Relations Between Polarized Structure Functions

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Notes On Hilbert's 12th Problem

a rX iv:mat h /6511v1[mat h.NT]3M a y261NOTES ON HILBERT’S 12TH PROBLEM SIXIN ZENG Abstract.In this note we will study the Hilbert’s 12th problem for a primitive CM ﬁeld,and the corresponding Stark’s ing the idea of “Mirror Symmetry”,we will show how to generate all the class ﬁelds of a given primitive CM ﬁeld,thus complete the work of Shimura-Taniyama-Weil.Introduction 0.1.Let K be a number ﬁeld,H K be the ideal class group of K ,and let K 0be the Hilbert class ﬁeld of K .The class ﬁeld theory tells us there is a canonical isomorphism Gal (K 0/K )≃H K .In general given any integral ideal d ,let K d be the maximal abelian extension of K unramiﬁed outside d ,and let H d be the generalized ideal class group relative to d ,then a similar isomorphism holds as well:Gal (K d /K )≃H d .On the other hand,Hilbert’s 12th problem asking for an explicit generation of all the abelian extension of K ,more precisely it is asking for ﬁnding a special transcendental function,whose values at some special points would generate all the abelian extension of K .When K is the rational ﬁeld Q ,the transcendental function is the exponen-tial function and the special points are the division points on the unit circle.Indeed by the classical Kronecker-Weber theorem,all the abelian extension of Q can be obtained by adding the roots of unity to Q .When K is an imaginery quadratic ﬁeld,this problem is answered by the theory of complex multiplications.As this theory has very much inﬂuenced the later thinking about this problem,let’s recall some details.So let K be such a ﬁeld,consider the set of elliptic curves E satisfying End (E )⊗Q ≃K ,i.e.,the set of elliptic curves with complex multiplications by K .All such elliptic curves can be constructed by the following way:take the caononical embeddingι:K −→C ,let a be an integral ideal of K ,then a is a rank 2Z -module in C ,i.e.,ι(a )is a lattice in C ,so we can take the quotient C /ι(a )which is an2SIXIN ZENGelliptic curve with complex multiplication by K.From the construction it is clear that the ideal group of K operates on this set of elliptic curves,indeed for any representative b of an ideal class,we have the isogeny:C/ι(a)→C/ι(ab). On the other hand all such elliptic curves are deﬁned over some numberﬁeld, and they are represented by the moduli points on the moduli space of all elliptic curves M1=H/P SL2(Z).In particular the Galois group acts naturally on the moduli points of this set of elliptic curves.So if E=C/ι(a),let℘be a prime of K,E(℘)=C/ι(a℘),s(℘)∈H K≃Gal(K0/K)the Artin symbol,then the main result of the complex multiplication is E s(℘)≃E(℘).In more concrete terms if j is the classical modular function,and let p E be the moduli point of E on M1=H/P SL2(Z),then j(p E)is an algebraic number and generates the Hilbert classﬁeld of K.Similar statements hold for all the ray classﬁelds of K.The idea behind the theory of complex multiplication can be summerized as the following:•Given the numberﬁeld K,in order to solve the Hilbert’s12th problem,ﬁrst we need toﬁnd a suitable class of algebro-geometric objects X,say varieties;•X should be closely related to K such that the ideal classes of K cannaturally act on them;•All such X should be deﬁned over some numberﬁeld,all such X shouldbe living on some natural moduli space so theirﬁeld of moduli are givenby the coordinate function evaluated at the moduli points.The Galoisgroup can naturally acts on them;•Moreover the action of the ideal classes of K and the action of Galoisgroup on the moduli points should be related by the reciprocity law ofclassﬁeld theory and the Kronecker congruence relations;If we canﬁnd such a class of X,then theﬁeld of moduli of X give the answer to Hilbert’s12th problem.For a general numberﬁeld besides imaginery quadratic,this philosophy is diﬃcult to apply.So from now on we will concentrate on a special case.From now on let K be a primitive CMﬁeld of degree2n,n≥1.K is then an imaginery quadratic extension of a totally realﬁeld F,with[F:Q]=n. Further let K∗be the reﬂexiveﬁeld of K,F∗be the maximal totally real subﬁeld of K∗,then K∗is also a primitive CMﬁeld of degree2n,[K∗:F∗]=2. Let{σ1,σ2,···,σn}be the set of archemidean primes of F,lifted to K as the CM types,and letρbe the complex conjugate.When n>1,according to the above philosophy,we naturally consider the set of abelian varieties of dimension n with complex multiplication by K with CM type{σ1,σ2,···,σn},i.e.,those abelian varieties satisfying End(X)⊗Q≃NOTES ON HILBERT’S12TH PROBLEM3 K,and moreover for anyα∈K the action ofαon the space of holomorphic diﬀerentials of X is given as the diagonal action of diag(ασ1,···,ασn).All such X can be constructed in the following way:letι:K→C n be the embedding ι(x)=(xσ1,···,xσn),then for any integral ideal a,ι(a)⊂C n as a Z-lattice is of rank2n,hence the quotient X=C n/ι(a)is an abelian variety,one see that such X satisfying the above conditions.From the construction such set of abelian varieties are naturally associated to K,with the integral ideals of K act on them as isogenies.This is exactly the same as elliptic curves.So next we shall consider the moduli space for these X and their arithmetic properties,it is here some complication arised,let’s be careful.Let O K and O F be the rings of integers of K and F,and let a be an integral ideal of O K.Regarding a as a module of O F,it is of rank2.More precisely we have the following(see for example,Yoshita’s book[19]):Lemma0.1.For any ideal a of O K we have the isomorphism as O F-module: a≃b·ω1⊕O F·ω2,whereω1,ω2∈K,b