p-Adic Mathematics

How to represent number fields as function fields ... and prove RH a la Weil!

p-adic analysis and math-physics

(ideas from Vladimirov, Volovich & Zelenov's book)

- "Tradition": physical processes take place in space-time, with coordinates assumed to be real numbers

- Quantum gravity minimal length due to uncertainty: dx²>=hG/c³; the inequality is stronger than Heisenberg's (and has a different meaning: space is discrete!)

- Archimedian axiom: refers to measurement; valid at classical / macroscopic level (human size' not micro nor "megasize"/ cosmologic).

​

analysis ...

- One should not cary over analysis to Qp; these are function spaces, so functionals are important (integration, duality, distributions etc.):

Q=F(P;Z) -> Sum/Q Fp[[x]]= algebric adeles

- Still, Qp is locally compact group (additive), so Haar measure and FT can be defined.

- What is the relation of FT on Qp, as duality, with "full" duality of (Q,+)? Q->A->Q*

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quantum mechanics

- Harmonic oscillator's role is to implement a discrete speectrum; no need anymore, if the theory is discrete from the begining; Hodge theory on graphs (Maxwell's methods for the flow of qubits; SU(2)-gauge theory on coalgebras of graphs)

- The main point: what is the physical role of primes, as factors of the principal quantum number n of hydrogen; how is the energetic spectrum of light (additive) related to the multiplicative one? (& quark form factors? generations? b/c of Fermat primes etc.)

modern string theory

- Forget landscapes (ambient space-time); just Riemann surfaces / function fields

- Why is Veneziano amplictude so important!?

A(a,b)=G(a)G(b) / G(a+b)=Int_[0,1]x^a-1(1-x)^b-1dx

Euler 2nd integrals; cocycle of Gamma(s) (symmetry functional? n!=|Aut(Sn)|)

- It is also a convolution x^a * x^b of characters:

chi(x,a)=x^a: (Q,x) x (C,+)->(C,x)

- Why is the symmetry s -> 1-s important? (Position vector in a symmetric space?)

Generalized Veneziano amplitude

- Koba-Nielsen amplitude

An=Int prod_i<j chi(a_ij; x_i-x_j) dx1...dxn

(generalize to POSets?)

- Freund and Witten adelic formula (p-adic dual amplitudes of strings over K=Qp):

Prod _p Int_Qp ||x^a(1-x^b)||_p dx=1

- Relate to p-cycles hierarchy and time-crystals ... and spectrum of H-atom.

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p-adic analysis & Quantum Groups

(loc. cit. p.xvii)

- "There are surprising connections of p-analysis with q-analysis and quantum groups"; naturally: Zp=Fp[[x]] are deformations of formal series.

​

- Haar measure on SUq(2) as a q-integral:

Int_[0,1] f(x)d_qx=(1-q)Sum f(qn)qn

is equal to the Haar integral on Qp if q=1/p, and proportional to DS_pf(s) (Dirichlet series) if q=p^s.

Misc.

- Continue with Connes article on Trace formula

- Present from Poisson formula to selberg trace formula (easy introduction)

Math-Phys Seminar

11/22/13

1) The rational circle & projective integers

2) Representing duality in Ab

3) Q is a function field

4) p-adic numbers as functions

5) The reals as the other p-adic completion

6) Riemann-Roch Theorem (Wiki version)

6. Riemann-Roch Theorem

- Tool for computing dim of meromorphic functions with prescribed zeros & poles (obvious inversion / multiplicative “duality”)

X=compact connected Riemann surface

D=divisor (C₀(X) chain)

(f)=principal divisor of a merm. fn.

l(D)=dim of merom. fn. (f)>=-D

- Riemann’s inequality: l(D)>=deg(D)-g+1

- Roch’s part: the “defect” is l(K-D), where

K=canonical divisor (divisor of merom. forms)

Some details … Topology, homology and cohomology (duality)

- Genus g=number of handles=½ Betti nr.

- Euler characteristic E(X)=2-2g (sum d: 1-2g+1)

- “Points” Divisors / chains C₀(X); d:C₁(X)->C₀(X)

- “Functions”: d:C⁰(X)->C¹(X); “meromorphic”, i.e. D.N.M. the field, but the projective space CP¹ (My “bifield”; proj. sp.; J-structure/Jordan algebra; symmetry prevails!)

- Principal divisors of mer.fn. (f) & C₀(X)/(f)

- Canonical divisors of diff. forms (w) & PD-equiv.

Some facts

- Hodge theory => g=C-dim of holom. 1-forms

- There is a canonical divisor K=(w) mod (f)

(divisor of f & of df: (d(1+zⁿ))=n-1, (1+zⁿ)=0 ?)

- The de Rham complex and homology;

- The augmentation: Deg:C₀(X)->Z

- The dimensions (dim=Euler-Poincare mapping):

l(D)=dim {f’s | (f)>=-D} (Note: - sign!!)

