispaper,westudy
0 theq-adicvaluationsofthereciprocalrootsinCp ofL-functionsassociatedto
3 charactersoftheGaloisgroupofP. WeshowthatforallcoversP suchthatthe
genusofCnisaquadraticpolynomialinpnfornlarge,thevaluationsofthese

] reciprocal roots are uniformly distributed in the interval [0,1]. Furthermore,
T weshowthatforalargeclassofsuchcoversP,thevaluationsofthereciprocal
N rootsinfactformafiniteunionofarithmeticprogressions.
.
h
Contents
t
a
m 1. Introduction 1
1.1. Setup 1
[
1.2. Genus stability and slope uniformity 2

1 1.3. Slope stability 3
v
1.4. Approach 3
3
1.5. Open problems 4
3
7 1.6. Acknowledgments 5
8 2. Artin-Hasse exponential 5
0 3. Genus stable Z -covers 6
p
.
1 4. Extrapolating L-functions 7
0 4.1. The π-adic valuation 7
7
4.2. The π-adic L-function 9
1
5. Upper and lower bounds on C∗(χ,s) 10
:
v 6. Slope uniformity 12
i
X 7. Slope stability 13
Appendix A. Dwork’s theory 16
r
a References 20
1. Introduction
1.1. Setup. Let k = F be a finite field of cardinality q = pa with p prime. Let
q
C = P1 be the projective line over k. Let P : ··· → C → C → C be a Z -
0 k 2 1 0 p
coverofsmoothprojectivegeometricallyirreduciblecurvesoverk. Thismeansthat
Gal(C → C ) ∼= Z/piZ. We identify Gal(··· → C → C → C ) with Z . In this
i 0 2 1 0 p
Date:January31,2017.
2010 Mathematics Subject Classification. 11T23(primary),11L07,13F35,11R58.
1
2 MICHIELKOSTERSANDHUIJUNEZHU
paper we always assume that the tower is ramified at exactly one rational point,
and is unramified at other points. Without loss of generality, we assume this point
to be ∞. By |P1| we denote the set of closed points of the projective line. For
k
x∈|P1|\{∞}=|A1| we denote by Frob(x)∈Z its Frobenius element.
k k p
Let Z be the ring of integers of Q . Let χ:Z →C∗ be a non-trivial character
p p p p
of conductor (order) pmχ for some mχ ≥1. We consider the L-function
(cid:89) 1
L(χ,s)= ∈1+sZ [[s]]⊂C [[s]].
1−χ(Frob(x))sdeg(x) p p
x∈|A1|
k
By the Weil Conjectures, L(χ,s) is a polynomial ([1, Theorem A]). We can write
the polynomial L(χ,s) as
deg(L(χ,s))
(cid:89)
L(χ,s)= (1−α s),
χ,i
i=1
whereα ∈Z . Byv wedenotethevaluationonC normalizedbyv (q)=1. By
χ,i p q p q
the Weil conjectures, one has 0≤v (α )≤1 and the leading term of L(χ,s) has
q χ,i
q-valuation deg(L(χ,s))/2 (by the functional equation they come in pairs which
multiply to q; [1, Theorem A]). We would like to understand the distribution of
the v (α ) in the interval [0,1]. One can easily show that the set of v (α ) only
q χ,i q χ,i
depends on m , and not on the choice of χ.
χ
1.2. Genus stability and slope uniformity. Wewillfirstdiscusstwoproperties
of P.
Definition 1.1. We say that P is genus stable if the degree of L(χ,s) is a linear
polynomial in pmχ for large enough mχ.
OnecanshowthatP isgenusstableifandonlyifthegenusofC isaquadratic
n
polynomialinpn forlargeenoughn(see[5,Proposition5.5]foraproofandfurther
discussions). The genus is always bounded below by a quadratic polynomial in pn,
so genus stable covers are covers for which the genus grows very slowly. In general,
the genus of C can grow arbitrarily fast. If P is not genus stable, not much is
n
currently known on the behavior of the slopes of the corresponding L-functions.
Definition 1.2. We say that the tower P is slope uniform if the following holds.
For every interval [a,b]⊆[0,1] one has
#{i=1,2,...,deg(L(χ,s)): v (α )∈[a,b]}
lim q χ,i =b−a
mχ→∞ deg(L(χ,s))
where the limit is over any finite non-trivial characters χ of conductor m .