a fractional ideal of O F.Moreover the ideal class of b is c·N K/F(a)=c·aaρ,where c is a fractional ideal of F, independent of a,and c2=D K/F.In particular O K=c·ω1⊕O F·ω2,and if a satisfying a·aρ=(µ),then a=c·ω1⊕O F·ω2.Now let X=C n/ι(a),by Shimura the polarization of X is given by the Riemann form E(x,y)which is E(x,y)=T r K/Q(ζxyρ),withζ∈K satisfying ζρ=−ζ,Im(ζσi)>0,any i.The“type”of this polarization,by deﬁnition,is the ideal class ofζd K/F aaρ,where d K/F is the relative diﬀerent of K over F. From the above lemma it is clear that b is the type of X.For any moduli problem,in order to have a good moduli space we need to ﬁx a polarization of X.For our class of X,we observe that they all have a large ample cone,in fact the dimension of the ample cone are all of n,hence it makes sense to consider the moduli problem with all the ample classesﬁxed. This is the moduli space of“type polarized abelian varieties”.Concretely for any fractional ideal b of O F,we can constructed a moduli space M n(b)which parametrizes families of abelian varieties with the form C n/v n·ι(b)⊕ι(O F),where v n is a vector in the product of upper half plane H n,and the dot product is deﬁned component-wise.It is well-known that M n(b)=H n/Γ(b)withΓ(b)={α∈SL2(O F)|α≡1mod(b)}.If X is the above abelian variety of type b,the X=C n/ι(a)with a=b·ω1⊕O F·ω2, up to isomorphism we can chooseω1,ω2∈K such thatωX=ω1/ω2satisfyingIm(ωσiX )>0for any i,i.e.(ωσ1X,···,ωσnX)is a vector in H n,hence deﬁnes amoduli point of X in M n(b).4SIXIN ZENGSo in summary given the class of CM abelian varieties with aﬁxed CM type,we can construct a natural moduli space M n(b)for aﬁxed type of X,i.e.,ifX1,X2are of the same type,they live on the same moduli space,but diﬀerenttypes can produce diﬀerent moduli spaces.So although the ideal classes of Kcan act on all of the X,the Galois group can only act on the subclass of Xwith the same type.On the other hand it is easy to show that all X are deﬁned over aﬁniteextension of the reﬂexﬁeld K∗.In fact for an ideal a∗of K∗,we deﬁnea=g(a∗)= τi a∗τi,where{τ1,···,τn}is a CM type of K∗,then g(a∗) acting on X will not change the type of X.It is from this point of view thatin1955,Shimura-Taniyama-Weil(see Shimura’s book[12])established that theﬁeld of moduli of X will generate part of the classﬁeld of K∗,precisely it isthe classﬁeld corresponding to the subgroup of the ideal class group H0K∗= {a∈H K∗|g(a)=(µ),µ∈K}.0.2.In this note we will try to extend the work of Shimura-Taniyama-Weil to cover all the classﬁelds of K.There are actually two problems here.First, the appearance of the reﬂexﬁeld K∗is quite inconvenient,since our abelian varieties X,their moduli spaces M n(a),and the CM points on the moduli are all constructed naturally from the given CMﬁeld K,it is natural for us to look for the invariants from these abelian varieties that directly generate the class ﬁelds of K,instead of the reﬂex K∗.Second and more important problem,is how can we deal with the classﬁelds that corresponding to the isogenies of abelian varieties that live on the diﬀerent type of moduli spaces.Theﬁrst problem can be solved in the following way.Since theﬁelds K and K∗are reﬂex to each other,by Shimura-Taniyama-Weil theory,the classﬁelds of K is generated by the abelian varieties associated to K∗.So the problem is toﬁnd invariants on X that somehow related to Shimura varieties associated to K∗.For this purpose Iﬁnd the following notion of“cone polarized Hodge Structure”quite useful.First we observed the for all the abelian varieties of the CM type the Kahler cone is very large.Indeed as the polarizations are determined by the Riemann form E(x,y)=T r K/Q(ζxyρ),so the polarizations are determined byζ∈K such thatζρ=−ζ,Im(ζσi)>0,any i.Suchζform a cone of dimension n, hence the Kahler cone is of n dimensional.We can also see this by the fact that since End(X)is an order in K,the automorphism group of X is then the unit group of the order,which is of the form Z n−1⊕T orsion.So the Kahler cone of X is necessarily n dimensional.Given such Kahler cone C X of X we can consider the primitive classes of X in the middle dimension relateive to this cone,i.e.,those classes of dimension n such that are annilated by any element in the Kahler cone.NOTES ON HILBERT’S12TH PROBLEM5{α∈H n(X,C) α·x=0,∀x∈C X}This is the generalization of the notion“transcendental lattice”in the theory of K3surface.The Kahler cone and these primitive classes relative to this cone should be considered as the most basic Hodge theoretical invariants of our abelian varieties.So to understand these abelian varieties,we need to construct the appropriate moduli spaces for these invariants.First let’sﬁx the Kahler cone and consider the primitive classes of the middle degree relative to this cone.Such primitive classes carries a natural Hodge structure,indeed for abelian varieties with F-multiplication the primi-tive classes can be characterized as the invariant classes of the automorphism group U F,it carries a Hodge structure of weight n,with the i-th Hodge number to be i n =n!6SIXIN ZENGTheorem0.2.