Formal Groups,Formal Groups,Integrable Systems and Integrable Systems and

Number TheoryNumber Theory

Universidad Complutense, Universidad Complutense,

Madrid, Spain andMadrid, Spain and

Scuola Normale Superiore, Pisa, ItalyScuola Normale Superiore, Pisa, Italy

Piergiulio Tempesta

Gallipoli, June 18, 2008

Outline: the main characters

• Finite operator theory

• Number Theory

Exact (and Quasi Exact) Solvability

Formal solutions of linear difference equations

Application: Superintegrability

Integrals of motion

Generalized Riemann zeta functions

New Appell polynomias of Bernoulli type

• Formal groups: a brief introduction

Symmetry preserving Discretization of PDE’s

Finite Operator Calculus

Formal group laws

Algebraic Topology

Riemann-typezeta functions

Bernoulli-typepolynomials

Delta operators

Symmetry preserving discretizations

P. Tempesta., C. Rend. Acad. Sci. Paris, 345, P. Tempesta., C. Rend. Acad. Sci. Paris, 345, 20072007

P. Tempesta., J. Math. Anal. Appl. P. Tempesta., J. Math. Anal. Appl. 20082008

S. Marmi, P. Tempesta, generalized Lipschitz S. Marmi, P. Tempesta, generalized Lipschitz summation formulae and hyperfunctions summation formulae and hyperfunctions 20082008, , submittedsubmitted

P. Tempesta, L - series and Hurwitz zeta functions P. Tempesta, L - series and Hurwitz zeta functions associated with formal groups and universal associated with formal groups and universal Bernoulli polynomials (Bernoulli polynomials (20082008))

P. Tempesta, A. Turbiner, P. Winternitz, J. Math. Phys, 2002

D. Levi, P. Tempesta. P. Winternitz, J. Math. Phys., 2004

D. Levi, P. Tempesta. P. Winternitz, Phys Rev D, 2005

Formal group lawsFormal group lawsLet R be a commutative ring with identityLet R be a commutative ring with identity

,..., 21 xxR be the ring of formal power series with coefficients in R

Def 1. A one-dimensional formal group law over R is a formal power series yxRyx ,, s.t.s.t.

xxx ,00,

zyxzyx ,,,,

When xyyx ,, the formal group is said to be commutative.

a unique formal series xRx such that 0, xx

Def 2. An n-dimensional formal group law over R is a collection of n formal power series

nnj yyxx ,...,,,..., 11 in 2n variables, such that

xxΦ 0,

z,yx,ΦΦzy,Φx,Φ

yxyx ,

xyyxyx ,

1) The additive formal group law

2) The multiplicative formal group law

3) The hyperbolic one ( addition of velocities in special relativity)

)1/(, xyyxyx

2244 1/11, yxxyyxyx

4) The formal group for elliptic integrals (Euler)

Indeed:

y yxx

t

dt

t

dt

t

dt0

,

0 440 4 111

ExamplesExamples

Connection with Lie groups and Connection with Lie groups and algebrasalgebras

...,,, 32 yxyxyxyx

yx,2

Let us write:

Any n- dimensional formal group law gives an n dimensional Lie algebra over the ring R,

defined in terms of the quadratic part :

xyΦyxΦyx 22 ,,,

• More generally, we can construct a formal group law of dimension n from any algebraic group or Lie group of the same dimension n, by taking coordinates at the identity and writing down the formal power series expansion of the product map. An important special case of this is the formal group law of an elliptic curve (or abelian variety)

• Viceversa, given a formal group law we can construct a Lie algebra.

Algebraic groups Formal group laws Lie algebras

• Bochner, 1946

• Serre, 1970 -

• Novikov, Bukhstaber, 1965 -

The associated formal group exponential is defined by

...6

232

3

221

2

1 t

cct

cttG

so that ttGF

2121, sFsFGss Def 4. The formal group defined by

the Lazard Universal Formal Group

The Lazard Ring is the subring of ,..., 21 ccQ generated by the coefficients of the

power series 21 sFsFG

Bukhstaber, Mischenko and Novikov : All fundamental facts of the theory of unitary cobordisms, both modern and classical, can be expressed by means of Lazard’s formal group.

• Algebraic topology: cobordism theory• Analytic number theory• Combinatorics

is called

Def. 3. Let ,..., 21 cc be indeterminates over QThe formal group logarithm is

...32

3

2

2

1 s

cs

cssF

Given a function G(t), there is always a delta difference operatorwith specific properties whose representative is G(t)

Main ideaMain idea The theory of formal groups is naturally connected The theory of formal groups is naturally connected

with finite operator theory. with finite operator theory.