χ
Intuitively, slope uniformity means that the q-adic valuations of the α for a
χ,i
characterχwithconductorpmχ approachauniformdistributionon[0,1]whenmχ
goes to infinity. Our first main result is the following.
Theorem 1.3. Assume that P is genus stable. Then P is slope uniform.
Actually, we prove a stronger statement about the distribution of the v (α ).
q χ,i
See Theorem 6.1.
SLOPES OF Zp-COVERS 3
1.3. Slope stability. We will now study a much stronger property of P.
Definition 1.4. We say that P is slope stable if there exists an integer m(cid:48) ∈ Z
≥1
such that the following holds. Let χ be a character with m = m(cid:48). Then for
0 χ0
every character χ with m ≥m(cid:48) the multiset {v (α ):i=1,2,...,deg(L(χ,s))}
χ q χ,i
is equal to the multiset
pmχ(cid:91)−m(cid:48)−1(cid:26) i ,vq(αχ0,1)+i,vq(αχ0,2)+i,...,vq(αχ0,deg(L(χ0,s)))+i(cid:27)−{0}.
pmχ−m(cid:48) pmχ−m(cid:48) pmχ−m(cid:48) pmχ−m(cid:48)
i=0
Intheliterature(see[4]forexample),slopestableiscommonlydescribedasforming
a finite union of arithmetic progressions. Intherestofthepaperwewritem(cid:48)-slope
stable when we want to emphasize m(cid:48).
We remark that slope stable implies genus stable. Indeed, if P is slope stable,
then for mχ ≥mχ0 the L-function deg(L(χ,s))=(degL(χ0,s)+1)pmχ−mχ0 −1 is
a linear polynomial in pmχ, hence P is genus stable.
We list some recent progress in the direction of our next result in Theorem 1.7
below. In order to do so, we first discuss a way to represent our tower P.
Any Z -cover P can be explicitly given by f ,f ,...,∈ F [X] with p (cid:45) deg(f )
p 0 1 q i
using Witt-vector equations using Artin-Schreier-Witt theory (see Section 3 for
details). The main theorem of [4, Theorem 1.2] can be phrased in the following
way:
Theorem 1.5 (Davis-Wan-Xiao). Assume that f = 0 for i ≥ 1. Then P is slope
i
stable.
This result was generalized by [6, Theorem 1] as follows:
Theorem 1.6 (Li). Assume that {deg(f ):i∈Z } is bounded. Then P is slope
i ≥0
stable.
(cid:110) (cid:111)
The cover P is genus stable if and only if δ(P) := max deg(fi) :i=0,1,...
pi
exists (see Section 3). In that case, one has deg(f )≤δpi for all i. Our main result
i
is the following.
Theorem 1.7. Fix δ ∈Q . There is a constant C =C(δ,q)∈R such that for
>0 ≥0
any P genus stable with δ =δ(P) and deg(f )≤δpi−C for all i large enough, P
i
is slope stable.
FirstofallTheorem1.7impliesTheorem1.6andTheorem1.5above. Theproof
of Theorem 1.7 can be found in Section 7. Our C in Theorem 1.7 is explicit and
quitesmall. Itisstillanopenproblemwhethergenusstabilityisequivalenttoslope
stability, equivalently, whether we can take C =0 in Theorem 1.7.
As an intermediate result, we also study how much the L-functions depend on
the defining equations, the f , of the Z -cover. We prove that many coefficients of
i p
the f do not influence the Newton polygon of the various L-functions (Theorem
i
7.4).
1.4. Approach. Let us discuss the approaches for studying these L-functions of
Z -towers.
p
In [7] the T-adic L-function of P, L(T,s) ∈ C [[T]][[s]], was introduced. This
p
function specializes to L(χ,s) for a certain value for T depending on the character
4 MICHIELKOSTERSANDHUIJUNEZHU
χ : Z → C∗. The authors of [4] study the T-adic Newton polygon L(T,s) as
p p
follows. They construct an upper and lower bound on the T-adic Newton polygon
ofafunctionsimilartoL(T,s). TheupperboundscomefromtheWeilconjectures,
whereas the lower bounds come from Dwork’s trace formula. These Newton poly-
gons touch at various points, and this turns out to be enough to prove their main
theorem. The author of [6] improves the approach of [7] by taking into account
both the p-adic and T-adic valuation of coefficients of L(T,s).