​

The “MRA”: Spec & filtration

- The “Hadamard product decomposition” (proj.)

Sum M^(P)(X) ----> M(X) (f=Prod f_P surj.?)

<d_P, .>: C₀(X;Z)=Z Spec(C(X)) ----> Z

- The filtration on mer. fn. M(X) over D’s

- The filtration of M^(P): … <V_-1<V₀<V₁…

(compare with finite p-adic case: (Q_p, nu_p) & Q^(p))

- Low dimensions: V_-1=0, V₀=C (Liouville’s Th.)

- poles at P: n>=0, l(nP) increasing: 1, …

- M(X) a fibered product?; rel. w. Laurent series?

Understanding l(D)=dim_C(V_D)

- It is dim_C of the vector space of meromorphic fn. f(z) with (f)+D>=0: the pole at any z₀ is no worse than the corresponding coefficient z₀(D) of D at z.

0) Generically z₀(D)=0;

+) If n=z₀(D)>0 then f has a pole at z₀ of order

n or less: f(z)=a_-nz^-n+... (locally).

-) If -n=z₀(D)<0 then f has a zero at z0 of order at least n: f(z)=a_nzⁿ+... (loc.)

​

- Linearly equivalent divisors D~D’ mod PD =>

M_f: V_D ~ V_D’ iso, M_f(g(z))=f(z)g(z) (so …?)

Riemann-Roch Theorem

- Riemann’s inequality: l(D)>=deg(D)-g+1

- Roch’s part: the excess is l(K-D) (always >=0)

- RR Th. computes an Euler characteristic (LHS):

l(D)-l(K-D)=deg(D)+(2-2g)/2 (EC=2-2g).

- Cor. deg(K)=2g-2=EC(X). Proof: D=0 => l(D)=1 by Liouville Th. (holom. fn. are constant).

- Cor. deg(D)>=2g-1 => l(D)=deg(D) + EC/2.

Proof: deg(K-D)=deg(K)-deg(D)<=-1 =>

f in l(K-D) is holom. with a zero => f=0.

Reinterpreting … maybe :)

- l=dim , deg=augmentation morphism

- substitute deg(K)=2g-2 & K=(2g-2)P (point=inf?)

dim(V_D)- dim(V_K-D)=deg(D)-deg(K)/2

=deg(D-K)+EC/2

=> dim(V_D)-deg(D)-[dim(V_D’)-deg(D’)]=EC/2

where D’=K-D; or using d(D)=dim(V_D)-deg(D):

d(D)-d(D’)=EC/2, K=D+D’ (balance eq.)

- Is it true in general that D⁺+D^-=D’⁺+D’^- =>

d(D⁺)-d(D^-)=d(D’⁺)-d(D’^-) ? (“additive fractions”)

g=0

- R-sphere=CP1, charts C0 & Cinf, transition map J(z)=1/z (link with duality, FT & partial fractions decomposition)

- w=dz on C=C0, d(1/z)= -1/z² dz has a double pole at infinity (d J(z)= - J² dz …? pole at infinity = zero in a chart at infinity?)

- canonical divisor K=div(w)=-2P, P=infinity=J(0)

- Th. => l(nP)=1,2,3, … (any point P)

- Topological meaning / visual? (wrapping the shere on itself)

​

…

g=1

Riemann-Hurwitz Formula (H’s Th.)

Relates topological info for morphisms of RS:

f:S(g)->S’(g’)

Index of a ramification point: locally ~ zⁿ

1) If f is unramified of degree N (calc. EC):

EC(S)=N EC(S’)

2) If f has ramification indexes eP at P:

N EC(S’)-EC(S)=Sum (e_P-1)

(looks like a discrete Stokes th.)

because of loss of eP-1 of P above f(P) when computing the Euler characteristic.

​

Examples

- Weierstrass P-function:RS(1)->RS(0) is a double cover (N=2), with ramif. at 4 points of e=2 (picture?): 0=2.2-Sum 1

- The “characters” zn:S(0)->S(0) have ramification index n at 0; to find e_inf, solve:

2=n.2-(n-1)-(e_inf-1)

=> e_inf=n.

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What are Riemann Morphisms?

- Meromorphic = “holomorphic characters”

f: RS(g)->RS(0)

- At zeros f wrapps the RS one way, and at poles it wrapp (near infinity) the other way, counter-clockwise like z-n:T->T ? role of z->1/z (symmetry; but Laurent series are NOT: sum zn=1/(1-z) … why? what does it mean? convol...)

- From characters / covering maps R->T & powers zⁿ:nZ->R->T, to “Riemann-Roch Th.

- Riemann-Hurwitz Formula for RS(g)->RS(g’)? e.g. g’=1? (later)

Dualities & Zeta, Gamma etc

- d/dx ln Zeta & Gamma <-> lattice periodization as in Poisson formula

- Z->R->T & Gamma=MT(Z-transform 1/n!)