(1)We have a natural isomorphism M P H≃H n/SL2(O F∗);(2)The natural morphisms M n(a)→M C→M P H are allﬁnite.By the theory of Shimura-Taniyama-Weil,the moduli points of X on this moduli space will generate the classﬁeld of K.The natural modular function on the Hilbert modular variety of K∗,when pulled back to M n(a),will become a natural modular function on M n(a).In this way,by directly considering the geometry of X,we can have the classﬁelds of K generated.This answers our ﬁrst question.Note that in our approach we do not emphasis on the use of“ﬁeld of moduli”, indeed since our moduli space of primitive Hodge structures can actually be regarded as the moduli space of abelian varieties with multiplication by F∗,the natural“ﬁeld of moduli”somehow lose its meaning on this moduli space(1). We only need the natural modular functions on the moduli spaces,which give the coordinates of the CM points.In any case our approach is more natural in view of Hodge theory,and serve the purpose of generating classﬁelds of K well.0.3.The second question is more diﬃcult.We need toﬁnd a natural way to interpolate the moduli spaces of diﬀerent types.For this purpose we will use the idea of Mirror Symmetry.Precisely if R1and R2are two ideals suchthat R1R−12is a real ideal,then although the abelian varieties X R1and X R2deﬁned by R1and R2are living on the diﬀerent moduli space,we will showtheir“Mirror partners”X′R1and X′R2are living on a single moduli,and theMirrors’ﬁeld of moduli can be used to generate the classﬁelds.To motivate our idea,let’s consider another approach to the Hilbert’s12th problem,namely the Stark’s conjectures([16]).The point of departure is to consider the Dedekind zeta function and Hecke L-functions for the numberﬁelds.Hilbert’s problem asks for a natural tran-scendental function,indeed for the abelian extension point of view,nothing can be more natural than these L-functions,as they transformed under the Galois group ually the Stark’s conjectures are formulated and studied for a totally realﬁeld,but as we shall see,it is more natural and sim-ple to study it for the CMﬁeld,because all theﬁnite extension of CMﬁelds are necessarily CM,i.e.,any extension of K has no real inﬁnity.The remarkable fact about the Stark’s conjectures is that it can be for-mulated on any classﬁelds of K uniformly,not just those classﬁelds in the Shimura-Taniyama-Weil theory,so in the explicit form it can be used as a guide for us to search for the solution of missing classﬁelds of the old theory.NOTES ON HILBERT’S12TH PROBLEM7 So let K as above,let R be an ideal class of K,recall that the Dedekind zeta function is deﬁned asζK(s)= a1s−1κK+ρK+O(s−1)whereκK=(2π)n R KN(a)s .NatuallyζK(s)= RζK(s,R),with the limitformula:ζK(s,R)=κw Ks n−1(1+δK(R)s)+O(s n+1) thenδK(R)=nγ+nlog2π−log|D K|−w K|D K|1/2ρK(R−1)w Ks n−1(1+δK s)+O(s n+1) withδK= RδK(R)•ifχ=χ0is not trivial,thenL K(s,χ)=ζK(s,R)=−R Kw K0=(−1)m(R K8SIXIN ZENGThe Stark’s conjecture predicts that if we write L K(s,χ)=R K(χ)s n+ O(s n+1),then R K(χ)also has the form of regulators,i.e.it is a determinent of a matrix whose entries are linear combination of logarithm of units of K0. Moreover we should expect R K(χ)σ=R K(χσ)for anyσ∈Aut(C).In our present case K is a primitive CMﬁeld,in this case Stark’s conjec-ture is somehow simple in the following sense:the regulator R K of K is a determinent of(n−1)×(n−1)matrix,and the regulator R Kof K0is a determinent of(mn−1)×(mn−1)matrix.Stark’s conjecture in fact says that the quotient R K/R K is a determinent of(mn−n)×(mn−n)matrix, and this matrix should be diagonalized into m−1blocks,with each block a n×n matrix,further in this case we can use the R K matrix to simplify these blocks,so in this extension there are only m−1essential new units.In other words,U Kas a module of U K is free of rank m−1.This is very much similar to the classical case of imaginery quadraticﬁelds.By some elementry argument we can show that R K=(R K)m·vol(S), where vol(S)is a determinent of(m−1)×(m−1)matrix whose entries are linear combinations of logarithm of units in K0.These are the basis of U Kas the module of U K.On the other hand by the Frobenius determinent formula, χ=χ( Rχ(R)δK(R))=det R1,R2=1(δK(R1)−δK(R2)) Hence we should expect:•δK(R)=log|ηK(R)|;•δK(R1)−δK(R2)=log|ηK(R1)ηK(R2)is a unit in K0.More overthese units should transform under the Galois group according to thereciprocity law.0.4.To prove things like this we need to have a good expression forδK(R), this can be done by using the theory of GL(2)Eisenstein series over the totally realﬁeld F.The Eisenstein series is deﬁned asE(w,s;a)=(c,d)∈(a⊕O F)/U F,(c,d)=(0)ni=1y s i|cσi z i+dσi|−2sThe idea is a classical one,sinceζK(s,R)= a∈R1w2satisfyingIm wσi>0,∀i.Then we haveNOTES ON HILBERT’S 12TH PROBLEM 9ζK (s,R )=N (a 1)s α∈a 1/U F ,α=012n −2πn h F RF [χF (d F )L F (2,χ−1F )i Im(ωi )+πn D −3/2F 0=b ∈d −1F a σ1,χ(bda )|N (b )|−1exp (2πi (n j =1b j ℜ(w j )+i |b j Im(w j )|))]where (1)a,d ∈A ×F such that div (a −1)=a and div (d )=d F ;(2)σs,χ(x )is a function deﬁned asσs,χ(s )= v ∈(finite primes ) 1+χv (w v )q s v +···+(χv (w v )q s v )ord v (x v )if x v ∈d v ,0if x v /∈d v .Here d v denotes the ring of integers of F v ,w v is the prime elementof F v and q v =|d v /w v d v |.So we haveζK (s,R )=−R K10SIXIN ZENGδK(R)=h(ω;a)−log i Im(w i)−logN(a)We reminded that a is the type of R−1,w=(w1,···,w n)∈H n is the CM point deﬁned by R,and h(w,a)is the complicated function deﬁned above.So our basic task is to understand this function.0.5.We notice that hχFsatisfying the following modular properties([19]):for γ∈Γa we have(1)hχF (γω;a)=hχF(ω,a)ifχF is not trivial;(2)h1(γω;a)−log( i Im(γω)i)=h1(ω,a)−log( i Imωi)So in particular we haveχFχF(a)hχF(γω;a)−log( i Im(γω)i)= χFχF(a)hχF(ω,a)−log( i Imωi)which suggests that we may actually have a Hilbert modular formηK(w;a) of parallel weight such that h(w;a)=log|ηK(w;a)|.Classically in case F= Q this is indeed the case asηK is the classical Dedekind eta functionη,it is well-known thatηhas an inﬁnite product expression,which when taking the logarithm translated into a Fourier expansion,which is exactly the above Fourier series.In the higher dimensional case it is not that easy2.Besides the fact that the Fourier expansion is too complicated and