Zeta functions of local orders [Elektronische Ressource] / vorgelegt von Siamak Firouzian Bandpey

Sujets

• Mathematics

Informations

• Publié par: universitat_regensburg
• Publié le: 01 janvier 2006
• Nombre de lectures: 5
• Langue: English

Extrait

Zeta Functions
of
Local Orders
DISSERTATION ZUR ERLANGUNG DES DOKTORGERADES DER-
˜NATURWISSENSCHAFTEN (DR. RER. NAT) DER FAKULTAT
˜NWF I-MATHEMATIK DER UNIVERSITAT REGENSBURG
vorgelegt von
Siamak Firouzian Bandpey
Regensburg, Januar 2006Promotionsgesuch eingereicht am: 10. Januar 2006
Die Arbeit wurde angeleitet von Prof. Dr.Jannsen
Prufungsaussc˜ huss: Vorsitzender : Prof. Dr. Finster
1.Gutachter : Prof. Dr. Jannsen
2.Gutachter : Prof. Dr. Schmidt
weiter Prufer˜ : Prof. Dr. GoetteContents
Introduction 4
1 Some background from commutative algebra and algebraic
geometry 10
2 Zeta functions of orders: deflnition and basic properties 24
3 A formula for the zeta function and the functional equation 35
4 Comparison with Galkin’s zeta function 46
5 A concrete example 49
6 The rational unibranch case I 59
7 Two more examples 67
8 The rational unibranch case II 74
9 On the Riemann hypothesis 79
Notation 90
Bibliography 93Introduction
The zeta-functions associated with algebraic curves over flnite flelds encode
many arithmetic properties of the curves. In the non-singular case the the-
ory is well-known. It is analogous to the theory of zeta-functions for number
flelds and culminates in the Hasse-Weil theorem about the Riemann hypoth-
esis for curves. In the singular case, which will be the main topic of this
thesis, the theory is more di–cult and less explored. First of all, one does
not deal with Dedekind rings anymore, but with orders, i.e., certain subring
ofthem. Thecorrespondingtheoryof(fractional)idealsbecomesmuchmore
complicated. Secondly, there are various candidates for the zeta-function.
In 1973 Galkin[G] published a paper which deals with the zeta-function of a
localringOofapossiblysingular,complete,geometricalirreduciblealgebraic
curve X deflned over a flnite fleld k =F of q elements. His isq
deflned in the half-plane fs2Cj Re(s) > 0g by the absolutely convergent
Dirichlet series
X
¡s‡ (s)= #(O=a) ;O
a O
where the sum is taken over the (non-zero ) ideals a in the ring O. Hence
it is formally deflned in the same way as the classical zeta-functions and it
encodes the numbers of ideals with given norms. Galkin also treated the
arithmetic case, whereO is the local ring of an order of an algebraic number
fleld. Healsodeflnedglobalzeta-functionsthisway,butitturnsoutthatthey
donothaveanyfunctionalequation,unlesstheconsideredringisGorenstein.
Green [Gr] deflned another zeta-function which always satisfles a functional
equation, but whichis not deflned interms of local conditions. In particular,Introduction 5
it does not possess an Euler product in the global case.
Finally, St˜ohr [St1],[St2] deflned a modifled zeta-function which both has a
functional equation and a purely local deflnition.The key point is to consider
all (fractional) ideals a that are positive, in the sense that they contain the
ringO, instead of considering the integral ideals, which are contained inO,
and so to deflne
X
¡s‡(O;s):= #(a=O) ; Re(s)>0:
a¶O
It is this zeta function that we will mainly consider in this paper. We want
to investigate its calculation and its properties, and for this it su–ces to
regard the local case, i.e., the case where O is a local ring. More precisely,
O will be a local order, i.e., a local integral domain of dimension 1, whose
e enormalization (integral closure)O is flnite overO. This implies thatO is a
semi-local Dedekind ring. Of course we have to assume that the residue fleld
of O is flnite (so that the groups a=O are flnite). Moreover, as in St˜ohr’s
paper we will restrict to the ‘geometric’ situation and assume that O is a
k-algebra for a flnite fleld k.
Now we discuss the plan of this thesis in more detail.
In the flrst section we will recall some facts from commutative algebra and
algebraic geometry. These will be used later, in part also for the motivation
of our investigation.
In section 2 we will introduce generalized zeta functions
X
¡s‡(d;s)= #(a=d)
a¶d
foreveryfractionalideal dinanorderO,andassociatedpartialzetafunction
X
¡s‡(d;b;s)= #(a=d) ;
a¶d
a»b
where bisanotherfractionalO-ideal,andthesumisoverallfractionalideals
a which contain d and which are equivalent to b (a=fi¢b for some fi2K).
Byintroducingthedegreeoffractionalideals,wecanwritethiszetafunction
as a power series inZ[[t]],Introduction 6
X
dega¡degdZ(d;b;t)= t ;
a¶d
a»b
¡swhere t = q . We deduce a simple reciprocity formula relating Z(d;b;t)