- d/ds ln Zeta & elliptic curves, via extension of s.e.s. of exp from R to C.

- So, study: a) elliptic curves, Abel-Jacobi; b) w=ex & ln w (Path int. construction of RS); c) characters (cover maps zn), RS morphisms (RR-Hurwitz); d) p-adic curves & new adelic duality; e) EC & Baas / state-sums of dualities

​

FT & partial fractions

- Convolution alg. (AQG) <-> Cauchy residues (analytic side) (FT <-> Riemann-Roch; G =G^^ <-> M= + V(D); p-adic MRA for rationals)

- Infinite determinants / Hadamard - Weierstrass products & Euler product

​

Spring 2014

(Math-Phys Fridays)

RH & PNT

“Chomolungma & the Base Camp”

1/24 News on the RH

- GRH for L(X)

- RZF and periodization op. Z:

(P. Garrett, R. Meyer; L.I. dito!)

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Riemann Zeta fn. fades aways ...

- GEH: any L-function associated to a Dirichlet character X (the analytic continuation of the Dirichlet L-series DS(X)) has the zeroes on Re(s)=½.

- Ralf Meyer: the zeta operator on Schwartz fns S

(Zf)(x)=Sum_{n in Z} f(nx) (reg. reprez. of Z & R₊)

- LI: It is the multiplicative periodization operator (orbit integral), analog from Poisson Summ. F.; and multipliers & distributions point towards Algebraic Quantum Groups duality!!

Distributions everywhere ...

- RM: Zf=sum_{n in Z} l_n^-1 f(x) (reg. reprez. of R₊ on S)

- LI: (additive) Poisson S.F. distribution form

(Orb.Int.) <f, 1>_Z=<FT(f), 1>=<f, FT*(1)>

FT*= IFT ?, IFT & FT(1)=delta fn. distr. (for R)

(idea of distributions and duality; fundam. sol. & Laplacian ...)

​

RM: Zeta function ⇔ Zeta operator

- Distribution Zeta’=Sum d_n^-1, f -> Sum_N f(n^-1)

FT(Zeta’)(s)=Sum n^-s=zeta(s), Z=Int_N l(zeta’)

with the integrated form (?) of the reg. reprez.

Int_R+ l(h) f(x)=Int_(0,inf) h(t)f(t^-1x)d^xt=(h*f)(x)

(what is l(h) / “integrated form”?)

- RM: the “data” Z, zeta & zeta’ are equivalent!

(RZF fades even more!)

- RM: inverse of Z: Z^-1f(x)=sum mu(n)f(nx)

- LI: DS(distributions)? DS(mu)(1)=Sum_N mu(n) l(n)^-1

(relation with/ from zeta^-1=DS(mu)?)

A “deja vu”: Analytic vs. Algebraic

- The use of distributions to implement duality to study RZF & RH when there is an incipient algebraic approach using AQG (mainly LI), is very similar to the evolution of Deformation Theory: Kodaira, Woronovich, Jumbo, Drinfeld’.

- LI to various sponsors: crossing paths with other researchers at some key points (but no time for details … yet!).

Forget the Zeta function!!

- From Paul Garrett: Riemann’s Explicit / Exact formula, [2.2];

- Schwartz fn. f(x): “dummy variables” (dito!)

- Jacobi’s Theta function

theta_f(y)=sum_Z f(yn), y>0 (~Z-op; why Z, not N?)

- Associated gamma function:

G_f(s)=MT(f) (Mellin transform on R₊)

- Integral representation of the zeta function:

[2.2.1] Prop. MT(Av.(theta_f))=G_f(s) zeta(s)

(“Av(F)” is the usual Fourier Series average for pointwise convergence at jump discontinuities)

… and my comments

- Is this an analog of “Abel’s Multiplicative Formula”? What is the Haar integral version for a general Pontryagin duality!?

- RHS should be MT of a convolution:

MT(f) DMT(id)=MT(f * id) …?

should the DS be reinterpreted as a Discrete MT?

Sum f(n) n-s=Sum f(n)n n^-s dn/n=DMT(g)

in particular zeta(s)=DMT(Id) (not 1 !).

Euler’s factor for p=infinity

- Euler’s factor at infinity: pi^-s/2 G_e^-x(s/2)

- PG: with Riemann’s choice of “dummy” f(x)= exp(-pi x²)(Gauss distribution/eigenvalue of FT):

1) Simplest Jac.-theta fn.=sum_{n in N} exp(-pi y n²);

2) G_f(s)=½ pi^-s/2 G_e^-x(s/2) (Euler’s Gamma fn.)

- The general complete zeta function is:

CZeta(s)=G_f(s) Zeta(s)=MT(Av(Z_+/-(f)))

- LI: its rather a completed Gamma function! (var=f); the completion “at inf.” is the zeta(s) factor!