diﬃcult to work with,we can not expect by directly exponenciate the above expression of h to get a meaningful function,precisely because of the inﬁnite unit group of F,as shown as the regulator term R F in the leading coeﬃcients of the Fourier expansion.The regulator R F is not a rational number,in fact we expect it to be transcendental, so even if we exponenciate h we can not get anything useful for the arithmetic purpose.In particular we can not expect to get an inﬁnite product expression as the classicalηfunction.0.6.What should I do?It turns out although we can not exponenciate theh function,can not get the explicit formula forηK(w;a),we still can say something quantitatively about it.The idea is to consider a twisted version of Eisentein series:E u,v(w,s;a)=(c,d)∈(a⊕O F)/U F,(c,d)=(0)ni=1e2πi(cσi u i+dσi v i)(Im w i)s|cσi w i+dσi|−2sNOTES ON HILBERT’S12TH PROBLEM11 where(u,v)∈R n⊕R n.This is entirely adopted from the classical Kronecker’s second limit for-mula(see[18]).To explain why we need to develop the twisted Eisenstein series,let’s recall the classical situation,i.e.,when F=Q.In this case the Eisenstein series isE(w,s)=m,n∈Z,(m,n)=0(Im(w))s2[1+(CONST+log Im(w)−4log|η(w)|)s]+O(s2)The twisted Eisenstein series is:E u,v(w,s)=m,n∈Z,(m,n)=0e2πi(mu+nv)(Im(w))s24w∞n=1(1−q n w)and g u,v is the Siegel’s function:g u,v=−q1q zφ(w,z) and we haveg u,v=q112SIXIN ZENG(3)g u,v is also closely related to Dedekindη.In fact as a function of zg u,v has a simple zero at z=0withη(w)as the coeﬃcient,i.e.,|g u,v|=|η(w)|2|q z−1|+O(z2)This shows that the absolute value|η(w)|is not a theta null,but rather a“derivative theta null”.But we note our theta functions arenormalized at z=0to be zero,φ(w,0)=0,if the theta functionnormalized in this way,their derivatives can also be used as the coor-dinates on the moduli space.By abuse of notation we still call|η(w)|a theta null,hence it gives a modular function on the moduli space.From the deﬁnition,E u+1,v=E u,v+1=E u,v,hence g u+1,v=g u,v+1= g u,v,but since|g−v,u|=|q12B2(−v)w g−v,u(w)Then from the periodic condition of g u,v we immediately see that|φ(w,z+1)=φ(w,z);|φ(w,z+w)|=|q−1z|·|φ(w,z)| That is,|φ(w,z)|satisfying the characterization of a theta function,moreover since we know it is an analytic function,it then has to be a theta function itself.This is the idea we would follow in the higher dimensional case,as we observed before,since in the higher dimension we can not expect any explicit inﬁnite product formula for the functionηK(w;a),but we still have all the periodic properties as the1-dimensional case.Now we go back to the higher dimensional case,the Eisenstein series is:E(w,s;a)=(c,d)∈(a⊕O F))/U F,(c,d)=(0)ni=1(Im(w i))s|cσi w i+dσi|−2sWe have the limit formula:E(w,s;a)=−2n−2h F R F s n−1[1+(CONST+logN(a)+log( i Im(w i))−h(w;a))s]+O(s n+1) The twisted Eisenstein series is:NOTES ON HILBERT’S12TH PROBLEM13E u,v(w,s;a)=(c,d)∈(a⊕O F)/U F,(c,d)=(0)ni=1e2πi(cσi u i+dσi v i)(Im w i)s|cσi w i+dσi|−2sand we have the limit formula:E u,v(w,s;a)=−2n−2hF R F log|g−v,u(w;a)|s n+O(s n+1)where log|g−v,u(w;a)|has an explicit Fourier expansion,just like h(w;a).Then we argue as the following:(1)Recall that u=(u1,···,u n)∈R n,v=(v1,···,v n)∈R n,and O F⊂R n as a lattice.From the deﬁnition,for anyα∈O F,we have E u+α,v=E u,v+α=E u,v,i.e.,translate invariant under O F,hence|g−v+α,u|=|g−v,u+α|=|g−v,u|.(2)Now write z=u−vw,i.e.,z=(z1,···,z n),z i=u i−v i w i,∀i.We try to write g−v,u as a function of(w,z),so we deﬁneφ(w,z)=q−12B2(−v)w= i q−112wφ(w,z)|−log(|ηK(w;a)|2 i|z i|)}=0that islim z→0|q1|ηK(w;a)|2 i|z i|=1Also we verify that our theta function is normalized at z=0to be 0.This implies thatηK(w;a)is a theta null.By the classical theory of theta function,theta null naturally gives rise to the modular forms on the moduli space.In fact this should be more or less expected.By Mumford’s theory of algebraic theta function,we may further conclude that these theta nulls in fact deﬁnes the moduli space as an integral scheme over Z,hence have all the expected integral properties.14SIXIN ZENG0.7.In summary we have found the explicit form of the functionδK(R).δK(R)=logN(a)+log i Im(w i)−h(w;a)=logN(a)+log i Im(w i)−log|ηK(w;a)|2=log[N(a) i Im(w i)|ηK(w;a)−2|]where R is an ideal class of O K,a its type,w its CM point on M n(a),and ηk(w;a)is the theta null:∂∂z nφ(w,z)|z=0=ηK(w;a)2By the Stark’s conjecture,we need to understandδK(R1)−δK(R2),i.e., we need to understand the quotientηK(w1;a1)ηK(w2;a)is meaningful,as the modular function evaluating at the CMpoints,and the natural action of Galois group on them is prescribed by the reciprocity law.But when R1and R2are of the diﬀerent type,for example, when R1R−12is a real ideal,thenηK(w;a1)andηK(w;a2)are on the diﬀerent moduli spaces,thus their quotient becomes meaningless.This is precisely the limit of Shimura-Taniyama-Weil’s theory.So what can we do?To go further we need toﬁnd a natural way to inter-polate the diﬀerent moduli spaces,and it is here the idea of Mirror symmetry comes.As we shall see,in this case this quotient will have a meaning similar to the classical one if we consider the complexiﬁed Kahler moduli of the abelian varieties.Why should we interpolate the diﬀerent moduli spaces?The functionδK(R) is deﬁned not on a single moduli space of theﬁxed type,but rather automati-cally been deﬁned on all the moduli spaces,as the type a can vary accordingly. Likewise the modular formηK is a Hilbert modular form on all the type-ﬁxed Hilbert modular varieties,and when the type vary,can be regarded as a mod-ular form on all the moduli spaces.This strongly suggests that we should have a natural way to interpolate all these moduli spaces of