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\input mamath.cmm
\head{Functional equations for zeta\fonote{$\zeta$ is a Greek letter}
functions of
non-Gorenstein orders in global fields}
\subhead{Functional equations for zeta\fonote{$\zeta$ is a Greek
letter} functions\newline of
non-Gorenstein orders in global fields}
\author{Barry Green\fonote{Financial assistance from the University of
Stellenbosch, South Africa, is gratefully acknowledged.}}
\authorrunning{Green}
\abstract{
This is a small test file for the new Springer-Verlag \TeX-macro
support
for {\tt manuscripta mathematica}. It is the beginning of an article of
Barry W. Green and it is taken out of
Vol.\ 64 Fasc.\ 4, 1989. We have modified the text to
demonstrate the mamath macro package. }
\titlea{1}{Introduction}
In 1973 Galkin published a paper [8], which deals with zeta function of
a non-maximal order in an algebraic number field or function field in
one variable over a finite field of constants. By using the methods of
Haar measure and duality he was able to establish a functional equation
for non-maximal orders in a closed form involving only zeta functions
together with an elementary factor, provided they were Gorenstein. He
also gave examples to show that this functional equation does not hold
in general in the absence of the Gorenstein condition. In the proof of
the functional equation a key role is played by the dualizing module
(also called the canonical module by Kunz in [9]) for the order. For
Gorenstein orders the dualizing module may be chosen to coincide with
the ring and so no distinction between the two is needed. However for
non-Gorenstein orders they do not coincide.
 
In this paper we show the zeta function of a non-maximal order may be
redefined so that in the absence of the Gorenstein condition one still
obtains a functional equation. Recently much work has been done on the
zeta and $L$-functions of arithmetic orders, particularly for orders in
a finite dimensional semisimple algebra over the rational number field,
by Solomon [12], Bushnell and Reiner [5], [6] and [7]. However for
non-maximal non-Gorenstein orders in general it appears that little if
any work has been done.
\titlea{2}{Basic definitions}
Throughout the paper $K$ will be either the field of rational numbers
$\bbbq$, or a field of rational functions of one-variable over a finite
field of constants $F$. Let $O_0$ be either the ring of integers $\bbbz$
or the polynomial ring $F[x]$, respectively.
 
Given $L,M\in {\rm M}(O)$ with $L\subset M,\ [M:L]$ will denote the
index of $L$ in $M$. The index is extended to all fractional $O$-ideals
by defining $[L:M]=[L:W]/[M:W]$ for any $W\in {\rm M}(O)$ and
$W\subset L\cap
M$. This definition is independent of the choice of $W$ and one easily
sees that $[L:M]=[M:L]^{-1}$ and $[L:M][M:N]=[L:N]$
\definition{}{Let $M\in {\rm M}(O)$ and $s$ be a complex
variable. The
zeta function of $O$ relative to $M$ is defined as
$$
\zeta_O(M,s)=\sum_{\scriptstyle L\in {\rm M}(O)\atop
\scriptstyle L\subseteq
M}\Vert L\Vert^s_O.
$$}
 
For some $r\in E, rM\subseteq O$ and is follows that these zeta
functions
converge absolutely in the same region as $\zeta_O(s)=\zeta_O(O,s)$,
that is for ${\rm Re}(s)>1$. Consequently they are analytic in the
half-plane ${\rm Re}(s)>1$.
\titled{Euler product identity} Given $M\in {\rm M}(O)$,
for each prime ideal
$P$ of $O$ we let $\zeta_{O_P}(M_P,s)$ denote the zeta function of the
localisation, relative to $M_P$. Then
$$
\zeta_O(M,s)=\prod_P\zeta_{O_P}(M_P,s)
$$
under the assumption that at least one side of the equation is
absolutely convergent. This result is proven by similar method to that
of [8], p. 4. See also [5], p. 135 and [12], p. 316.
\titlea{3}{Main results}
\titled{Local case} Let $O$ be an order in $E$ and $P$ a prime ideal.
Then:
\theorem{3.1}{For each dualizing $O_P$-module $J_P$,
$$
\zeta_{O_P}(J_P,s)/\zeta_{\bar O_P}(s)\in\bbbz[u,u^{-1}]
$$
where $u=\Vert P\Vert^s_{O_P}$ and $\bar O_P$ denotes the integral
closure of $O_P$ in $E$.\fonote{For the case of the basefield $K=\bbbq$,
theorem 3.1 is a special case of Bushnell and Reiner's theorem 1 of
[5]}}
\remark{}{As a consequence of 3.1 and the result that $\zeta_{\bar
O_P}(s)$ admits an analytic continuation to a meromorphic function of
the complex plane this is also true of $\zeta_{O_P}(J_P,s)$.}
 