(usually f(x)=Gauss distrib. is fixed).

Riemann Zeta Function as a “renormalization” factor?

LI: so, reinterpreting PG [2.2],

MT(f) . DMT(id) = MT (Av(theta_f))

now reduce from Z to N (no reason to double):

1) theta_f(y) -> (Zf)(y) Zeta operator;

2) Is Av(theta_f) needed? 0 plays no role!

3) MT(f) DMT(id)= MT(Z(f)) (Eq.(13) in Meyer)

so zeta(s)=DMT(id)=MT(Z(f))/MT(f) (like an expectation value?), independent of f(x)!

​

RM Spectral interpretation of zeroes

- RM: Zeroes <-> spectrum of the scaling invariant vector field Df(x)=x f’(x) (a restriction D_- of D=D_- + D₊);

- Quantum Mechanics interpretation?

Pf(x)=d/dx f, Qf(x)=xf(x), [P,Q]=hi

D=QP (PQ & Hausdorff? Dyson? formulas)

(see Razvan Gelca - QM, Weil quantization, and non-commutative theta functions .. Qp / deform.)

- What is the relation between Z, integral representation of l(n) (reg. rep.) & D?

1/31

- The “holographic picture’’: Q.Sys. of RH, fractal manifolds & H-atom / qubit modes of int.

- Divisors & Q analog of Riemann-Roch Th.!?

- Iwasawa-Tate set-up & Weil’s suggestion

(homogeneous distributions & QG; dito again!)

Highlights from: Iwasawa-Tate set-up, S. Haran’s & J.-F. Burnol’s

- Euler=Hadamard products of DS(f) (zeta fn.) =>

1) Riemann-Mangold exact f. via log & d/ds (Weierstrass form) & Perron’s formula (IMT), retracing a multiplicative proof of Poisson S.F.(?)

2) Weil’s exact formula via log & MT (distribution sense);

- Iwasawa-Tate set-up: look at adeles to relate homogeneous distributions & zeroes;

- Haran-Weil exact form: uniform w.r.t. primes;

(later)

The Z-operator & Connes’ approach

- Recall: DMT(id)=MT(Zf)/MT(f);

- In additive case, Poisson Summation Formula: Zf=Z(FT(f))=FT(Zf) => FT(Zf)/FT(f)=Zf/FT(f) =?

- Cones uses the same operator (U, distributions & adeles to construct a “dynamical system”, to represent the zeroes of the Zeta f.); more later: Paula Tretkoff, Cones etc. (compare with Burnol and Mayer).

- Note: Euler’s factor & Area of unit sphere in R^m

A(||x||=1)=2 G(½)m G(m/2) (m-> s C!?)

(Terras: Harm. Anal. p.5)

The “Whole” Picture!

1) The initial impetus: primes are fundamental modes of resonance (<-> fine str. ct., quarks etc.)

2) The basic unit (lego block of the universe): qubit (QC abstract nonsense) <-> H-atom (physics: concrete nonsense; Schrod. Eq.& SM :)

3) H-atom is not understood: Schrodinger (basic model), Zeeman effect, Lamb shift, Fock’s SO(4)-model etc. (Terras: H.A. on Symm. Sp. p.94)( Does F-PI QED give a better spectrum?)

4) Goal: the “Hilbert-Polya” Q. dyn. sys. for RZF should be H-atom (FPI on “fractal manifolds”)

Fractal Manifolds

From Kontsevich formal pointed manifolds (dg-coalgebras of graphs) as the “local model”, to fractal manifolds obtained by gluing p-adic curves (Qp completed at inf. with a p-graded copy of the reals, via the inversion op. J), using the symmetry flow of Ab as a dynamical system (correlation of primes <-> propagator; hierarchy of primes <-> grading; star product on primes as a fusion rule; g(P) & Q₊ as formal groups etc. Q via sum Q(p) & Aut-gluing?)

From the RH side ...

- Spec H-atom is represented as the eigenvalues of Schrodinger Eq. (Laplacian with Coulomb / harmonic potential; h,m,e):

E(k)=-½ me⁴/h² . 1/k²

(=> fine str. constant as an “average”)

- Back to Heisenberg & QC: represent the spectral lines from an INTERACTION of TWO H-atoms … g=2 RS/string interaction? The Veneziano amplitude for strings is Euler’s Beta Int. (“cocycle of Gamma fn.”) … hmmm

- There should be a simple NT model! (God ->P)

Quantum Numbers (Terras p.94)

1) Principal Q-number: k

2) Azimuthal Q-number: n (why n(n+1)? Sets)

3) Magnetic Q-nr.: p; |p|<=n (2n+1 values/spins)

(representation theory of SU(2): qubit)

4) Total nr. of states for fixed k & n:

Sum (2n+1)=k² (hence E_k~1/k² somehow).

