nLab
Laurent series

Contents

Idea

Laurent series generalize power series by allowing both positive and negative powers. In particular, Laurent series with complex coefficients generalize Taylor series of analytic functions to meromorphic functions. A Laurent series for a meromorphic function f(z)f(z) at finite zz\in\mathbb{C} has the form

f(z)= n=k f nz n, f(z) = \sum_{n=k}^{\infty}f_n z^n,

where kk is merely constrained to be finite and is often negative.

Or, in some contexts one wants to take k=k = -\infty. Here such a formal sum (with powers extending infinitely in both directions) is suggestive notation for an element belonging to the dual nRz n\prod_{n \in \mathbb{Z}} R \cdot z^n of a ring nRz n\oplus_{n \in \mathbb{Z}} R \cdot z^n (see Laurent polynomials below). In such contexts, Laurent series can be likened to distributions, i.e., functionals on the algebra of functions R\mathbb{Z} \to R with compact support.

Definition

Definition

A Laurent series in one variable zz over a commutative unital ring kk is a doubly infinite series

f(z)= n= f nz n, f(z) = \sum_{n=-\infty}^{\infty} f_n z^n ,

where f nkf_n\in k. Equivalently: a Laurent series is a function k:nf n\mathbb{Z} \to k: n \mapsto f_n. The kk-module of Laurent series is denoted k[[z,z 1]]k[ [z, z^{-1}] ].

A Laurent polynomial is a Laurent series for which all but finitely many f nf_n are zero. Laurent polynomials form a ring which may be described as k[z,z 1]k[z, z^{-1}] or abstractly as k[x,y]/(xy1)k[x, y]/(x y - 1). Observe that Laurent series in the generality discussed here do not analogously form a ring: the obvious definition of the coefficients of the product h(z)=f(z)g(z)h(z) = f(z)g(z) of two Laurent series f(z),g(z)f(z), g(z),

h n= kf kg nk,h_n = \sum_{k \in \mathbb{Z}} f_k g_{n-k},

doesn’t make sense in general (although it could sometimes make sense in topological contexts where some such infinite sums can converge, as in the case k=k = \mathbb{C}).

Remark

The kk-vector space of Laurent series does however form a module over the ring of Laurent polynomials, i.e., if f(z)k[z,z 1]f(z) \in k[z, z^{-1}] and g(z)g(z) is a Laurent series, then the product h(z)=f(z)g(z)h(z) = f(z)g(z) as defined above always makes sense.

In general, questions of convergence are treated as separate issues. In complex analysis, the Laurent series na nz n\sum_{n \in \mathbb{Z}} a_n z^n describes a meromorphic function in a neighborhood around the point z=0z = 0 (possibly with a pole there) if all but finitely many negatively indexed terms are zero. Similarly, series of the form n=N a n(za) n\sum_{n = -N}^{\infty} a_n (z-a)^n describe meromorphic functions in a neighborhood of z=az=a with poles of order at most NN.

On the other hand, in algebra one often hears of the ring of formal Laurent series. Here, the presence of the word “ring” signifies that we are restricting the coefficients so that multiplication makes sense. Thus,

Definition

The ring of formal Laurent series over a commutative ring AA in an indeterminate xx consists of Laurent series nf nz n\sum_{n \in \mathbb{Z}} f_n z^n, with f nAf_n \in A but where all but finitely many f nf_n for n<0n \lt 0 vanish.

Multiplication defined as above clearly makes sense. If AA is a field kk, then this ring is usually denoted k((x))k((x)) and is in fact a field; indeed it is the field of fractions of the ring k[[x]]k[ [x] ] of formal power series, where k[[x]]k[ [x] ] is often viewed as a discrete valuation ring.

Remark

It would perhaps be clearer if we used the term “restricted Laurent series” to cover the Laurent series considered in Definition 2, and let “Laurent series” be the term that covers doubly infinite series.

Properties

Laurent series as distributions

Another point of view on Laurent series is given in the following alternative definition.

Definition

A Laurent series over kk is a kk-linear functional

ϕ:k[z,z 1]k\phi: k[z, z^{-1}] \to k

on the algebra of Laurent polynomials.

