formal groups, integrable systems and number theory
DESCRIPTION
Formal Groups, Integrable Systems and Number Theory. Piergiulio Tempesta. Universidad Complutense, Madrid, Spain and Scuola Normale Superiore, Pisa, Italy. Gallipoli, June 18, 2008. Outline: the main characters. Formal groups: a brief introduction. Symmetry preserving - PowerPoint PPT PresentationTRANSCRIPT
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
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