In our approach, we construct a π-adic L-function, in infinitely many variables
(thereisavariableforeachcoefficientofthedefiningpolynomialsf ),whichspecial-
i
izes to L(T,s) and to the L(χ,s). This function behaves much better than L(T,s),
sinceitincorporatesspecialp-adicpropertiesofelementsofp-thpowerorderinC∗.
p
Furthermore,wepickatopologyontheringthisfunctionlivesinwhichreflectsthe
structure of L-functions much better. We then follow a similar approach as in [4]
to prove our results.
1.5. Open problems. We would like to state some open problems regarding Z -
p
covers of a curve over a finite field in characteristic p.
Problem 1. Generalize the results of this paper to any Z -cover of the projective
p
line. For example, it is easy to do this when only 2 rational points are ramified.
One should be able to solve this problem by using the methods of [9].
Problem 2. GeneralizetheresultstoZ -coversofcurvesotherthantheprojective
p
line. This seems to be quite tricky, since one needs a better trace formula.
Problem 3. Is genus stability equivalent to slope stability? It might be possible
to improve the results from this paper, or it might be the case that there are genus
stable Z -covers which are not slope stable.
p
Problem 4. Let P be a non genus stable cover. Note that P is not slope stable.
Is P slope uniform? It might be possible to apply our techniques when the genus
of C is bounded above by a quadratic polynomial in pn (‘almost genus stable’).
n
Almostgenusstabilityisequivalenttosayingthattheset{deg(f )/pi :i=0,1,...}
i
has a supremum. It seems that our theory has no handle on other cases.
Problem 5. Set K = k(X). There are various constructions in algebraic geom-
etry, for example through ´etale cohomology, which give rise to continuous Galois
representations
Gal(K/K)→GL (Z )∼=Z∗.
1 p p
For p = 2, one has Z∗ ∼= Z/2Z×Z , and for p > 2, one has Z∗ ∼= F∗×Z . Hence
p p p p p
such representations give rise to continuous morphisms Gal(K/K) → Z , which
p
if non-trivial, correspond to Z -covers of P1. Are Z -covers obtained in this way
p k p
genusstable/slopestableorslopeuniform? Itseemstobetrueincertaincases(see
[3]).
Problem 6. Assume that P is a Z -cover given by f ∈W(k(X)) (see Section 3).
p
Then for c ∈ Z , one can consider the ‘twisted’ Z -cover given by cf. Let χ be a
q p
non-trivial character of finite order. Are the q-adic Newton polygons of L (χ,s)
f
and L (χ,s) the same? In [8], Ren, Wan, Xiao and Yu show that these Newton
cf
polygons are similar, but it is unknown even for χ of conductor p if these Newton
polygons are actually the same.
SLOPES OF Zp-COVERS 5
Problem7. FindanefficientalgorithmforcomputingL(χ,s)ortheq-adicNewton
polygon of L(χ,s) for a Z -cover and a finite non-trivial character χ.
p
Problem 8. Let P be a Z -cover. What can one say about the complex numbers
p
αχ,i when mχ → ∞? One can write αχ,i = q1/2e2πiθχ,i with θχ,i ∈ R/Z for
i = 1,...,deg(L(χ,s)). Is the multiset {θ : i = 1,...,deg(L(χ,s))} uniformly
χ,i
distributed in R/Z when m →∞?
χ
1.6. Acknowledgments. We would like to thank Daqing Wan for introducing us
to this topic, for his suggestions and for his interesting problems and conjectures.
2. Artin-Hasse exponential
Define the Artin-Hasse exponential by the formal power series
(cid:32)(cid:88)∞ Tpi(cid:33)
E(T)=exp ∈1+T +T2Z [[T]]
pi (p)
i=0
whereZ isthelocalizationofZattheprimep(oneeasilyobtainsE(T)∈Q[[T]],
(p)
and it is well-known that E(T) ∈ Z [[T]]). Let S be either Z , Z or F . We
(p) (p) p p
can view E(T) as an element of S[[T]].
