Download From Symmetries to Number Theory via Finite Operator Theory

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
Discretization of PDE’s
• Finite operator theory
Formal solutions of linear
difference equations
Exact (and Quasi Exact) Solvability
Application: Superintegrability
Integrals of motion
Generalized Riemann zeta functions
• Number Theory
New Appell polynomias of Bernoulli type
Riemann-type
zeta functions
Finite Operator Calculus
Delta operators
Symmetry preserving
discretizations
Formal group laws
Algebraic Topology
Bernoulli-type
polynomials

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

P. Tempesta., C. Rend. Acad. Sci. Paris, 345, 2007

P. Tempesta., J. Math. Anal. Appl. 2008

S. Marmi, P. Tempesta, generalized Lipschitz summation formulae
and hyperfunctions 2008, submitted

P. Tempesta, L - series and Hurwitz zeta functions associated with
formal groups and universal Bernoulli polynomials (2008)
Formal group laws
Let R be a commutative ring with identity
Rx1, x2 ,...
be the ring of formal power series with coefficients in R
, y s.t.
Def 1. A one-dimensional formal group law over R is a formal power series x, y  Rxs.t.
x,0  0, x   x
x, y , z   x,  y, z 
When x, y    y, x  the formal group is said to be commutative.
 a unique formal series  x  Rx
such that x, x   0
Def 2. An n-dimensional formal group law over R is a collection of n formal power series
 j  x1 ,..., xn , y1 ,..., yn  in 2n variables, such that
Φx,0  x
Φx, Φy, z  ΦΦx, y , z
Examples
1) The additive formal group law
x, y   x  y
2) The multiplicative formal group law
x, y   x  y  xy
3) The hyperbolic one ( addition of velocities in special relativity)
x, y   x  y  /(1  xy)
4) The formal group for elliptic integrals (Euler)


 x, y   x 1  y 4  y 1  x 4 / 1  x 2 y 2 
Indeed:

x
0
dt
1  t4

y
0
dt
1  t4

 x, y 
0
dt
1  t4
Connection with Lie groups and algebras
• 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.
Let us write:
x, y   x  y   2 x, y   3 x, y   ...
defined in terms of the quadratic part 2 x, y  :
Any n- dimensional formal group law gives an n dimensional Lie algebra over the ring R,
x, y  Φ2 x, y  Φ2 y, x
Algebraic groups
• Bochner, 1946
• Serre, 1970 • Novikov, Bukhstaber, 1965 -
Formal group laws
Lie algebras
Def. 3. Let c1, c2 ,... be indeterminates over Q The formal group logarithm is
s2
s3
F s   s  c1  c2  ...
2
3
The associated formal group exponential is defined by
so that F Gt   t
t2
t3
2
G t   t  c1  3c1  2c2   ...
2
6
Def 4. The formal group defined by s1, s2   GF s1   F s2  is called
the Lazard Universal Formal Group
The Lazard Ring is the subring of Qc1, c2 ,... generated by the coefficients of the
power series GF s1   F s2 
• Algebraic topology: cobordism theory
• Analytic number theory
• Combinatorics
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.
Given a function G(t), there is always a delta difference operator
with specific properties whose representative is G(t)
Main idea

The theory of formal groups is naturally connected with
finite operator theory.

It provides an elegant approach to discretize continuous
systems, in particular superintegrable systems, in a
symmetry preserving way

Such discretizations correspond with a class of interesting
number theoretical structures (Appell polynomials of
Bernoulli type, zeta functions), related to the theory of
formal groups.
Introduction to finite operator theory
• Silvester, Cayley, Boole, Heaviside, Bell,..
x n  nx n1
Dx n  nx n 1
x n  xx  1 ...x  n  1 
n
x  a     a k x nk
k 0  k 
n
Umbral Calculus
x  a n    a k x nk


n
k 0  k 
• G. C. Rota and coll., M.I.T., 1965• Di Bucchianico, Loeb (Electr.J. Comb., 2001, survey)
F  F[[t]],
P  P[t]
f t  x n  an
nk x nk , k  n
t x 
k n
 0,
k
n
s