diﬀerent types,such that these functions can naturally deﬁned.In other words,when weﬁx the type a,we get the Hilbert modular forms,what then happens if weﬁx the CM pointsωand let the type vary?When we look the explicit form ofδK(R)and h,we note the apparent symmetric roles played by the quantities σIm(ωσi)and N(a).Indeed if we ﬁx the CM pointsωand let a vary,we should have a meaning for the quantity N(a).This can be achieved by considering the Kahler moduli of our abelian varieties.0.8.Mirror Symmetry is usually formulated for the Calabi-Yau varieties, roughly it asserts that Calabi-Yau always come in pairs,X and X′,with theNOTES ON HILBERT’S12TH PROBLEM15“complex moduli”and“Kahler moduli”exchanged.In terms of the Hodgenumber,it means the Hodge diamond of X′is a rotation of X.For abelian varieties,Mirror Symmetry is generally regarded as“trivial”,as the underlying topological type would not change.Nevertheless we canstill talk about it.There are several constructions of Mirror manifold for theabelian varieties,the simplest one I believe,is given by Manin([9]).It goes asthe following:let k be any completeﬁeld,X an abelian variety over k,T bethe algebraic torus of dimension n over k,T≃(k×)n.Then the multiplicativeuniformization is0→P X→T→X→0,where P X is a free abelian groupof rank n,P X is called the period of X.Under this uniformization the Mirrorpartner X′is then0→P X→T∨→X′→0,i.e.,we explicitly indentify theperiods in T and T∨.When k≃C one verify that the complex moduli andKahler moduli of the two are exchanged.In our situation,given the Mirror pair X and X′we will be mainly concernabout the relations of theta functions on them.Since the underlying topologicaltype would not change,we may regard the Mirror transform as a“rotation”of complex structure of X.So to compare the theta functions on X and X′we have toﬁx the underlying real structures.We begin with X=C n/(w·a⊕O F)≃R2n/Z2n,any polarization of X isgiven by an integral skew-symmetric bilinear form on R2n.Given such a formω,we canﬁnd an integral basis{λ1,···,λ2n}of the integral lattice such thatif{x1,···,x2n}is the dual basis,thenω= n i=1δi dx i∧dx n+i,withδ1|δ2|···the elementary divisors.Note in our case the biliear form is given by the trace T r K/Q(ζxyρ)withthe admissibleζ∈K such thatζρ=−ζ,Im(ζσi)>0.Thus we may regard(x1,···,x n)as an integral basis of O F,and(x n+1,···,x2n)as an integralbasis of a.In particular(x n+1,···,x2n)depends on a.In the following we willdenote it as x n+i(a)if we need to use this dependence.Next we introduce the complex structure,so X becomes a complex tori,and we can introduce the complex coordinates.To do this let e i=λi/δi,i=1,2,···,n,and let{z i}be the complex dual of{e i}.Consider the changeof coordinates transform:Ω·(x1,···,x2n)T=(z1,···,z n)TThenΩ=(∆δ,Z)with∆δ=diag(δ1,···,δn)the diagonal matrix,and Zsymmetrical,Im(Z)>0.We recogonize that∆−1δZ is the period matrix of X.Note in our case for abelian varieties with CM by K,the peiod martix Z isnecessarily diagonal∆−1δZ=diag(w1,···,w n),with w=(w1,···,w n)∈H n.In particular we have z i=δi·x i+δi·w i·x n+i(a),we may regard it as the transform from the underlying real coordinates to the complex coordinates.16SIXIN ZENGNow recall the theta function on X is characterized by the periodic condi-tion:θ(z+λi)=θ(z);θ(z+λn+i)=e−2πiz iθ(z)Taking absolute values we have:|θ(z+λi)|=|θ(z)|;|θ(z+λn+i)|=e2πIm(z i)|θ(z)|However from the above coordinates transformation,Im(z i)=δi·Im(w i)·x n+i(a)The Mirror symmetry transform says that we can exchange the complex moduli with the Kahler moduli,while the coordinates w=(w1,···,w n)∈H n can be regarded as the complex moduli of X,where is the Kahler moduli?Our Kahler moduli coordinates are actually in the variables(x n+1(a),···,x2n(a)). Since they are depend on the type a,we want to write the dependency explic-itly,in order to understand the transformation of types.For this end let’s write x n+i=x n+i(O F).The type ideal a as a Z module,is a submodule of O F of full rank,i.e.,if weﬁx an integral basis of O F,then O F/a≃⊕n i=1Z/t i Z,with x n+i(a)=t i x n+i and i t i=N(a)/D F.Thus the positive rational numbers(t1,···,t n)can be conviniently regarded as the coordinates of the ideal a,and under appropriate identiﬁcation,can be regardedas the coordinates of of the Kahler class in the the Kahler moduli.So in particular we haveIm(z i)=δi·Im(w i)·t i·x n+iBut from the above formula,when we exchange the complex moduli(w1,···,w n) and Kahler moduli(t1,···,t n),it’s not going to change the multiplier e2πIm(z i)=e2πδi·t i·Im(w i)·x n+i.Since the absolute value of theta functions can be regardedas a real analytic function on R2n,thus we conclude that for the given Mir-ror pair X and X′,their theta functions’absolute values satisfying the same periodic condition,hence must be only diﬀered by a constant!Recall our previous puzzle,when two ideal classes R1and R2of O K sat-isfying R1R−12is a real ideal,then R1and R2are of the diﬀerent type,sothe corresponding abelian varieties X1and X2living on the diﬀerent moduli spaces,soηK(w,a1)/ηK(w,a2)has no meaning.But we knowηK(w,a)is the theta null of X,thus by the above formula,ηK(w,a)is also the theta null of the Mirror X′.So although X1and X2living on the diﬀerent moduli space,if their Mirror X′1and X′2are in the same moduli space,thenηK(w,a1)/ηK(w,a2) would be meaningful!This is indeed the case,as X′1and X′2are on the sin-gle moduli space,the complexiﬁed Kahler moduli space of X.This is the underlying rationale for us to use the Mirror symmetry.Note from this relation of theta functions we also see that if X is deﬁnes over a numberﬁeld,then the Mirror X′also is deﬁned over that numberﬁeld.。