Lemma 2.1. For i∈Z the map
≥1
E(·):TiS[[T]]→1+TiS[[T]]
is a bijection. Furthermore, for any s ∈ S this map sends sTi +Ti+1S[[T]] to
1+sTi+Ti+1S[[T]].
Proof. One easily sees that the maps are defined. Let s ∈ 1+TiS[[T]]. We show
by induction that for j ≥ i there is a h ∈ TiS[[T]], unique modulo TjS[[T]]],
j
such that E(h )−s ∈ TjS[[T]]. For i = j, the statement holds. Assume that the
j
statement holds for j. Then there is h ∈ TiS[[T]], unique modulo TjS[[T]], with
j
E(h )−s−cTj ∈Tj+1S[[T]] for some c∈S. Then one has E(h −dTj)−s−(c−
j j
d)Tj ∈Tj+1S[[T]] and hence only for c=d one has E(h −cTj)−s∈Tj+1S[[T]].
j
Hence we have to set h =h −cTj, modulo Tj+1S[[T]]. The first result follows.
j+1 j
The proof of the second result is easy. (cid:3)
Define π (T)∈TZ [[T]] for i∈Z by Lemma 2.1:
i p ≥0
E(π (T))=(1+T)pi.
i
Letx∈C withv (x)>0. Thenforalli≥0onehas, v (π (x))>0. Ontheother
p p p i
hand, by Lemma 2.1, one has E(x)−1∈xZ [[x]] and hence
p
(1) v (π (x))=v (E(π (x))−1)=v ((1+x)pi −1).
p i p i p
Below we shall study v (π (x)) via this equality.
p i
Lemma 2.2. Let m∈Z . Let x=1+ζ ∈Q where ζ is a primitive pm-th
≥1 pm p pm
root of unity. Then one has
v (π (x))=piv (x)
p i p
for 0≤i<m and π (x)=0 for i≥m.
i
6 MICHIELKOSTERSANDHUIJUNEZHU
Proof. For 0≤i<m, we have v (ζpi −1)=piv ((ζ −1); hence by (1) we have
p pm p pm
v (π (x)) = v (ζpi −1) = piv (x). For i > m, E(π (x)) = ζpi −1 = 0, hence
p i p pm p i pm
π (x)=0 by the bijection of Lemma 2.1. (cid:3)
i
A lemma as above does not hold for other x ∈ C . This has as a consequence
p
that later results only hold for characters of finite order. See Remark 7.7 for more
details.
We will now analyze the reduction of π (T) modulo p.
i
Lemma 2.3. For i∈Z one has
≥0
π (T)∈Tpi(1+TZ [[T]])+pZ [[T]].
i p p
Proof. The following diagram commutes, where the vertical maps are the natural
projections:
TZ [[T]] E(·)(cid:47)(cid:47)1+TZ [[T]]
p p
(cid:15)(cid:15)(cid:15)(cid:15)τ1 (cid:15)(cid:15)(cid:15)(cid:15)τ2
TF [[T]] E(·)(cid:47)(cid:47)1+TF [[T]].
p p
Hence we find
(cid:16) (cid:17)
E(τ (π (T))=τ (E(π (T)))=τ (1+T)pi =1+Tpi.
1 i 2 i 2
FromLemma2.1oneobtainsτ (π (T))∈Tpi(1+TF [[T]]). Theresultfollows. (cid:3)
1 i p
3. Genus stable Z -covers
p
Letk =F beafinitefieldofcardinalityq =pa. ByF wedenotetheextension
q qi
ofF ofdegreeiinafixedalgebraicclosureofF . WeletZ betheringofintegers
q q q
oftheunramifiedextensionofQ withresiduefieldF . LetK =k(X),thefunction
p q
fieldofP1. WeareinterestedinstudyingZ -coversofP1. Suchacoverisalsocalled
k p k
an Artin-Schreier-Witt cover. We restrict without loss of generality to Z -covers
p
whichramifyonlyat∞,andwhicharetotallyramifiedat∞. ByW(K)wedenote
the p-typical Witt-vectors of K (see [5] for an introduction on Witt vectors). We
use the symbol [ ] to represent the Teichmu¨ller map. Such a tower can be given
(almost uniquely) by
(cid:88)
f =f(X)=c + c [X]i ∈W(K)
0 i
i≥1,(i,p)=1
with c ∈ Z and v (c ) → ∞ as i → ∞ and min{v (c ) : (i,p) = 1} = 0
i q p i p i
(see [5, Proposition 4.3]). Let P : ... → C → C → C = P1 be the corre-
2 1 0 k
sponding Z -cover. More concretely, in terms of function fields one has k(C ) =
p i
K(y ,y ,...,y ) where y = (y ,y ,...) ∈ W(K) is such that (yp,yp,...) −
0 1 i−1 0 1 0 1
(y ,y ,...) = f ∈ W(K) (see [5]). We identify the Galois group of P and Z
0 1 p
by the following isomorphism:
Gal(P)=Gal(K(y ,y ,...,)/K)→Z
0 1 p
σ (cid:55)→σy−y.