f t   
f F
t k x n  n! n ,k
ak k
t
k  0 k!
n
f t x    ak x n k
k 0  k 
n
n
algebra of f.p.s.
F
algebra isomorphic to P*
subalgebra of L (P)
F : subalgebra of shift-invariant operators
s
Tf  x   f  x   
S ,T   0
pn x  : polynomial in x of degree n.
Def 5. Q  F s is a delta operator if Q x = c  0.
Def 6. pn x nΝ is a sequence of basic polynomials for Q if
Qpn x   npn1 x 
pn 0  0 n
p0 x   1
pn x nΝ
Q  Fs
Def 7. An umbral operator R
is an operator mapping basic sequences into basic sequences:
pn xnΝ Q  qn xnΝ Q
1
2
 Finite operator theory and Algebraic Topology
E: complex orientable spectrum
.
D2
Di
  D  c1
 ...  ci 1
 ...
2!
i!
E
Appell polynomials
an x nN
 x an x   nan1 x 
a0 x   c  0
Additional structure in Fs : Heisenberg-Weyl algebra
D. Levi, P. T. and P. Winternitz, J. Math. Phys. 2004,
D. Levi, P. T. and P. Winternitz, Phys. Rev. D, 2004
Q: delta operator,   Fs ,
1
Lemma
a)  = Q ' ,

[Q, x  ] = 1
Q' Q, x


 1
b) Q, x    x 
x  
n
: basic sequence of operators for Q
nN
R: L(P)
x  
n
1
Q1
nN
R
x  
x 
Q2
x
n
2
R
L(P)
nN
n
x  
R
n
nN
nN
R

Delta operators, formal groups and basic sequences
1
, x   1
   a T l, m  Z
 a  0  ka  1
m
k

q
k l
k
k
k
k
k
(Formal group exponentials)
Q  x  = 1
Simplest example:
Discrete derivatives:
T 1
Q   
  T 1
pn  x n
pn x   x n  xx   ...x  n  1 

q 
n
Theorem 1: The sequence of polynomials Pn x  xn  x  1 satisfies:
 q xn
xn
s
q 
q 
q 
q 
 nxn1
p0 x   1
n
pn 0  0
x n   S q  n, k xk
n
  s q  n, k x k
q 
k 0
n, k  : generalized Stirling numbers of first kind
k 0
t n  q 
q 
S n, k  

n!
k!
nk

S q n, k  : generalized Stirling numbers of second kind
x  y 
N
q 
 n  q 
q 
    xk yn k
k 0  k 
N
q 
(Appell property)
 s  n, k S   k , m   S   n, k s  k , m  
q
k
q
q
k
q
m,n
k
Finite operator theory and Lie Symmetries
S:
Ea x, u, u x , u xx ,..., unx   0,
x  R p , u  R q , a  1,..., s
X̂ : generator of a symmetry group
p
q
i 1
 1
Xˆ   i x, u  xi   x, u  u
• Invariance condition (Lie’s theorem):
pr n  Xˆ Ea E ... E 0  0, a  1,..., s
1
s
I) Generate classes of exact solutions from known ones.
II) Perform Symmetry Reduction:
a) reduce the number of variables in a PDE and obtain particular solutions,
satisfying certain boundary conditions: group invariant solutions.
b) reduce the order of an ODE.
III) Identify equations with isomorphic symmetry groups.
They may be transformed into each other.
Many kinds of continuous symmetries are known:
group invariant sol.
•Classical Lie-point symmetries
part. invariant sol.
contact symmetries
•Higher-order symmetries
generalized symmetries
master symmetries
conditional symmetries
•Nonclassical symmetries
•Approximate symmetries
partial symmetries
 symmetries
•Nonlocal symmetries (potential symmetries, theory of coverings,
WE prolongation structures, pseudopotentials, ghost symmetries…)
(A. Grundland, P. T. and P. Winternitz, J. Math. Phys. (2003))
etc.
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
Difference Equations
• D. Levi, P. T. and P. Winternitz, JMP, 2004
Reduce to classical point symmetries in the continuum limit.
Operator equation
R

, x   1
Differential equation
x 
n
x
nN
R
Family of linear difference
equations
x  
n
nN
R


R : E x ,u   E x ,u~ 
~
:
x n
 Pn x
Theorem 2
Let E be a linear PDE of order n  2 or a linear ODE of order n  3 with constants or
~
polynomial coefficients and E = R E be the corresponding operator equation. All
difference equations obtained by specializing and projecting E~ possess a subalgebra of
Lie point or higher-order symmetries isomorphic to the Lie algebra of symmetries of E.
n
 ck  x f  x   0
• Differential equation
k
k 0
n
 c Q f x   0
• Operator equation
k
k 0
k
• Family of difference equations
n
k
Q  q
 ck  q F  x   0
k 0
F x   f x 1  f Pn x 
Pn x nN : basic sequence for  q
Consequence: two classes of symmetries for linear P  Es
Generalized point symmetries
Purely discrete symmetries
R
Isom. to cont. symm.
No continuum limit
Superintegrable Systems in Quantum Mechanics
• Classical mechanics
• Integral of motion:
• Quantum mechanics
• Integral of motion:
H , F  0
F
0
t