It provides an elegant approach to discretize It provides an elegant approach to discretize continuous systems, in particular superintegrable continuous systems, in particular superintegrable systems, in asystems, in a symmetry preserving waysymmetry preserving way

Such discretizations correspond with a class of Such discretizations correspond with a class of interestinginteresting number theoretical structuresnumber theoretical structures (Appell (Appell polynomials of Bernoulli type, zeta functions), polynomials of Bernoulli type, zeta functions), related to the theory of formal groups.related to the theory of formal groups.

Introduction to finite operator theoryIntroduction to finite operator theory

• G. C. Rota and coll., M.I.T., 1965-

Umbral Calculus• Silvester, Cayley, Boole, Heaviside, Bell,.. 1 nn nxDx 1 nn xnx

0k

knkn xak

nax

0kknkn xa

k

nax

11 nx...xxx n

F F[[t]], P P[t]

0 !k

kk tk

atf

knnk nxt ,!

F

• Di Bucchianico, Loeb (Electr.J. Comb., 2001, survey)

algebra of f.p.s.

algebra isomorphic to P*

subalgebra of L (P)

sf F

nn axtf

nk

nkxnxt

knknk

,0

,

n

k

knk

n xak

nxtf

0

s F : subalgebra of shift-invariant operators

0

T,S

xfxTf

Def 5. Q F is a delta operator if Q x = c 0.

: polynomial in x of degree n.

Def 6. is a sequence of basic polynomials for Q if

xpn

xnpxQp nn 1

Νnn xp

10 xp npn 00

Q F Νnn xp

Def 7. An umbral operator R is an operator mapping basic sequences into basic sequences:

21 QnnQnn xqxp ΝΝ

s

s

Finite operator theory and Algebraic Topology

.

...!

...!2 1

2

1 i

Dc

DcD

i

iEE: complex orientable spectrum

Nnn xa xnaxa nnx 1 00 cxa

Appell polynomialsAppell polynomials

Additional structure in F : Heisenberg-WeylHeisenberg-Weyl algebras

F , s [Q, x ] = 1Q: delta operator,

D. Levi, P. T. and P. Winternitz, J. Math. Phys. 2004, D. Levi, P. T. and P. Winternitz, Phys. Rev. D, 2004

Lemma a) = , 1' Q

b) 1, xxQ

Nnnx : basic sequence of operators for Q

1Q 2Q

Nn

nx 1 Nn

nx 2

R: L(P) L(P)

Nnnx

x

R

R

R

R

xQQ ,'

Nnnx

Delta operators, formal groups and basic sequencesDelta operators, formal groups and basic sequences

k

ka 0 k

kka 1

m

lk

kkq Ta

1

Zml , 1, x

1 nqnn xxxP

qn

qnq nxx 1 10 xp 00 np

Theorem 1: The sequence of polynomials satisfies:

: generalized Stirling numbers of first kind

n

k

qk

qn xknSx0

,

n

k

kqqn xknsx

0

,

: generalized Stirling numbers of second kind

kns q ,

knS q ,

kn

kq

nq

kn

tknS

!!,

N

k

qkn

qk

Nq qyx

k

nyx

0

(Appell property)

k k

nmqqqq mksknSmkSkns ,,,,,

1

TQ

Simplest example: xQ nn xp

1... nxxxxxp nn

= 1

1T

Discrete derivatives:

(Formal group exponentials)

Finite operator theory and Lie SymmetriesFinite operator theory and Lie Symmetries

S: saRuRxuuuuxE qpnxxxxa ,...,1,,,0,...,,,,

u

p

i

q

xi uxuxXi

1 1

,,ˆ: generator of a symmetry groupX̂

saEXprsEEa

n ,...,1,0ˆ0...1

• Invariance condition (Lie’s theorem):

I) Generate classes of exact solutions from known ones.

III) Identify equations with isomorphic symmetry groups. They may be transformed into each other.

II) Perform Symmetry Reduction:

b) reduce the order of an ODE.

a) reduce the number of variables in a PDE and obtain particular solutions, satisfying certain boundary conditions: group invariant solutions.

•Classical Lie-point symmetries

•Higher-order symmetries

•Approximate symmetries

partial symmetries

contact symmetries

•Nonlocal symmetries (potential symmetries, theory of coverings, WE prolongation structures, pseudopotentials, ghost symmetries…)

symmetries

•Nonclassical symmetries

master symmetries

Many kinds of continuous symmetries are known:

etc.

generalized symmetries

conditional symmetries

group invariant sol.

part. invariant sol.

(A. Grundland, P. T. and P. Winternitz, J. Math. Phys. (2003))

Problems: how to extend the theory of Lie symmetries to Difference Equations?

how to discretize a differential equation in such a way that its symmetry properties are preserved?

Generalized point symmetries of Linear Generalized point symmetries of Linear Difference EquationsDifference Equations

• D. Levi, P. T. and P. Winternitz, JMP, 2004

Reduce to classical point symmetries in the continuum limit.