heterogeneous interfacial structure

heterogeneous interfacial structure英文版Heterogeneous Interfacial StructureHeterogeneous interfacial structure refers to the structural differences that exist at the boundary between two different materials or phases. This structure plays a crucial role in determining the physical and chemical properties of the interface, as well as its stability and reactivity.At the interface between two materials, the atomic arrangement, bonding configuration, and electronic structure can all differ significantly from the bulk materials on either side. This heterogeneity can lead to a range of unique properties, such as charge accumulation, bond formation, and catalytic activity. For example, in the field of materials science, heterogeneous interfaces are often exploited to enhance the performance of devices such as solar cells and fuel cells.The study of heterogeneous interfacial structure is challenging due to the complexity of the interactions involved. Experimental techniques such as scanning probe microscopy, spectroscopy, and diffraction methods can provide insights into the atomic-scale structure and electronic properties of interfaces. Computational modeling is also an important tool for understanding and predicting interfacial behavior.In recent years, there has been increasing interest in the use of heterogeneous interfacial structures in nanotechnology and materials science. This interest is driven by the potential for novel materials with enhanced properties, as well as the development of new technologies such as nanodevices and sensors.In conclusion, heterogeneous interfacial structure is a crucial aspect of materials science and nanotechnology. Its understanding and control offer the potential for the development of novel materials and devices with enhanced performance and functionality.中文版异质界面结构异质界面结构指的是两种不同材料或相之间的边界处存在的结构差异。

the 类蜂窝状结构完全带隙二维光子晶体3 guide download

下面考虑更小的填充率β= 49 %的情况。由于 θ= 1°在附近存在着一个比较大的完全带隙 ,所以这里给出了在θ= 1°时光子带隙随 d 变化的关系 ,如图 2 ( b) 所示。比较图 2 (a) 和图 2 ( b) ,可以看出带隙的数目有所下降 ,而且有些带隙的宽度变窄 ,例如 E1 、
H1 、H7 、H9 ,但在图 2 ( b) 中出现了一个由 E5 带隙和 H4 带隙之间相互重叠而产生较大完全带隙。进一步可看出 , E5 - H4 带隙在 d = 0. 27 a 位置出现 , 在 d = 0. 38 a 位置达到 6. 5 %的最大宽度 ,其中心频率为ωa/ (2πc) = 0. 49 。图 3 为完全带隙最大时所在位置的结构 ,可以看到两个介质柱之间存在互相重叠的部分。
大值为 60 % ,即介质柱的半径 r ≤ 3 a/ 6 。考虑在背景介质为 GaA s (εb = 13. 6) 中填充空气柱 (εa = 1) 和在背景介质为空气中填充 GaA s 介质柱两种情况。这里选择具有代表性的半导体 GaA s 是因为它在红外区存在人们感兴趣的光学性质 ,从而可推广到其他类似系统。 3 . 1 空气柱在 Ga As 中
3 国家自然科学基金重大研究计划 (90401008) 、新世纪优秀人才支持计划、教育部科学技术研究重点项目 (104144) 、高等学校博士学科点专项科研基金 (20040558009) 和广州市科技计划项目 ———纳米专项 (2004Z32D2051) 资助课题。
作者简介 : 林旭彬 (1981～) ,男 ,广东人 ,硕士研究生 ,主要从事二维光子晶体波导器件的研究。
摘要 : 通过在二维三角晶格中引入两个完全一样的介质圆柱构成了类蜂窝状结构光子晶体 ,并对其光子能带进行了频域计算。借助数值方法分析了介质柱位置改变对光子能带的影响 ,计算结果表明 ,这种类蜂窝状结构二维光子晶体可以产生很宽的带隙 ,而且在一定填充率下 ,可以通过调整介质柱的位置使完全光子带隙达到最大化。关键词 : 光电子学 ; 光子晶体 ; 完全带隙 ; 频域方法 ; 蜂窝状结构 ; 三角晶格中图分类号 : O488 文献标识码 : A

