Rademeister torsion - Illinois State University

ON REIDEMEISTER TORSION OF FINITE FIELDS
LUCIAN M. IONESCU
Abstract. Reidemeister torsion for groups [1] is used to study the structure
of basic finite fields, in relation with the POSet of prime numbers recently
defined by the author [2].
A generalized Lefschetz-Hasse-Weil zeta function is defined, and used to
study the Reidemeister zeta function.
Some connections with Ruelle’s dynamical zeta function are explored.
Contents
1. Introduction
2. Reidemeister Zeta Function and Torsion
3. Generating / Zeta functions: an interface
3.1. A Zeta Function Interface
3.2. Primon Model as a Fynmann Path Integral Quantum System
4. Structure of finite fields
References
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1. Introduction
To understand prime numbers, as one of the most present task in mathematics,
the structure of basic finite fields can be probed using the Reidemeister torsion for
groups as defined in [1], p.39.
The relation with the natural POSET structure on primes numbers, induced by
the rezolution of symmetries of basic finite fields, is studied in §4.
To better understand generating functions of the zeta function type, in §3 an
interface for Hasse-Weil zeta function is extracted from [3].
...
2. Reidemeister Zeta Function and Torsion
We follow [1]. Given an endomorphism φ : G → G of a group G, the Reidemeister
number of φ, is the number of phi-conjugacy classes, denoted R(φ). If φ is eventually
commutative, i.e. there is n such that φn (G) is abelian, then
R(φ) = R(H1 (φ)) = #Coker(1 − H1 (φ)),
so it is an effective tool to study the case of abelian groups.
Date: September 6, 2014.
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LUCIAN M. IONESCU
The Reidemeister zeta function (RZF) is the generating function for Reidemeister
numbers, as an analog of the Lefschetz / Hasse-Weil zeta function (LHW-zeta
function), as explained in [1], p.10 (see also [3], p.5):
X
Rφ (z) = exp(
R(φn )z n /n).
n
For finite abelian groups, in our case the multiplicative group of a basic finite field
(Fp×, ·), the Reidemeister number is rational:
Y
1
Rφ (z) =
,
1 − z #[γ]
where the (Euler) product is taken over the periodic orbits of φ in G.
3. Generating / Zeta functions: an interface
The relation between the zeta function as a generating function a la LepschetzHasse-Weil with the Riemann zeta function, with its defining (Dirichlet) series,
Euler and Hadamard product forms is explored, at an abstract level, independent
on context, as much as possible, as an interface.
3.1. A Zeta Function Interface. Let deg : X → Z be a function (“degree”),
grading the set X, which is partitioned into disjoint sets Xd = deg−1 (d), with
a(d) = |Xd | elements each (“closed points”; the notation parallels that from [3]).
Interpreted as lattice spaces deg : (X, <<, dm) → (Z, |, card) (“degree/mass”
function), with | the divisibility relation in (N, ·) 1, the number of points with
multiplicatity (arithmetic) function is
X
X
X
N (m) =
Deg(Xd ) =
d|Xd | =
d · a(d).
d|m
d|m
We interpret the lattice as a path space:
d|m
↔
d : 1 → m,
with a(d) the action.
Linearize the degree function on the free module of divisors generated by the set
X:
X
X
deg : ZX → Z, deg(
ni x i =
ni deg(xi ).
The zeta function (ZF) of the “path space” (X, deg, <) is
X
Z(X, t) = exp(
N (n)tn /n).
n∈N
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In general (when X is finite ), the zeta function is a rational function (following
[3], p.5).
Proposition 3.1.
Z(X, t) =
Y
x∈X
1
.
1 − tdeg(x)
1... with a path integral application in mind ...
2Is the compact case, with Haar measure, e.g. in Pontryagin duality, similar?
(1)
ON REIDEMEISTER TORSION OF FINITE FIELDS
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Proof.
ln Z(X, t) =
X
N (n)tn /n =
n∈N
=
X
X
d≥1
XX
da(d)tn /n =
n∈N d|n
(−a(d)) ln(1 − td ) =
d≥1
XX
ttd /l
d∈N l≥1