SLOPES OF Zp-COVERS 7
For x∈|A1| one has ([5, Lemma 5.8])
k
 
(cid:88)
Frob(x)=TrZqdeg(x)/Zpc0+ ci[x(cid:48)]i∈Zp,
i≥1,(i,p)=1
where x(cid:48) ∈k is any representative of x.
The conductor of C →C satisfies ([5, Proposition 4.14])
m(cid:48) 0
(cid:110) (cid:111)
f(Cm(cid:48) →C0)=max 1+ipmχ−vp(ci)−1 :i≥1,(i,p)=1, vp(ci)<m(cid:48) ·∞.
Let χ : Z → C∗ be a finite character. By the Weil Conjectures (see [1, Theorem
p p
A] and [5, Proposition 4.14]) L(χ,s) is a polynomial of degree
(cid:26) (cid:27)
i
deg(L(χ,s))=−1+pmχ−1max :i≥1,(i,p)=1, v (c )<m .
pvp(ci) p i χ
There are unique polynomials
(cid:88)
f = a Xj ∈F [X]
i ij q
j
such that
∞
(cid:88) (cid:88)
(2) f = pi [a ][X]j,
ij
i=0 j
where [ ] denotes the Teichmu¨ller lift. Write d = deg(f ). One has a = 0 if
i i ij
j ∈ pZ and deg(f ) ≥ 1. The tower P is genus stable if and only if the set
≥1 0
(cid:110) (cid:111)
di :i=0,1,... has a maximum δ, and in that case there is a unique m with
pi
(cid:26) (cid:27)
d d
δ = m =max i :i=0,1,... .
pm pi
Assume from now on that P is genus stable with the above notation. One has
deg(f )≤δpi. Furthermore,form(cid:48) >monehasf(C →C )=(1+d pm(cid:48)−m−1)∞.
i m(cid:48) 0 m
and for a character χ with m >m one has
χ
deg(L(χ,s))=d pmχ−m−1−1=δpmχ−1−1.
m
For the rest of the paper we assume that we are given a genus stable
Z -cover P of P1 with δ =d /pm and coefficients (a ), introduced in this
p F m ij
q
section.
4. Extrapolating L-functions
4.1. The π-adic valuation. In this section we will introduce a ring R with valu-
ation for a given genus stable P with δ. Set
X={(i,j)∈Z2 :j ≤δpi =d pi−m, j (cid:54)∈pZ }⊂Z2 .
≥0 m ≥1 ≥0
Consider the formal power series ring
R=Z [[π :x∈X]].
q x
Let U := {u : X → Z : u(x) = 0 for almost all x}, which is monoid under
≥0
addition. For each u∈U we set
(cid:89)
πu = πu(x).
x
x∈X
8 MICHIELKOSTERSANDHUIJUNEZHU
Any r ∈ R can now be written in the form r = (cid:80) a πu where a ∈ Z . For
u∈U u u q
u∈U we set
1 (cid:88) 1
w(u)= j·u((i,j))∈ Z .
δ δ ≥0
(i,j)∈X
For r =(cid:80) a πu ∈R\{0} we set
u u
Z
v(r):=min{w(u):a (cid:54)=0}∈ .
u δ
We set v(0):=∞. Note that v(π )= j. For (i,j)∈X we set
(i,j) δ
∆ :=pi−v(π ).