部分偏振光的表示方法

部分偏振光的表示方法王晓,石顺祥,马琳(西安电子科技大学技术物理学院,陕西西安710071)摘要随着光纤通信和光纤传感的快速发展,人们对光的偏振态提出了越来越高的要求。

光是一种横波,其偏振态大致分为:完全偏振光、部分偏振光和自然光。

文中介绍了完全偏振光的几种表示方法,给出了几种部分偏振光的描述方法。

并阐述了它们之间的差异与联系。

关键词偏振态;完全偏振光;部分偏振光中图分类号TP212·14文献标识码A文章编号1007-7820(2009)06-078-03SeveralRepresentations of State ofPolarization ofLightWangXiao, Shi Shunxiang, Ma Lin(School ofTechnicalPhysics, XidianUniversity, Xi an 710071, China)Abstract Alongwith the rapid developmentof fiber optic communications and optical fiber sensing, thedemands on the polarization state of light are higher and higher·Light is a transversewave, whose polarization state is classified into the following kinds: complete polarized light, partially polarized light and natural light This article introduces several expressive ways of complete polarized light and partially polarized light, and elaborates the differences and relations between them·Keywords state ofpolarization; complete polarized light; partially polarized light 随着科学技术的发展,光纤通信和光纤传感越来越受到人们的关注。

(光学专业优秀论文)光学梳状滤波器技术研究

摘
要
摘要
波分复用技术（ＷＤＭ）对于光通信技术的发展起到了巨大的推动作用，以ＤＷＤＭ技术为核心的光传送网将成为电信网的主要传送平台。在密集波分复用
技术中，随着信道复用密度的提高，一种薪型光器件一一光学梳状滤波器
（Ｉｎｔｅｒｌｅａｖｅｒ）应运而生。本文对实现光学梳状滤波器的几种主要技术方案进行了较为深入的理论分析，对ＭＧＴＩ型光学梳状滤波器进行了实验研究，取得了较好的结果。１、分析了ＭＧＴＩ型Ｉｎｔｅｒｌｅａｖｅｒ滤波器的基本原理，讨论了输出光谱参数和设计参数之间的关系，获得了实现矩形化光谱的最佳设计参数；根据ＤＷＤＭ系统的实际要求，具体给出了信道阳Ｊ隔为５０ＧＨｚ的ＭＧＴＩ型Ｉｎｔｅｒｌｅａｖｅｒ滤波器设计参数及其误差分析；从多镜Ｆ—Ｐ腔光谱特性出发，讨论了Ｇ—Ｔ腔的特性，进而分析了ＭＧＴＩ型Ｉｎｔｅｒｌｅａｖｅｒ滤波器的色散特性：理论结合实验，分析了Ｉｎｔｅｒｌｅａｖｅｒ滤波器中偏振分光镜偏振效应对器件隔离度的影响。２、设计了信道间隔为５０ＧＨｚ的ＭＧＴＩ型Ｉｎｔｅｒｌｅａｖｅｒ滤波器，获得了该滤波器的原型器件；实验得到了该器件在ｃ波段两组交错分波的矩形化光谱，其纹波小于０．１ｄＢ，通过对器件时延的测量，间接获得了其色散特性；首次提出ＭＦＰＩ型不等带宽５０ＧＨｚ的Ｉｎｔｅｒｌｅａｖｅｒ滤波器技术方案。３、分析了ＰＢＩ型Ｉｎｔｅｒｌｅａｖｅｒ滤波器的基本原理，给出了一种ＰＢＩ型Ｉｎｔｅｒｌｅａｖｅｒ滤波器结构；设计了５０ＧＨｚ光学梳状滤波器，实验得到了该器件两组交错分波类余弦输出光谱；采用不同光程差的晶体级联方案，实现了输出光谱平顶化。４、分析了ＦＭＺＩ型Ｉｎｔｅｒｌｅａｖｅｒ滤波器的工作原理，其输出光谱曲线为类余弦函数；采用级联ＦＭＺＩ方案，实现了Ｉｎｔｅｒｌｅａｖｅｒ滤波器的光谱平顶光谱输出；最后，基于偏振光干涉和多光束干涉原理，分析了ＢＧＴＩ型Ｉｎｔｅｒｌｅａｖｅｒ滤波器工作原理，模拟了Ｇ．Ｔ腔部分反射系数ｒ与输出光谱的关系，获得了５０ＧＨｚＢＧＴＩ型Ｉｎｔｅｒｌｅａｖｅｒ滤波器平顶化输出光谱。