Hilbert space: L2 R n , 
H , X   0

X
0
t
Integrable
A system is said to be
M , 
Symplectic manifold
I n
Superintegrable I  n
• minimally superintegrable if
I  n 1
• maximally superintegrable if I  2n 1
Stationary Schroedinger equation (in E2)
1
H    2  V  x, y 
2
H  E 
Generalized symmetries
Superintegrability
Exact solvability
• M.B. Sheftel, P. T. and P. Winternitz, J. Math. Phys. (2001)
• A. Turbiner, P. T. and P. Winternitz, J. Math. Phys (2001).
There are four superintegrable potentials admitting two integrals of motion which are
second order polynomials in the momenta:


a
b
VI   x  y  2  2
x
y
a 1  b  c cos  
VIII   2 

r r  sin 2  
2
2
2


VII   2 4 x 2  y 2 
VIV 
a
 bx
2
y
2a  b  c
 2 2
Smorodinski-Winternitz potentials
They are superseparable
General structure of the integrals of motion

X  aL3  bL3 P1  P1L3   cL3 P2  P2 L3   d P1  P2
2
2
2
 2eP1P2  L3  P2   x, y 
H , X   0
with
L3  y x  x y
P2   y
P1   x
The umbral correspondence immediately provides us
with discrete versions of these systems.
HI
D




a
b
2
2
1 2
2
2
2
2





x


y

y
x


x x   y y
  x   y 
2
2
2
2
 1 2
D
2
2   1
2
2
2 
X 1    x   2 x x   ax x      y   2  y y   b y y  
 2
  2

H
D
I

, X D1  0

Exact solvability in quantum mechanics
Spectral 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.
Its Hilbert space S of bound states consists of a flag of finite dimensional vector spaces
So  S1  S2 ...  S n  ...
preserved by the Hamiltonian:
HS i  Si
The bound state eigenfunctions are given by

 
 n x   g x Pn s 
The Hamiltonian can be written as:
H  ghg 1
h  ai J i  bij J i J j
hPn  En Pn
J  generate aff(n,R)
Generalized harmonic oscillator


VI   2 x 2  y 2 
Gauge factor:

a
b

x2 y2

  x2  y 2 
g  x y exp 

2


p1
p2
a  p1  p1 1
b  p2  p2 1
After a change of variables, the first superintegrable Hamiltonian becomes
h  2 J 3 J1  2 J 4 J 2  2 J 3  2 J 4  2 p1  1J1  2 p2  1J 2
where
J1   s1 , J 2   s2 , J 3  s1 s1 , J 4  s2 s2 , J 5  s2 s1 , J 6  s1 s2
It preserves the flag of polynomials
Pn s1 , s2   s1 
N1
s2 N
2
0  N1  N 2  n
The solutions of the eigenvalue problem are Laguerre polynomials
HPmn  Emn Pmn
Pmn x, y   Ln
1/ 2 p1 
x L 
2
m
1/ 2 p2 
y 
2
Discretization preserving the H-W algebra
~~
~~
~
~
~
~
h  2 J 3 J1  2J 4 J 2  2 J 3  2 J 4  2 p1  1J1  2 p2  1J 2
~
~
~
~
J1   s1 , J 2   s2 , J 3  s11  s1 , J 4  s2  2  s2
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
L x , x  x    x 

L, x  x    x 
  x    ak x
k 0
k

  x  1   ak xk q 
k 0
All the discrete versions of the e.s.hamiltonians obtained preserving the HeisenbergWeyl algebra possess at least formally the same energy spectrum. All the polynomial
eigenfunctions can be algebraically computed.
Applications in Algebraic
Number Theory:
Generalized Riemann zeta
functions
and
New Bernoulli – type
Polynomials
Formal groups and finite operator theory


To each delta operator it corresponds a realization of the
universal formal group law
Given a symmetry preserving discretization, we can associate
it with a formal group law, a Riemann-type zeta function and
a class of Appell polynomials
Symmetry preserving
dscretization
Generalized Bernoulli
structures
Formal groups
Hyperfunctions
Zeta Functions
Formal groups and number theory

We will construct L - series attached to formal
group exponential laws.