- Must learn first modular functions, DFT & DMT, L-functions, theta functions modular group etc. (SL(2;Z) is the key; Q->P¹Z) (maybe tomorrow :)

H-atom & Strings

- When H-atom “moves” from k1->k2 … model: change of complex structure on g=2 RS? or a symmetry of the “Riemann glasses” (g=2 is not a pretzel); Aut(g=2) and Fp, Fq correlation …

- “Energy picture”: tangent / linearized / log of probability <- Feynman amplitudes (path integrals / Riemann surfaces or field extensions).

- Study RS algebraically, without embedding them as “strings” in a “landscape”.

​

2/7

Dynamical Zeta Functions

(Reidemeister & Finite Fields)

0.1.4 Hasse-Weil Zeta Function

(see Alexander Fel’shtyn: DZF, Nielsen theory & Reidemeister torsion)

- V non-singular proj. algebraic variety over k=Fq,

V: P_i(x₀, …, x_m)=0;

- It is invariant under Frobenius map F:k^m->k^m

F(x₀,...,x_m)=(x₀^q, …, x_m^q).

- Fact: number of points of V in finite extensions K/k = number of points of V fixed by Fⁿ (…?)

- Picture: variety in various ext. & its “shadows”;

Frob. as a “discrete time” (Markov step); the variety V(K/k) partitioned into orbits “g” of the Frobenius Fⁿ (time “t=n” “attractors”?)

Def. Hasse-Weil Zeta Function

- Def. Z(z)=exp[ sum #Fix(Fⁿ)/n zⁿ]

- Structure: generating function of number of points; log - like series (exp^log (?)=?);

- Lemma ZF(z)=prod _{g in Orb(F)} 1/(1-z^#g)

(compare with Wikipedia; also Mustata / later on)

- Product of zeta functions of the Discrete Fourier Transform for the n-circle Zn (& cyclotomic ext.):

1) DFT(d₁)=Sum_k=1..inf z^nk=1/(1-zⁿ) (comp. DS(1))

2) Problem: understand “DFT / Simple fractions” relationship (primary decomposition of modules)

Relation with Riemann Zeta Function

- Change variables z=q^-s?(q=|Fq|; EC & e^z:(C,+)->(C,x)?)

- from Weil’s conjectures:

log Z(z)=Sum_l=0,...,2m (-1)^l+1 P_l(z) (P_l(z) poly.)

i.e. like an Euler char. (or Chern char. class?) in 1st homology of genus g=m …

(“ Pl(z)=det(1-z F*|H_l(V)) the characteristic polynomials of induced action of the Frobenius morphism in etale cohomology” … too much :)

2/14

- Recall Hasse-Weil Zeta

- The Whole Picture :)

- Overview of Zetas

Zeta Partition Fn/ H-atom / Fractal Manifolds

- Polya-Hilbert: QM / Hilbert space for R.H.

- Connes a.a.: C*-KMS states (abstract QM)

- These are “old” ways (before Feynman’s Era)

- The Golden Triad:

1) H-atom as QC / QM a la Heisenberg;

2) Number Theory Model: gluing basic fields => fractal (formal) manifold (a la Kontsevich/ LI)

3) Adeles: gluing (blow-up) p-adic curves (Zp^k vs. Fp^k) / the dynamical system (Ab_f, Aut).

Zeta Functions

- POSet: 1 & mu ... conv. & FT => zeta + & -.

- (N,.) 1 & mu ... convol. alg. & DS => RZF etc.

- Main example: P primes or Ab (dyn. sys.)

​

Lecture 1 - Overview of Zeta Functions

- The Hasse-Weil ZF, although def. as a gen. fn., it seems to have a cohomological “global” interpretation: Weil’s Conj. (rational fn, funct. eq., rel. with Betti numbers (!!), analog of RH), proved by Grothendieck & Deligne for smooth alg. var.

- Must study: the p-adic curve Zp=lim Zpk <-> p-adic fn. field Cl(Fp)=lim Fpk, as a formal manifold (Calculus + deformation quantization?)

​

From Lecture 2 - Hasse-Weil, Mustata

1. Review of finite fields

- Notations: k=F_q, q=p^e, [K:k]=r

- Fr(x)=x^p in Gal(k*/k) “arithmetic Frobenius element” (its inverse: geometric Fr. el. …?)

- K=Fix Fr^r the unique subextension of k* of degree r over k; in particular for k=F_q.

- Gal(K/k)=<Fr> cyclic

- Canonical (?) iso.: Gal(k*/k)=projlim G(K/k)

= projlim Z/rZ= Z^hat (& sum_p Zp? in adeles)

with Fr topological generator (?).