⁄ ⁄ ⁄and Z(b ;d ;t), where a = c: a for a dualizing ideal c ofO.
Here b : a =fx2 K j xa? bg for fractional ideals a and b. We also relate
Z(d;b;t) to Z(O;b : d;t) by simple formula. Therefore it su–ces to study
the case d = O. Most of this material is contained in St˜ohr’s paper [St1],
but we have fllled in some proofs.
Insection3weintroduceanimportantinvariantofanorderO,thesemigroup
mS(O)=f(ord (x);:::;ord (x))jx2Onf0gg?N ;p p1 m 0
ewhere p ;¢¢¢ ;p are the maximal ideals of O, and ord in the normalized1 m pi
mdiscrete valuation associated to p . We associate a similar set S(b)?Z toi
any fractional O-ideal b, and use it to give a formula for the zeta function
Z(O;b;t) (Theorem 3.10). We use this formula to show that (Theorem 3.6)
L(O;b;t) eZ(O;b;t)= =L(O;b;t)¢Z(O;t);m diƒ (1¡t )i=1
ewhere d = dim O=p and L(O;b;t) is a polynomial inZ[t] of degree • 2–i k i
e e(– = deg O = dim O=O the singularity degree of O), which satisfles thek
functional equation
¡– ¡– ⁄t L(O;b;t)=(1=qt) L(O;b ;1=qt):
We give some flrst properties of the polynomial L(O;b;t). By summing up
over the (flnitely many) representatives of the ideal classes (b) ofO, we get
similar results for
L(O;t)
Z(O;t)= m diƒ (1¡t )i=1
with
X
L(O;t)= L(O;b;t):
(b)
Again,theseresultsaremostlycontainedin[St1],wherewehaveaddedsome
proofs.Introduction 7
In the short section 4 we show that Galkin’s zeta function can be related to
St˜ohr’s (generalized, local) zeta functions. Therefore we will concentrate on
the latter in the remaining part.
In section 5 we use the mentioned explicit formula of the previous section to
calculate Z(O;t) (and hence L(O;t)) for a flrst concrete example, namely
2 3O =k[[x;y]]=(y ¡x ), which is the singularity of a cusp.
In the remaining sections, which constitute the main part the thesis, we
econcentrate on the rational unibranch case, i.e. , the case where m=1(O is
eagain a local ring) and d = 1 (k is equal to the residue flelds ofO andO).
(This situation arises, e.g., for a singularity of a curve at a totally rational
point, which just has one branch.) In this case the semigroup S(O) is a
subsemigroup ofN , and it is determined by the flnite set0
N nS(O)0
of gaps ofO, i.e., the natural numbers not contained in S(O).
Insection6wedevelopfurthertoolsforthecomputationofthezetafunctions.
For any (fractional) ideal b we deflne the numerical conductor f(b) and the
f(b)conductor F(b)= p , and we prove a formula
f(b)degb X(qt) iL(O;b;t)= n (b)ti(U :U )b O i=0
where the integers n (b) only depend on S(O) (more precisely on the gapsi
of S(b)) in a simple way. This generalizes a result of St˜ohr, who treated the
case b =O. Next we introduce another invariant of b, the ring O = b : bb
0 0 e(which is the biggest orderO;O?O ?O; operating on b), and prove the
useful formula
degObL(O;b;t)=t L(O ;b;t):b
We apply both results in section 7, where we calculate the zeta functions of
3 4 5 2 5the orders O = k[[x ;x ;x ]]? k[[X]] and O = k[[x ;x ]]? k[[X]] with12 13
gapsf1;2g andf1;3g, respectively.
In section 8 we develop further tools for the computation of the polynomials
L(O;b;t) and the zeta polynomial L(O;t) of O itself. Our strategy is toIntroduction 8
deduce information just from the semigroup S(O). We succeed in this in the
(n)case of orders with S(O) = S =f0;2;4;6;¢¢¢ ;2n;2n+1;¢¢¢g (i.e., with
gapsf1;3;5;¢¢¢ ;2n¡1g) which we call balanced. We prove for these
2 nL(O;t)=1+X +X +¢¢¢+X
2where X =qt :
In section 9 we come to the main objective of this thesis - the investigation
whentheconsideredzetafunctionssatisfytheRiemannhypothesis,i.e.,have
all zeroes on the line Re(s) = 1=2. For Z(O;t) this means that all zeroes
¡1=2fi of L(O;t) have the property jfij = q . First of all, by the functional
equation, this can only hold if O is Gorenstein (i.e., when O is a dualizing
ideal). But St˜ohr gave examples of orders which do not satisfiy
the Riemann hypothesis.
We study this more systematically. First of all we show (Theorems 9.5 and
9.6) that for balanced orders, the Riemann hypothesis holds for Z(O;t) and
the ‘principal zeta function’ Z(O;O;t) which was more often studied in the
literature. Z(O;t) was studied less often, because in general it is di–cult
to flnd all equivalence classes of ideals. Here we study it for all orders of
singularity degree • 3 and flnd that the Riemann hypothesis for Z(O;t)
only holds in the balanced cases. In the same vein, we show the following
for the principal zeta function and arbitrary (rational, unibranch) orders O
(Theorem 9.9): If S(O) is not balanced, then Z(O;O;t) does not satisfy the
Riemann hypothesis for q >>0.
We close with a sp