Differential equation

Operator equation

Family of linear difference equations

u~,xE~

u,xE:

xPx nn :

R

Nnnx

x

R

R

Nnnx

1, x

R

Let E be a linear PDE of order n 2 or a linear ODE of order n 3 with constants or polynomial coefficients and = R E be the corresponding operator equation. All difference equations obtained by specializing and projecting possess a subalgebra of Lie point or higher-order symmetries isomorphic to the Lie algebra of symmetries of E.

Theorem 2

• Differential equation

n

k

kxk xfc

0

0

• Operator equation

n

k

kk xfQc

0

0

• Family of difference equations

n

k

kqk xFc

0

0 xPfxfxF n 1

Nnn xP : basic sequence for q

qQ

Consequence: two classes of symmetries for linear P Es

Generalized point symmetries

Purely discrete symmetries

Isom. to cont. symm.

No continuum limit

R

E~

E~

Superintegrable Systems in Quantum MechanicsSuperintegrable Systems in Quantum Mechanics

• Classical mechanics

0, FH 0

t

F• Integral of motion:

Symplectic manifold ,M

• Quantum mechanics

• Integral of motion: 0, XH 0

t

X

Hilbert space:

A system is said to be

nI

nI

Integrable

Superintegrable

• minimally superintegrable if 1nI

12 nI• maximally superintegrable if

,2 nRL

Stationary Schroedinger equation (in )2E

EH yxVH ,2

1 2

There are four superintegrable potentials admitting two integrals of motion which are second order polynomials in the momenta:

• M.B. Sheftel, P. T. and P. Winternitz, J. Math. Phys. (2001)

• A. Turbiner, P. T. and P. Winternitz, J. Math. Phys (2001).

Superintegrability

Exact solvability

Generalized symmetries

22

222

y

b

x

ayxVI bx

y

ayxVII

2222 4

22 sin

cos1 cb

rr

aVIII 22

2

cba

VIV

They are superseparable

Smorodinski-Winternitz potentials

General structureGeneral structure of the integrals of motionof the integrals of motion

with

The umbral correspondence immediately provides us with discrete versions of these systems.

22

2132233113

23 PPdLPPLcLPPLbaLX

yxPLPeP ,2 2321

xP 1 yP 2 yx xyL 3

222

22

22

1yxyx

DI yxH

22

22yy

bx

ax

22222222

1 2

1

2

1yyyxxx

D ybyxaxX

0, XH

0, 1 DDI XH

Exact solvability in quantum mechanicsExact solvability in quantum mechanics

Spectral properties and discretizationSpectral properties and discretization

Def 8. A quantum mechanical system with Hamiltonian H is called exactly solvable if its complete energy spectrum can be calculated algebraically.

......21 no SSSS

preserved by the Hamiltonian:

ii SHS

The bound state eigenfunctions are given by sPxgx nn

The Hamiltonian can be written as:1ghgH nnn PEhP

jiijii JJbJah generate aff(n,R)J

Its Hilbert space S of bound states consists of a flag of finite dimensional vector spaces

Generalized harmonic oscillatorGeneralized harmonic oscillator

Gauge factor:

After a change of variables, the first superintegrable Hamiltonian becomes

It preserves the flag of polynomials

The solutions of the eigenvalue problem are Laguerre polynomials

where

2exp

2221

yxyxg pp 122 ppb 111 ppa

2211432413 12122222 JpJpJJJJJJh

212121 1625241321 ,,,,, ssssss sJsJsJsJJJ

nNNssssP NNn 212121 0, 21

22/122/1 21, yLxLyxP pm

pnmn mnmnmn PEHP

22

222

y

b

x

ayxVI

Discretization preserving the H-W algebraDiscretization preserving the H-W algebra

The commutation relations between integrals of motion as well as the spectrum and the polynomial solutions are preseved. No convergence problems arise.

Let us consider a linear spectral problem

2121 22411321

~,

~,

~,

~ssss sJsJJJ

2211432413

~12

~12

~2

~2

~~2

~~2 JpJpJJJJJJh

xxxL x ,

xxxL ,

0k

kk xax q

kk

k xax

0

1

All the discrete versions of the e.s.hamiltonians obtained preserving the Heisenberg-Weyl algebra possess at least formally the same energy spectrum. All the polynomial eigenfunctions can be algebraically computed.

Applications in Applications in Algebraic Number Algebraic Number

Theory:Theory:

Generalized Riemann Generalized Riemann zeta functionszeta functions

and and New Bernoulli – type New Bernoulli – type

Polynomials Polynomials

Formal groups and finite Formal groups and finite operator theoryoperator theory

To each delta operator it corresponds a To each delta operator it corresponds a realization of therealization of the universal formal group universal formal group lawlaw

Given a symmetry preserving discretization, Given a symmetry preserving discretization, we can associate it with a formal group law, a we can associate it with a formal group law, a Riemann-type zeta function and a class of Riemann-type zeta function and a class of Appell polynomialsAppell polynomialsSymmetry preserving

dscretization

Formal groups Zeta FunctionsGeneralized Bernoulli structures

Hyperfunctions

Formal groups and number Formal groups and number theorytheory

We will construct L - series attached We will construct L - series attached to formal group exponential laws. to formal group exponential laws.