​

2. Varieties over Finite Fields

- X variety over k: reduced scheme of finite type

- Closed points x in X_cl: O(X,x) local ring with residue field a finite extension of deg(x):=[k(x):k]

- K-valued points (k->K hom.):

X(K)=Hom_{Spec k}(Spec K, X)=U_{x in X} Hom(k(x),K)

- If r=[K:k] => X(K)=U_deg(x)|r Hom(k(x),K) (d. sum)

-If deg(x)=e|r => G(F(q^r)/Fq)=Zr acts transit. on

Xr(K)=U_deg(x)=e Hom(k(x),K)

and Stab(phi) = G(..)=Zr => |Hom(k(x),K)|=e.

- P. |X(K)|=Sum e|r |{x in X_cl of deg(x)=e}|

The Affine case: X->A_kⁿ

(George?)

Intermezzo: the “p-adic curves” Zp

- From F. Gouveia, ch.4, group of units of Zp (later)

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A Supply of Zeta Functions

(skipped 2/21/14)

Artin-Mazur, lefschetz, Ruelle, Nielsen, Reidemeister

0.1.5 Dynamical Zeta Functions

- A discrete dynamical system (DDS) is an automorphism f:X->X (in some category, e.g. Top, Manifolds, R-modules etc.), representing the discrete time-step change, similar to a Markov system.

- The Artin-Mazur zeta function is a generating function for the number of fixed points of a discrete dynamical system f:X->X in Top:

F_f(z)=exp(Sum_n F(fⁿ) zⁿ/n)

- Example: the Ihara zeta function of a graph can be interpreted as an A-M ZF (see wiki).

​

The Lefschetz Zeta Function

- Defined by Smale for a diffeomorphism of a manifold f:M->M (disc. dyn. sys. in Manifolds):

L_f(z)=exp(Sum_n L(fⁿ) zⁿ/n)

with L(g)=Sum_k=0..dim(X) (-1)^k Tr[g_*k:H_k(X;Q)->H_k(X;Q)]

(the index of the d-flow? Tr(g_*)=sum E-Val. of ?)

- LZF is a rational function (Prod. of char. poly.?)

L_f(z)=Prod_0..dim(X) det(I-f_*k . z)^{(-1)^(k+1)}

- Dold: Sum_d|n mu(d) L(f^n/d)=0 in Zn.

(Meaning? d|n <-> subgroups of Zn; mu*L(f) and mu*1=d? (L(f) as an arithmetic function: |Ind(fⁿ)|)

Nielsen Theory: another way to count fixed points of DDS

- p:U(X)->X universal cover of X; U(f) lift of f; two lifts are conjugate iff conjugated by a deck transformation in Pi₁(X).

- The fixed point class of U(f): p(Fix(U(f)))<Fix(f).

- Reidemeister number of f(x) is

R(f)= # of lifting classes of f(x)

- Nielsen number of f(x) is

N(f)= # of essential fixed point classes (non-zero index in Pi₁(X)).

Reidemeister Numbers in Groups

- Let f:G->G be a DDS in Groups, f in End(G) (Not necessarily an automorphism!)

- Definitions:

1) a & b are f-conjugate iff there is g in G such that b = g a f(g)^-1

2) Def. the number of f-conjugate classes is called the Reidemeister number of f, denoted by R(f).

3) The Reidemeister zeta function of f is:

R_f(z)=exp(Sum R(fⁿ) zⁿ/n)

Fluid Dynamics Interpretation

- Lagrangian specification: tracing a particle in time: g(n)=fⁿ(g), with g in G (“space”), n in Z (discrete time);

- This makes sense when G=A is Abelian (Z-module), i.e. it is a “discrete vector space”, not a “group of transformations”;

- Then, in additive notation:

f-conjugate: g(1)-g(0)=a-b

i.e. the “free vector” a-b is a velocity vector g(1)-g(0) (in particular it is parallel to the flow).

Applications to Finite Fields

- Let a in (F_p^x , .) and M_a(x)=ax: (Z_p,+)->(Z_p,+)

- Questions:

1) What is R(M_a)?

2) What is R_Ma(z)?

3) What is the relation with the orbit partition of Zp (of its projective space?)?

- Example p=7, F_p^x = Z₂ x Z₃

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Miscellaneous

2/28/14

- Methodology of research & teaching

- Cyclotomic poly & generic Zeta Functions

- Mobius transform & path integrals

A more efficient math-methodology?

Interactions between mathematicians A & B (&C :):

- Research: A applies his knowledge to B’s problem;

- Teaching: not downloading knowledge of A onto B (“learn & DIY”), but “translating” from (e.g.) “algebraic geometry” into a language of the B’s problem: “how would you rephrase classical algebraic geometry facts in this context” (that’s more than teaching; “variations on a theme of Paganini” by … )

- Collaboration: “here’s the idea; can WE prove it?” :) (is it worth the time? article? seminar talk? fun?)