The associated series is nϕ(z n)z n\sum_{n \in \mathbb{Z}} \phi(z^n)z^n. But here we emphasize the point of view that Laurent series have a distribution-like character, with Laurent polynomials being considered a space of functions k\mathbb{Z} \to k with compact (finite) support, via the evident inclusion

k[z,z 1] nkz n nkz n=k .k[z, z^{-1}] \cong \oplus_{n \in \mathbb{Z}} k \cdot z^n \hookrightarrow \prod_{n \in \mathbb{Z}} k \cdot z^n = k^{\mathbb{Z}}.

Of particular interest as a Laurent series is the formal Dirac distribution,

δ(z)= nz n,\delta(z) = \sum_{n \in \mathbb{Z}} z^n,

which intuitively is like the Fourier transform n= e inx\sum_{n = -\infty}^\infty e^{i n x} of the pointwise multiplicative identity 1\mathbf{1} given by 1(n)=1\mathbf{1}(n) = 1 for all nn. It has the following property:

  • For f(z)f(z) a Laurent polynomial, f(z)δ(z)=f(1)δ(z)f(z)\delta(z) = f(1)\delta(z).

Indeed, notice that δ(z)\delta(z) is the distribution pp(1)p \mapsto p(1) which evaluates a Laurent polynomial at the multiplicative identity z=1z=1.

More generally, for ak *a \in k^\ast an invertible unit, the series δ(az)= na nz z\delta(a z) = \sum_{n \in \mathbb{Z}} a^n z^z satisfies

  • f(z)δ(az)=f(a 1)δ(az)f(z)\delta(a z) = f(a^{-1})\delta(a z).

Just as it doesn’t make sense in general to multiply distributions (at least not without heavy qualifications, as in the theory of Colombeau), we cannot make sense of expressions like δ(z) 2\delta(z)^2. However, one can meaningfully work with derivatives of distributions, so we have for example

δ(z)= nnz n\delta'(z) = \sum_{n \in \mathbb{Z}} n z^n

which has the property

  • f(z)δ(z)=f(1)δ(z)f(1)δ(z)f(z)\delta'(z) = f(1)\delta'(z) - f'(1)\delta(z).

Indeed, there is a formal calculus of the Dirac distribution that plays an important role in the theory of vertex operator algebras. See for example the treatment in Frenkel, Lepowsky, Meurman.

One can also work with convolution products h(z)=(f*g)(z)h(z) = (f \ast g)(z), defined by multiplying coefficients degree-wise:

h(z)= nf ng nz n.h(z) = \sum_{n \in \mathbb{Z}} f_n g_n z^n.

Algebraic closure

Theorem

If KK is algebraically closed and has characteristic 0, then the algebraic closure of the field of (restricted) Laurent series K((x))K((x)) over KK is the field of Puiseux series over KK.

Here “restricted” refers to Remark 2. See Puiseux series for more details on this result.