These series are convergent and These series are convergent and generalized the Riemann zeta functiongeneralized the Riemann zeta function

The Hurwitz zeta function will also be The Hurwitz zeta function will also be generalizedgeneralized

Theorem 3Theorem 3.. Let G(t) be a formal group exponential of the form ( 2 ), such that 1/G(t)

is a function over , rapidly decreasing at infinity.

i) The function

defined for admits an holomorphic continuation to the whole and, for every we have

ii) Assume that G(t) is of the form ( 5 ). For the function L(G,s) has a representation in terms of a Dirichlet series

where the coefficients are obtained from the formal expansion

iii) Assuming that , the series for L(G,s) is absolutely and uniformly convergent

for and

Generalized Hurwitz Generalized Hurwitz functionsfunctions

Def. 9 Let G (t) be a formal group exponential of the type (4). The generalized Hurwitzzeta function associated with G is the function L(G, s, a) defined for Re s > 1 by

0

111

:,n

sns

ax

Gan

adxx

xG

e

sasL

Theorem 4.

1

,1

n

aBanL

nG

G

assLasLa GG ,1,

Lemma 1 (Hasse-type formula):

0

11

1

1

1

11log

1

1,

n

snn

sG a

nsa

sasL

Finite Operator Calculus

Formal group laws

Algebraic Topology

Riemann-typezeta functions

Bernoulli-typepolynomials

Delta operators

Symmetry preserving discretizations

Bernoulli polynomials and Bernoulli polynomials and numbersnumbers k

k

kxtt

tk

xBe

e

t

0 !1

x = 0 : Bernoulli numbers

• Theory of Riemann and Riemann-Hurwitz zeta functions

• Fermat’s Last Theorem and class field theory (Kummer)

• Measure theory in p-adic analysis (Mazur)

• Combinatorics of groups (V. I. Arnol’d)

• Ramanujan identities: QFT and Feynman diagrams

• GW invariants, soliton theory (Pandharipande, Veselov)

More than 1500 papers!

Interpolation theory (Boas and Buck)

Congruences and theory of algebraic equations

30

1,

42

1,

30

1,

6

1,

2

1,1 864210 BBBBBB

CongruencesCongruences

p

s

ns

pn

s1

1

11

If p is a prime number for which p-1 divides k, then

Zp

Bkp

k 1

212

Let m, n be positive even integers such that II. Kummer

(mod p-1),

pnm p

n

B

m

BZmod,

0nm

I. Clausen-von Staudt

where p is an odd prime. Then

Relation with the Riemann zeta function:

n

Bn n1

1

1,

nsan

as

0

1

1

1, dxx

e

e

sas s

x

ax

Hurwitz zeta function:

Integral representation:

Special values: 1

, 1

n

aBan n

Universal Bernoulli Universal Bernoulli polynomialspolynomials

xBccxB Gakn

Gak ,1, ...,...,,

Def. 10. Let ,..., 21 cc be indeterminates over Q. Consider the formal group logarithm

...32

3

2

2

1 s

cs

cssF

and the associated formal group exponential

...6

232

3

221

2

1 t

cct

cttG

so that ttGF The universal Bernoulli polynomials

are defined by

0, !k

kG

akxt

a

k

txBe

tG

t

Remark. When a = 1 and iic 1then we obtain the classical Bernoulli polynomials

Def. 11. The universal Bernoulli numbers are defined by (Clarke)

0, !k

kG

ak

a

k

tB

tG

t

( 1 )

( 2 )

( 3 )

( 4 )

Properties of UBPProperties of UBP

n

k

knGak

Gan xB

k

nxB

0,. 0

n

k

Gakn

Gak

n yBxBk

nyx

0,,

xBtG

tGxxB G

anG

an ,,1

'

Universal Clausen – von Staudt congruence (1990)

Generalized Raabe’s multiplication theorem

Theorem 4.

Assume that

Conclusions and future perspectives

• Semigroup theory of linear difference equations and finite operator theory

• Finite operator approach for describing symmetries of nonlinear difference equations

H-W algebra

Symmetry-preserving discretization of linear PDEs

class of Riemann zeta functions, Hurwitz zeta functions, Appell polynomials of Bernoulli-type

Main result: correspondence between delta operators, formal groups, Main result: correspondence between delta operators, formal groups, symmetry preserving discretizations and algebraic numbersymmetry preserving discretizations and algebraic number theorytheory

• q-estensions of the previous theory