(i,j) (i,j)
One has ∆ =pi− j ≤pi. We will now prove a lower bound on ∆ .
(i,j) δ (i,j)
Lemma 4.1. Let x ∈ X. Then one has ∆ ≥ 0 with ∆ = 0 if and only if
x x
x=(m,d ).
m
Proof. One has deg(f )≤δpi with equality if and only if i=m. Furthermore, one
i
has ∆ ≥ pi− deg(fi) with equality if and only if j = deg(f ). The result then
(i,j) δ i
follows easily. (cid:3)
It turns out that v induces a valuation on the quotient field Q(R) of R.
Proposition 4.2. The ring R is a domain. We can extend v to the quotient field
Q(R) of R by
v :Q(R)→Q(cid:116){∞}
a/b(cid:55)→v(a)−v(b).
Then (Q(R),v) is a valued field, that is, for r,s∈Q(R) one has
v(r+s)≥min{v(r),v(s)} and v(rs)=v(r)+v(s).
Proof. We show that for r,s ∈ R one has v(r+s) ≥ min{v(r),v(s)} and v(rs) =
v(r)+v(s). The first property follows directly. For the second property, assume
that r,s (cid:54)= 0. We first put a well order ≤ on X. We then put the following
monomial order ≤ on R. We write πu <πu(cid:48) if w(u)<w(u(cid:48)) or w(u)=w(u(cid:48)) and
for x(cid:48) = min{x ∈ X : u(x) (cid:54)= u(cid:48)(x)} we have u(x(cid:48)) < u(cid:48)(x(cid:48)) in Z≥0. If πu1 ≤ πu(cid:48)1
and πu2 ≤ πu(cid:48)2, then one has πu1u2 ≤ πu(cid:48)1u(cid:48)2, making it a monomial order. Write
r =(cid:80) a πu and s=(cid:80) b πu. Then there are minimal terms, with respect to ≤,
u u u u
such that au1,bu2 (cid:54)=0. Then au1bu2πu1+u2 is the minimal term of rs and one gets
v(rs)=v(r)+v(s).
Since the second property holds, it follows that R is a domain. We extend the
definition of v to Q(R). Then it is easy to see that (Q(R),v) is a valued field. (cid:3)
Werefertovastheπ-adicvaluationonRandQ(R)andwewillusethenotation
v in the rest of the paper. Note that the valuation ring of v strictly contains R,
since it contains expressions like π(i,j) .
π(i(cid:48),j)
Definition 4.3. Let K(cid:48) be a field and let v(cid:48) : K(cid:48) → R(cid:116){∞} be a valuation.
Let t ∈ K(cid:48) with v(cid:48)(t) > 0. We then normalize by setting v(cid:48) = v(cid:48)/v(cid:48)(t). Let
t
g = a +a s+a s2 +... ∈ 1+sK(cid:48)[[s]]. We then define t-adic Newton polygon
0 1 2
of g to be the lower convex hull of the set of points {(i,v(cid:48)(a ) : i = 0,1,...}. The
t i
SLOPES OF Zp-COVERS 9
slopes of such a polygon are the slopes of each of its width 1 segments, counted
with multiplicity and put in increasing order. Let g ,g ∈ K(cid:48)[[s]]. One sees, by
1 2
for example extending the valuation to an algebraic closure of K(cid:48), that the slopes
of the Newton polygon of g g are just the ordered slopes of g union those of g .
1 2 1 2
NotethattheonlyslopeoftheNewtonpolygon1−asisjustv(cid:48)(a). Inparticular,if
t
g isapolynomial,theslopesofitst(cid:48)-adicNewtonpolygoncorrespondtothet(cid:48)-adic
valuation of the reciprocal roots of g.
In this paper, we will study two different types of Newton polygons. Recall
that (Q(R),v) is a value field valuation with respect to v. We define the π-adic
Newton polygon of g ∈ 1+sQ(R)[[s]] with respect to this valuation v (without
normalization). The field (C ,v ) is also a valued field. Hence for g ∈1+sC [[s]]
p p p
and x with v (x)>0 we can consider the x-adic Newton polygon of g.
p
4.2. The π-adic L-function. ForagivengenusstablecoverP overF withδ and
q
coefficients(a ) ,weintroduceourπ-adicL-andC-functionswithcoefficients
ij (i,j)∈X
in R=Z [[π :(i,j)∈X]], which are deformation of classical p-adic and T-adic
q (i,j)
L- and C-functions, respectively. For any integer n, denote µµµ := {x ∈ C : xn =
n p
1}.