Cyclotomic polynomials

- Def. CPn(z)=Prod_ord(r)=n (x-r) its zeroes are generators of nth roots of unity Un

- exp:Zn->Un; Ab(n)=subgroups of Zn (Ab cutoff at level n) and lattice structure.

- Polynomials as group rings, so … functor .@k

- X=Spec kZn (or Un?) = {r1, …, r_phi(n)}

- The partition of Zn into generators &

n=Sum_d|n phi(d)

apply @k => CPn=prodd|n CPd

- Compare with affine varieties (Mustata) ...

Affine Varieties (Mustata / Lecture 2)

- X variety over a finite field, k=Fq, K/k field extension of degree r, K-valued closed point

X(K) = Hom_Spec(k) (Spec(K), X)

=U_{x in X} Hom_k-alg(k(x),K)

|X(K)|=Sum _e|r e . |{x in Xcl | deg(x)=e}

- What part of this algebraic geometry paradigm holds for Zn (Z-modules /discrete vector spaces)?

ideals <-> submodules, prime <-> minimal, k(x) <-> submodules gen. by x etc. (lattice struct.)

- Extensions are … extensions! fields are too restrictive!

back to finite fields

- Cyclotomic extensions <-> group rings, right?

- Then is Fp CPn(x)= Fp[x]/(CPn(x)) ?

- Example: F_p CP₂=Fp[x]/(x²+x+1) = F_p2, right?

- So we may “think” finite fields Fpⁿ are “0-dim affine varieties” … can we?

Generic Zeta Functions

Behind the Hasse-Weil ZF / Lecture 2, p.5:

- Given a degree function: deg:X->N (X finite), and the associated partition of X=UX_e, X_e= {x|deg(x)=e}, a(e)=|X_e|, pullback lattice of (N,.): “closed points” X(d), d|lcm(degrees) (X complete?),

N(m)=Sum _e|m Deg(X_e)=Sum _e|m e.a(e) (path int.)

(& convolution a*id? P.I. & kernel; & ind(H)=[G:H]?)

- Use divisors (and relate to kG vs. F(G)): a=Sum m_i x_i and linearize degrees deg(a)=Sum m_i deg(x_i);

Path integral interpretation … (Lagrangian & action)

The Generic ZF is rational

- The ZF of the “path space” (X, deg, <) is

Z(X,t)=exp(Sum N(m) t^m/m)

- Th. Z(X,t)=prod_{x in X} (1-t ^deg(x))^-1

proof: (following M/L2, p.5)

1) log(Z(X,t))=Sum_m N(m)t^m/m=

Sum_m Sum_r|m r a(r) t^m/m=...

=sum (-a(r))log(1-t^r)=log(1/prod (1-t^r)).

- Euler’s product form:

Z(X,t)=prod_{x in X}(1+t^deg(x)+t^2deg(x)+...)

Mobius Transform & Path Integrals

- Context: lattice, e.g. (N,.), (P, <) etc.; let F(n) arithmetic function:

(TF)(n)=Sum d|n F(d) mu(n/d)

so TF=F*mu is the derivative of F (wiki: “Mobius inverse of F”? why!)

- Its inverse is I(f)=f*1 (mu*1=d), the path integral

Int₁ⁿ f=Sum _d|n f(d)=(f*1)(n).

- From functions to power series (convol. alg):

Lambert series: Sum a_nxⁿ=Sum b_nxⁿ(1-xⁿ) (meaning? FT / z-transform?)

Apply to general Zeta Functions

- Context: lattice, derivative & path integral …

- Local degree as a “Lagrangean”, path integral as an action (Sum d|n deg(d)) …

- What’s Z(X,t)? partition function, somehow?

Prime Number Theorem and

Riemann Zeta Function

(Alg. Sem. 3/4/2012)

Contents

- Alternative formulations of PNT

- Riemann zeta function and PNT

- Statistical mechanics interpretation

- Exact formulas

- In search of a duality

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4.3 Equivalent Formulations of PNT

- Recall: functional model of rationals (Q+, ., |) vs. (N, +, <) (int=ln o Exp);

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- Part of a larger framework: Q₊ is an Algebraic Quantum Group (g, G, U(g), F(G) & AQG duality).

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- PNT & X=”is n prime?” as a random variable

X:(P, ln(p))->(N, 1/ln(n)),

PNT pi(x)/x ~ 1/ln(x) ⇔ fr(X) ~ P(1<X<x)

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A Physics Interpretation: Primon Gas

- A statistical mechanics interpretation:

1) Primes p are basic states of energy E_p=E₀ln(p)

2) A natural number is a population of particles with occupation numbers N(p), s.t. (FTA):

n=Prod _{p in P} p^N(p), 2^nd quantiz.: E(n)=E₀ ln(n).