ln(1 − td )−a(d) = ln 
Y

(1 − td ))−a(d)  .
d≥1
The rational form of the zeta function Z(X, t) (RHS / Euler’s product form) follows
by collecting expanding the product over X = ∪d≥1 Xd :
Y
Y
(1 − td )a(d) =
(1 − tdeg(x) ).
d≥1
x∈X
Remark 3.1. The proof should be reinterpreted in the context of convolution arithmetic functions, probably hiding the Mobius inversion formula.
Remark 3.2. The relation with DFT of deg : Z/N Z → N , where N = |X|, should
also be explored (Generating Functions [], Discrete and Continuum Book ... []??);
the RHS product form is a DFT of convolutions of step functions (characteristic
functions of Xd ), while the divisors d|N suggest the use of Chinese Remainder
Theorem (local to global principle).
As stated in [3], Remark 3.3., , p.5, the relation with RZF is via interpreting the
LHW-zeta function as a generating function of the “effectiveP0-cycles” (divisor), i.e.
elements of gX = ZX. Given such a 0-cycle / divisor α =
ni xi , with ni ≥ 0
Y
Z(X, t) =
(1 + tdeg(x) + t2deg(x) + ...
x∈X
=
X
tdeg(α).
α∈gX
If we apply this to the case X = P and deg(p) = ln p, then then gP = ZP , we
obtain the Riemann zeta function, as stated above (see Diagram 2):
X
X
X P
Z(P, t) =
t α(p) ln p =
tln n =
n−s = ζ(s)
n∈N
α∈ZP+
P
where α = p∈P α(p)p (formal sum), n = Exp(α) =
the “Mellin substitution”3: t = e−s .
n∈N
Q
pα(p), and tln n = n−s , with
3.2. Primon Model as a Fynmann Path Integral Quantum System. By
setting X = P = Spec(Z), x = p, and deg(x) = ln p, t = e−s , the above product is
analogous to the Euler product to the Riemann zeta function:
Y
X
1
Z(X, exp(−s)) =
=
n−s = ζ(s).
1 − p−s
p∈P
3Here one silently passes from formal series, in the context of convolution algebras, to real
parameters when convergent; then, extended via analytic continuation, when daring towards Pontryagin, Fourier and Mellin duality.
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LUCIAN M. IONESCU
So, our goal is to relate the above formalism of Leipchitz-Hasse-Weil zeta function,
with the statistical mechanics formalism, via the primon model (Riemann gas) [?]:
Energylevels : Ep = ln p,
X
X
Occupation numbers : n = exp(
e(p) ln p (divisor : D =
e(p)Xp ),
X
P artition function : Z(s) =
exp(sEn ), (s = 1/kT ).
The divisibility lattice should yield a Feynman path integral formalism, which in
turn will yield the partition function as a Feynman Path Integral (FPI):
X
F P I : Z(s) =
K(A, B)
A,B∈Obj(C)
F − Amplitude :
K(A, B) =
X
exp(−S(γ)), S(γ) =
γ∈Hom(A,B)
Z
L(p, q),
γ
where C is the category of states A ∈ Obj(C) and transitions γ (paths) between
states Hom(A, B).
Remark 3.3. The Hilbert-Polya historic suggestion of realizing Rymann Hypothesis
as a self-adjoint condition in a quantum mechanics model should be upgraded in
view of the success of FPI quantization, String Theory and Riemann surfaces relation, and Kontsevich’s tools for proving Formality Theorem (byproduct: exitence of
star products, a.k.a of deformation quantizations of Poisson manifolds / symplectic
mechanics).
When the POSET of primes is endowedQ
with the natural measure dm = ln pdp,
as in the functional model of rationals r = p∈P pk(p) :
exp
/ (Q× , ·)
g = ZP
KKK
KK
KK
ln
R
k(p)dp KKK
K% R
(2)
the above proof still holds for the Riemann zeta function, with its interpretation as
a LHW-zeta function.
** PROOF? **
...
4. Structure of finite fields
Our main example of Reidemeister zeta function corresponds to G = (Fp× , ·),
the multiplicative group of a basicx finite field (the symmetries of the prime cycle
Aut(Z/pZ, +)).
...
References
[1] Alexander Fel’shtyn, Dynamical zeta functions, Nielsen theory and Reidemeister torsion,
chao-dyn/9603017v2, 1996.
[2] L. Ionescu, The POSET of prime numbers, in preparation.
[3] M. Mustata, Lectures on Hasse-Weil zeta function, http://www.math.lsa.umich.edu/ mmustata/lecture2.pdf
ON REIDEMEISTER TORSION OF FINITE FIELDS
Department of Mathematics, Illinois State University, IL 61790-4520
E-mail address : [email protected]
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