The following theorem shows the way the zeta functions depend on the
choice of $J_P$.
\theorem{3.2}{Let $J_P$ and $J_{1P}$ be dualizing $O_P$-modules in $E$.
Then
$$
\zeta_{O_P}(J_P,s)=\Vert(J_P:J_{1P})\Vert^s_{O_P}\zeta_{O_P}(J_{1P},s).
$$}
\theorem{3.3}{For each dualizing $O_P$-module $J_P,\ \zeta_{O_P}(J_P,s)$
satisfies the functional equation
$$
\zeta_{O_P}(J_P,s)/\zeta_{\bar O_P}(s)=\Vert\bar O^*_P\Vert^{2s-1}_{O_P}
\zeta_{O_P}(J_P,1-s)/\zeta_{\bar O_P}(1-s)
$$
where $\bar O^*_P=(J_P:\bar O_P)$.}
\corollary{3.4}{There exists a dualizing $O_P$-modula $J_P$ such that
$\bar O^*_P={\cal F}_P$, where ${\cal F}_P$ is the conductor of $\bar
O_P$ in $O_P$.}
\titleb{3.1}{Basic definitions}
The group $SU$(5) -- 5 $\times$ 5 unimodular matrices of determimant
1 -- has $5^2 - 1 = 24$ free entries\dots
\theorem{}{The Strong Unconstrained  Convex Function\dots}
\lemma{3.5}{Let $z\in{\rm ex}_t(F)$ and $\varepsilon>0$. Then there
exists a simplex\dots}
Consider as an example the following theorem (1.1) (1.6).
Several slightly different forms of such theorems can be found in the
standard textbooks [1,3]. In general,
the theorems show that the causal Cauchy development of a 3-surface is
incomplete if \dots
\corollary{3.6}{ For any fixed integer $n \ge1,$ there exists\dots}
\proposition{3.7}{ For any fixed integer $n \ge1,$ there exists\dots}
\proof Since ${\rm ex}_t(F)\subseteq F$, it is trivial that
${\rm hull}_t\bigl ({\rm ex}_t(F)\bigr )\subseteq {\rm
hull}_t(F)$\qed
\remark{3.8}{All the strict inequalities appearing in the strong
versions\dots}
\example{3.9}{Let $K\subseteq \bbbr^n$ be a polytope defined
as the convex hull of a given finite set\dots }
\note{3.10}{Let $K\subseteq \bbbr^n$ be a polytope defined\dots}
\definition{3.11}{A convex set K is called circumscribed if\dots}
\problem{3.12}{A convex set K is called circumscribed if\dots}
\exercise{3.13}{A convex set K is called circumscribed if\dots}
\solution{3.14}{A convex set K is called circumscribed if\dots}
Somewhat larger masses are possible in particular models. One can, in a
general, qualitative way, argue as follows\dots
\begfig 2.8 cm
{\figure{1.1}{This is the first figure legend in this section}}
\endfig
{\it [The space for the figure and its legend may appear where it is
necessary for the page layout and not where it is coded in the input
file.]}
 
These were taken during August 1959 -- December 1976
and at the National Solar Observatory
(Kitt Peak) during December 1976 -- July 1984. As the sun rotates, the
longitude of the central meridian decreases.
\begfig 3 cm
\figure{1.2}{This is the legend of a figure. The space for the figure is
3 cm. The legend is longer than one line therefore it is set as a
paragraph}
\endfig
From each daily magnetogram, the magnetic fields around the central
meridian have been used to cover all latitudes within this particular
longitude band. During the course of one solar rotation, all longitudes
get covered, resulting in a synoptic map representing the observed
longitudinal (line-of-sight) magnetic field as a function of latitude
and longitude.
\smallskip
 
\item{a. } Supernovae of Type I (SNI) are believed to occur when a white
dwarf, consisting mainly of carbon and oxygen, is accreting matter in a
close binary system, and is pushed by the mass transfer beyond the
Chandrasekahar limit of stability\dots
\item{ b.} The ``double quasar" was discovered in 1979 as a pair of
quasars, 6 arcseconds apart, with identical redshifts and spectra. The
initial guess that these were two images of one quasar produced by the
gravitational lens effect\dots
\item{c. } The extragalactic distance scale still depends on the use of
standard candles and standard reference distances (such as the distances
of the Hyades star cluster)\dots
\begref{References}
\refno {1.} Diamond, J.: Hypergeometric series with a $p$-adic variable.
Pac. J. Math. {\bf 94}, 265-278 (1981)
\refno { 2.} Ditters, E.J.: On the classification of smooth commutative
formal groups. Higher Hasse-Witt matrices of an Abelian variety in
positive characteristic. Journ\'ees de G\'eometrie Alg\'ebrique de
Rennes. Ast\'erisque {\bf 63}, 67-72 (1979)
\refno {3. } Ditters, E.J.: The formal group of an Abelian variety,
defined over $W(k)$. Rapport 144, Vrije Universiteit Amsterdam, October
1980
\refno {  4.  }Ditters, E.J.: On the covariant Dieudonn\'e-module of an
Abelian variety of dimension two over $W(k)$. Math. Z. {\bf 176},
135-150 (1981)
\refno {5.} Ditters, E.J., Hoving, S.J.: On the connected part of the
covariant Tate $p$-divisible group and the $\zeta$-function of the
family of hyperelliptic curves $y^2=1+\mu x^N$ modulo various primes.
Rapport 355, Vrije Universiteit Amsterdam, December 1987. Math. Z. {\bf
200}, 245-264 (1989)
\refno {6.} Dwork, B.: A deformation theory for the zeta function of a
hypersurface. Proc. of the International Congress of Mathematicians,
pp. 247-259 (1962)
\refno {7.} Dwork, B.: Lectures on $p$-adic differential equations.
(Grundlehren der math. Wissenschaften, vol. 253) Berlin Heidelberg New
York: Springer 1982
\refno {8.} Gross, B.H., Koblitz, N.: Gauss sums and the $p$-adic
function. Ann. Math. {\bf 109}, 569-581 (1979)
\refno {9.} Hazewinkel, M.: Formal groups and applications. New York:
Academic Press 1987
\refno {10.} Honda, T.: On the theory of commutative formal groups. J.
Math. Soc. Japan {\bf 22}, 213-246 (1970)
\endref
\address{Barry W. Green
Mathematisches Institut
Universit\"at Heidelberg
6900 Heidelberg
Federal Republic of Germany}
\bye