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- The partition function (probability distribution) is … the Riemann Zeta Function (1/kT=b=s):

Z(b)=Sum_n e^-bEn=Sum n^-s=DS(1) (Dirichlet Series)

Other functions used to count primes

- Chebyshev’s functions:

theta(x)=sum _p<x ln(p), psi(x)=Sum_pk_<x ln(p);

… and relations with pi(x)=Sum _p<x 1

1) x>5 => pi(x)<=theta(x)<=psi(x)

2) psi(x)=Sum_n 1/n theta(x^1/n)

3) PNT ⇔ psi(x) ~ x ⇔ theta(x) ~ x

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The Riemann Zeta Function

- Dirichlet series of a_n is DS(a)=Sum a_n n^-s

- Mellin transform MT(f)=Int_R+f(x) x^-s dx/x

(“multiplicative Fourier transform”; via exp(x))

- Riemann zeta Function: Z(s)=DS(1)

- Euler’s Product: Z(s)=Prod _p 1/(1-p^-s)

- Thermodynamic Energy:

<E>= - d/ds ln Z(s) = DS(L(n))

like a Weierstrass Zeta Function related to Psi(x).

Relation with Chebyshev’s Psi(x)

- Compare Psi(x)=Sum_n<x L(n) with

<E>_[0,x] (0) = Sum n<x L(n) = Psi(x)

at infinite temperature “s=1/kT=0” (“no interactions”)

- Mellin transform: Energy vs. probability

<E>_s= - Z’/Z(s)=s MT₁(Psi(x))

- Density of energy: <E>_s/s = MT₁(Psi(x))

- Question: how MT affects psi(x) ~ x ?

Riemann-Mangoldt

Exact Formula

alg. sem. 3/18/14

… and Exact Formulas

(Multiplicative Poisson Summation Formulas)

1) Riemann-Mangold (via Inverse MT & Hadamard prod):

Psi(x)=x-Sum x^r/r + Z’(0)/Z(0)-½ ln(1-x^-2)

2) & Weierstrass zeta function (think Elliptic Curves!)

½ ln(1-x^-2)=Sum x^r/r, r=-2n trivial zeroes of Gamma function G_f(s)=MT(e^-z) (Mellin transform/EC).

3) The exact formula for thermodynamic energy:

<E>=-d/ds ln Z(s)=Sum _{c in CP} 1/(z-c)|₀^s

In search of a duality ...

- Q=ZP & primes <-dual-> F(G) & zeroes … G=?

Sum_{p in P} ln(p)/p . 1/(p^s-1) =<E>=Sum _{c in CP} 1/(s-c)

all poles and zeroes, trivial or not;

- Convergence is NOT an issue: use distributions framework (Weil) or formal series framework (QG):

DS(1)=MT(Z(f)) / MT(f) (Paul Garrett p. 5 & B. Julia p.1374)

- The “key”: “orbit integral” Z(f) and AQG (adeles).

The Discrete-Continuum Connection

- The discrete-continuum relation Z->R (Pontryagin duality) via Abel’s formula (Julia, p.281):

Sum₁^x a_n g(n)=A(x) g(x)-Int₁^x A(t)g(t) dt, A(x)=Sum₁^x a_n

- Examples with x=infinity & g(x)=x^-s (fn.)

1) a(n)=1 or mu(n) => Z+/- (fermi/bose); A(x) counts integers or is M(x) (super-symm. content);

2) a(n)=l(n) (=1/k at p^k) or L(n) => log(Z) or -Z’/Z, with A(x)= Pi(x)=Sum 1/k pi(x^1/k) or =psi(x) (& exact f.)

and the role of Mellin Transform

(loc. cit. B. Julia p.1374) Mellin Transform:

Modular Forms / additive generating functions -> Zeta / L-functions (multiplicative generating functions):

DS(A)=MT₁(<A|f>) / MT(f) (MT₁?)

(Is Int₁^inf a path integral on the Lie group?)

where <A|f>=Sum _{n in N} A(n) f(nx) is the distribution with kernel A(n) (f some Schwartz fn).

(multiplicative functions are “distributions”:

A(n)=Prod p in P A(p^k) (Q’ <-?-> S’)

from Pontryagin adelic duality for Q+ ...

Adelic duality & Fourier Transform

(Q,+) -> A -> (Q^, .) is related to the module decomposition of (Q,+): Euler product form <-> partial fractions decomposition, which corresponds to a Fourier transform of the associated arithmetic function etc. … how?

- p-adic numbers and characters of Q

- Discrete Fourier Transform via partial fractions (roots of unity & cyclotomic polynomials)

3/7/2014

- Generic Zeta Functions

- Generalized Mellin Transform

Generic Zeta Functions

- see outline

- Mellin transform & the representation:

(C,+)->End(C, .) (see TeX file)

and the polar form C^x= (R₊ , .) x (T,.) ...

- Poisson Summation Formula:

FT(Z(f))(0)=FT(F)(0) & FT(1)(0) …?

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