These series are convergent and generalized the
Riemann zeta function

The Hurwitz zeta function will also be
generalized
Theorem 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
and, for every
admits an holomorphic continuation to the whole
we have
ii) Assume that G(t) is of the form ( 5 ). For
in terms of a Dirichlet series
where the coefficients
iii) Assuming that
for
and
the function L(G,s) has a representation
are obtained from the formal expansion
, the series for L(G,s) is absolutely and uniformly convergent
Generalized Hurwitz functions
Def. 9 Let G (t) be a formal group exponential of the type (4). The generalized Hurwitz
zeta function associated with G is the function L(G, s, a) defined for Re s > 1 by

1  e x 1a  s 1
an
LG s, a  :
x
dx


s
s  0 G x 
n n  a 
Theorem 4.
BG n 1 a 
LG  n, a   
n 1

LG s, a    sLG s  1, a 
a
Lemma 1 (Hasse-type formula):
1 log 1    1 s
1   1 n 1 s
LG s, a  
a 
a

s 1

s  1 n 0 n1
n
Riemann-type
zeta functions
Finite Operator Calculus
Delta operators
Symmetry preserving
discretizations
Formal group laws
Algebraic Topology
Bernoulli-type
polynomials
Bernoulli polynomials and numbers

t
B x 
xt
e   k tk
t
e 1
k!
k 0
x = 0 : Bernoulli numbers
1
1
1
1
1
B0  1, B1   , B2  , B4   , B6 
, B8  
2
6
30
42
30
• Fermat’s Last Theorem and class field theory (Kummer)
• Theory of Riemann and Riemann-Hurwitz zeta functions
• Measure theory in p-adic analysis (Mazur)
Interpolation theory (Boas and Buck)
• Combinatorics of groups (V. I. Arnol’d)
Congruences and theory of algebraic equations
• Ramanujan identities: QFT and Feynman diagrams
• GW invariants, soliton theory (Pandharipande, Veselov)
More than 1500 papers!
Congruences
I. Clausen-von Staudt
If p is a prime number for which p-1 divides k, then
B2 k 

p 1 2 k
1
Z
p
II. Kummer
Let m, n be positive even integers such that m  n  0 (mod p-1),
where p is an odd prime. Then
Bm Bn

,
m
n
mod p Z p
Relation with the Riemann zeta function:

1
 s 1



1

p

s
n
n 1
p
 1  n   
 s   

Hurwitz zeta function:
 s, a   
Integral representation:
1
e ax s 1
 s, a  
x dx
s  0 1  e x
Special values:
  n, a   
1
s
n 1 n  a 

Bn 1 a 
n 1
Bn
n
Universal Bernoulli polynomials
Def. 10. Letc1, c2 ,... be indeterminates over Q . Consider the formal group logarithm
s2
s3
F s   s  c1  c2  ...
2
3
and the associated formal group exponential
so that F Gt   t
are defined by
t2
t3
2
G t   t  c1  3c1  2c2   ...
2
6
(1)
(2)
The universal Bernoulli polynomials BkG,a x, c1,..., cn ...  BkG,a x 
a
 t  xt
tk
G

 e   Bk ,a x 


G
t
k!
k 0


(3)
Remark. When a = 1 and ci   1i then we obtain the classical Bernoulli polynomials
Def. 11. The universal Bernoulli numbers are defined by (Clarke)
a
k
 t 
G t

   Bk ,a


G
t
k!
k 0


(4)
Properties of UBP
n
B x      BkG,a 0x n k
k 0  k 
n
G
n.a
x  y 
n
n
    BkG,a x BnGk ,  a  y 
k 0  k 
n
Generalized Raabe’s multiplication theorem

G ' t   G
BnG1,a x    x 
 Bn ,a  x 


G
t


Universal Clausen – von Staudt congruence (1990)
Theorem 4.
Assume that
Conclusions and future perspectives
Main result: correspondence between delta operators, formal groups,
symmetry preserving discretizations and algebraic number theory
Symmetry-preserving discretization of linear PDEs
H-W algebra
class of Riemann zeta functions, Hurwitz zeta
functions, Appell polynomials of Bernoulli-type
• Finite operator approach for describing symmetries of nonlinear difference equations
• Semigroup theory of linear difference equations and finite operator theory
• q-estensions of the previous theory