Function field analogy

function field analogy

number fields (“function fields of curves over F1”)function fields of curves over finite fields 𝔽 q\mathbb{F}_q (arithmetic curves)Riemann surfaces/complex curves
affine and projective line
\mathbb{Z} (integers)𝔽 q[z]\mathbb{F}_q[z] (polynomials, function algebra on affine line 𝔸 𝔽 q 1\mathbb{A}^1_{\mathbb{F}_q})𝒪 \mathcal{O}_{\mathbb{C}} (holomorphic functions on complex plane)
\mathbb{Q} (rational numbers)𝔽 q(z)\mathbb{F}_q(z) (rational functions)meromorphic functions on complex plane
pp (prime number/non-archimedean place)x𝔽 px \in \mathbb{F}_pxx \in \mathbb{C}
\infty (place at infinity)\infty
Spec()Spec(\mathbb{Z}) (Spec(Z))𝔸 𝔽 q 1\mathbb{A}^1_{\mathbb{F}_q} (affine line)complex plane
Spec()place Spec(\mathbb{Z}) \cup place_{\infty} 𝔽 q\mathbb{P}_{\mathbb{F}_q} (projective line)Riemann sphere
p() p()p\partial_p \coloneqq \frac{(-)^p - (-)}{p} (Fermat quotient)z\frac{\partial}{\partial z} (coordinate derivation)
genus of the rational numbers = 0genus of the Riemann sphere = 0
formal neighbourhoods
p\mathbb{Z}_p (p-adic integers)𝔽 q[[tx]]\mathbb{F}_q[ [ t -x ] ] (power series around xx)[[zx]]\mathbb{C}[ [z-x] ] (holomorphic functions on formal disk around xx)
Spf( p)×Spec()XSpf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X (“pp-arithmetic jet space” of XX at pp)formal disks in XX
p\mathbb{Q}_p (p-adic numbers)𝔽 q((zx))\mathbb{F}_q((z-x)) (Laurent series around xx)((zx))\mathbb{C}((z-x)) (holomorphic functions on punctured formal disk around xx)
𝔸 = pplace p\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p (ring of adeles)𝔸 𝔽 q((t))\mathbb{A}_{\mathbb{F}_q((t))} ( adeles of function field ) x((zx))\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x)) (restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)
𝕀 =GL 1(𝔸 )\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}}) (group of ideles)𝕀 𝔽 q((t))\mathbb{I}_{\mathbb{F}_q((t))} ( ideles of function field ) xGL 1(((zx)))\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))
theta functions
Jacobi theta function
zeta functions
Riemann zeta functionGoss zeta function
branched covering curves
KK a number field (K\mathbb{Q} \hookrightarrow K a possibly ramified finite dimensional field extension)KK a function field of an algebraic curve Σ\Sigma over 𝔽 p\mathbb{F}_pK ΣK_\Sigma (sheaf of rational functions on complex curve Σ\Sigma)
𝒪 K\mathcal{O}_K (ring of integers)𝒪 Σ\mathcal{O}_{\Sigma} (structure sheaf)
Spec an(𝒪 K)Spec()Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z}) (spectrum with archimedean places)Σ\Sigma (arithmetic curve)ΣP 1\Sigma \to \mathbb{C}P^1 (complex curve being branched cover of Riemann sphere)
() pΦ()p\frac{(-)^p - \Phi(-)}{p} (lift of Frobenius morphism/Lambda-ring structure)z\frac{\partial}{\partial z}
genus of a number fieldgenus of an algebraic curvegenus of a surface
formal neighbourhoods
vv prime ideal in ring of integers 𝒪 K\mathcal{O}_KxΣx \in \SigmaxΣx \in \Sigma
K vK_v (formal completion at vv)((z x))\mathbb{C}((z_x)) (function algebra on punctured formal disk around xx)
𝒪 K v\mathcal{O}_{K_v} (ring of integers of formal completion)[[z x]]\mathbb{C}[ [ z_x ] ] (function algebra on formal disk around xx)
𝔸 K\mathbb{A}_K (ring of adeles) xΣ ((z x))\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x)) (restricted product of function rings on all punctured formal disks around all points in Σ\Sigma)
𝒪\mathcal{O} xΣ[[z x]]\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ] (function ring on all formal disks around all points in Σ\Sigma)
𝕀 K=GL 1(𝔸 K)\mathbb{I}_K = GL_1(\mathbb{A}_K) (group of ideles) xΣ GL 1(((z x)))\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))
Galois theory
Galois groupπ 1(Σ)\pi_1(\Sigma) fundamental group
Galois representationflat connection (“local system”) on Σ\Sigma
class field theory
class field theorygeometric class field theory
Hilbert reciprocity lawArtin reciprocity lawWeil reciprocity law
GL 1(K)\GL 1(𝔸 K)GL_1(K)\backslash GL_1(\mathbb{A}_K) (idele class group)
GL 1(K)\GL 1(𝔸 K)/GL 1(𝒪)GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})Bun GL 1(Σ)Bun_{GL_1}(\Sigma) (moduli stack of line bundles, by Weil uniformization theorem)
non-abelian class field theory and automorphy
number field Langlands correspondencefunction field Langlands correspondencegeometric Langlands correspondence
GL n(K)\GL n(𝔸 K)//GL n(𝒪)GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O}) (constant sheaves on this stack form unramified automorphic representations)Bun GL n()(Σ)Bun_{GL_n(\mathbb{C})}(\Sigma) (moduli stack of bundles on the curve Σ\Sigma, by Weil uniformization theorem)
Tamagawa-Weil for number fieldsTamagawa-Weil for function fields
theta functions
Hecke theta function
zeta functions
Dedekind zeta functionWeil zeta functionzeta function of a Riemann surface

References

  • Igor Frenkel, James Lepowsky, and Arne Meurman, Vertex Operator Algebras and the Monster, Volume 134 in Pure and Applied Mathematics, Academic Press 1988.

See also

For discussion of products of distributions, see

  • J.F. Colombeau, Multiplication of distributions, Bull. Amer. Math. Soc. (N.S.) Volume 23, Number 2 (1990), 251-268. (web)
Revised on June 16, 2014 10:13:13 by Urs Schreiber (88.128.80.111)