Definition 4.4. For k ∈Z we set
≥1
S∗(k,π)= (cid:88) (cid:89) E(π(i,j))TrZqk/Zp([aij][x]j) ∈R.
x∈F∗ (i,j)∈X
qk
We define the π-adic L-functions of our Z -cover by
p
(cid:88)∞ sk
L∗(π,s)=exp( S∗(k,π) )
k
k=1
(cid:89) 1
= ∈1+sR[[s]];
x∈|A1k|\{0}1−sdeg(x)(cid:81)(i,j)∈XE(π(i,j))TrZqk/Zp([aij][x]j)
and
L∗(T,s)
L(π,s)= ∈1+sR[[s]].
1−sdeg(x)(cid:81)(i,j)∈X: j=0E(π(i,j))TrZqk/Zp([aij])
We define the π-adic characteristic function by
(cid:32)(cid:88)∞ 1 sk(cid:33)
C∗(π,s)=exp S∗(k,π) ∈1+sR[[s]].
1−qk k
k=1
One has
C∗(π,s)
C∗(π,s)=L∗(π,s)L∗(π,qs)L∗(π,q2s)··· , and L∗(π,s)= .
C∗(π,qs)
We have a Z -algebra morphism
p
ev :R[[s]]→Z [[T]][[s]]
T p
s(cid:55)→s
π (cid:55)→π (T).
(i,j) i
10 MICHIELKOSTERSANDHUIJUNEZHU
Furthermore, for c∈C with v (c)>0, one has a Z -algebra morphism
p p p
ev :Z [[T]][[s]]→C [[s]]
c p p
(cid:88) (cid:88)
a (T)si (cid:55)→ a (c)si.
i i
i i
Let χ : Zp → C∗p be a nontrivial character of conductor pmχ. Then πχ :=
χ(1)−1=ζpmχ −1.
Remark 4.5. Variousspecializationsofourπ-adicLfunctiongiveknownclassical
and T-adic L-functions. Most importantly, the classical L-function is given by
(cid:89) 1
L(χ,s)= =ev ◦ev (L(π,s)).
1−χ(Frob(x))sdeg(x) πχ T
x∈|A1|
k
The following functions are introduced in [4] to study L(χ,s):
S∗(k,T)=ev (S∗(k,π)),
T
L∗(T,s)=ev (L∗(π,s)),
T
C∗(T,s)=ev (C∗(π,s)).
T
5. Upper and lower bounds on C∗(χ,s)
We write C∗(π,s) = (cid:80) a si ∈ R[[s]] with a ∈ R. In this section, we will give
i i i
upper and lower bounds on the π -adic Newton polygon of C∗(χ,s).
χ
Proposition 5.1 (Hodge lower bound). The π-adic Newton polygon of C∗(π,s)
lies above the polygon whose slopes are
a(p−1) 2a(p−1)
0, , ,...,
δ δ
that is, above the polygon with vertices (k,a(p−1)k(k−1)) (for k =0,1,2,...).
2δ
Proof. See Appendix. (cid:3)
Lemma 5.2. Let χ:Z →C∗ be a non-trivial character of finite order. Then the
p p
following hold.
i) The Newton π -adic Newton polygon of C∗(χ,s) lies above the π-adic New-
χ
ton polygon of C∗(π,s).
(cid:0) (cid:1)
ii) Let r ∈ R\{0}. One has v ev ◦ev (r) ≥ v(r) with equality if and
πχ πχ T
only if the following hold:
• m >m or v(r)=0;
χ
• r = c·πe +(other terms) with c ∈ Z∗ and e ∈ Z such that
(m,dm) q ≥0
v(r)=v(πe ).
(m,dm)
Proof. We shall prove Part ii) first. Note that for (i,j) ∈ X one has (by Lemma
2.2):
v (ev ◦ev (π ))=v (π (π ))≥piv (π )=pi =v(π )+∆ .
πχ πχ T (i,j) πχ i χ πχ χ (i,j) (i,j)