Universal unfoldings of Laurent polynomials and tt structures

Abstract. This article surveys the relations between harmonic Higgs bundles and Saito structures which lead to tt∗ geometry on Frobenius manifolds. We give the main lines of the proof of the existence of a canonical tt∗ structure on the base space of the universal unfolding of convenient and nondegenerate Laurent polynomials.
c 0000 (copyright holder)
1
2
Hale Waihona Puke C. SABBAHIntroduction The notion of a tt structure on a holomorphic vector bundle is now understood, after the work of C. Hertling [7], as an enrichment of that of harmonic Higgs bundle previously introduced by N. Hitchin and C. Simpson. Given the Higgs ﬁeld Φ and the harmonic metric h on the holomorphic bundle E on a complex manifold M , the new ingredients needed for a tt∗ structure are a real structure on the associated C ∞ bundle H , a holomorphic endomorphism U of E and a C ∞ endomorphism Q of H subject to some compatibility relations. In the following, we relax the condition for h to be positive deﬁnite, and only ask that it is Hermitian and nondegenerate. When needed, we will emphasize the positive deﬁnite case. It has been much enlightening in two ways to interpret (cf. [31]) harmonic Higgs bundles as variations of polarized twistor structures of weight 0: ﬁrstly, it makes the analogy with variations of Hodge structures more transparent and, secondly, it enables one to do geometry with the external parameter z added for this purpose. From this point of view, the relations satisﬁed by the endomorphisms U and Q in a tt∗ structure appear as expressing the complete integrability of a connection on the twistor bundle. A nearby (purely holomorphic) notion, that of a Saito structure, has emerged from the work of K. Saito [23] and M. Saito [25] as a basic tool to produce Frobenius manifolds from singularities of holomorphic functions. While it is already present in [7], the bridge between these two notions is made more transparent in §1 by the introduction of a potential for the Higgs ﬁeld. When the tt∗ structure exists on the tangent bundle of M , we speak of tt∗ geometry, which is a generalization of special geometries on M . Of particular interest for us is the case where M is a Frobenius manifold. In such a case, a Saito structure exists on the tangent bundle together with supplementary symmetry properties, giving rise to a commutative and associative product with unit on T M . Adding a tt∗ structure in a compatible way (i.e., with the help of a potential for the Higgs ﬁeld) leads to the structure of harmonic Frobenius manifold. The main result we report here (cf. Theorem 4.7) is the existence of a canonical harmonic structure on the canonical Frobenius manifold attached to a convenient and nondegenerate Laurent polynomial. Moreover, the corresponding Hermitian form is positive deﬁnite. In this survey article, which contains no original result, we ﬁrst give (§§1 and 2) a quick overview of tt∗ structures, Saito structures and variations of twistor structures (a more detailed exposition can be found in [7] and [9]). In §3, we explain the Fourier-Laplace method for constructing polarized pure twistor structures starting from a variation of polarized Hodge structure. In §4, we show how to apply this technique to the Gauss-Manin connection of a Laurent polynomial, with the help of M. Saito’s mixed Hodge theory [26]. One can ﬁnd details for the results of §§3 and 4 in [19, 22], and many other results and applications in [9]. Acknowledgements. The author thanks Ron Donagi and Katrin Wendland, organizers of the conference “From tQFT to tt∗ and integrability”, for having given him the opportunity to talk about the contents of this article.

物理实验报告英文版7

iv
Table of Contents
Title Page Authorization Page Signature Page Acknowledgements Table of Contents List of Figures List of Tables Abstract Chapter1 Introduction 1.1 Structure of Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . 1.2 Electronic properties of Carbon Nanotubes . . . . . . . . . . . . . . Chapter2 Superconductivity in 0.4nm Carbon Nanotubes array 2.1 The band structure of 0.4nm Carbon Nanotubes . . . . . . . . . . 2.2 Meissner eﬀect in 0.4nm Carbon Nanotubes array . . . . . . . . . 2.3 The model of coupled one-dimensional superconducting wires . . . 2.4 Motivation and scope of the thesis . . . . . . . . . . . . . . . . . . i ii iii iv v vii xi xii 1 3 4 8 9 9 12 13
July 2008, Hong Kong
HKUST Library Reproduction is prohibited without the author’s prior written consent

离散数学中英文名词对照表

离散数学中英⽂名词对照表离散数学中英⽂名词对照表外⽂中⽂AAbel category Abel 范畴Abel group (commutative group) Abel 群(交换群)Abel semigroup Abel 半群accessibility relation 可达关系action 作⽤addition principle 加法原理adequate set of connectives 联结词的功能完备（全）集adjacent 相邻（邻接）adjacent matrix 邻接矩阵adjugate 伴随adjunction 接合affine plane 仿射平⾯algebraic closed field 代数闭域algebraic element 代数元素algebraic extension 代数扩域（代数扩张）almost equivalent ⼏乎相等的alternating group 三次交代群annihilator 零化⼦antecedent 前件anti symmetry 反对称性anti-isomorphism 反同构arboricity 荫度arc set 弧集arity 元数arrangement problem 布置问题associate 相伴元associative algebra 结合代数associator 结合⼦asymmetric 不对称的(⾮对称的)atom 原⼦atomic formula 原⼦公式augmenting digeon hole principle 加强的鸽⼦笼原理augmenting path 可增路automorphism ⾃同构automorphism group of graph 图的⾃同构群auxiliary symbol 辅助符号axiom of choice 选择公理axiom of equality 相等公理axiom of extensionality 外延公式axiom of infinity ⽆穷公理axiom of pairs 配对公理axiom of regularity 正则公理axiom of replacement for the formula Ф关于公式Ф的替换公式axiom of the empty set 空集存在公理axiom of union 并集公理Bbalanced imcomplete block design 平衡不完全区组设计barber paradox 理发师悖论base 基Bell number Bell 数Bernoulli number Bernoulli 数Berry paradox Berry 悖论bijective 双射bi-mdule 双模binary relation ⼆元关系binary symmetric channel ⼆进制对称信道binomial coefficient ⼆项式系数binomial theorem ⼆项式定理binomial transform ⼆项式变换bipartite graph ⼆分图block 块block 块图（区组）block code 分组码block design 区组设计Bondy theorem Bondy 定理Boole algebra Boole 代数Boole function Boole 函数Boole homomorophism Boole 同态Boole lattice Boole 格bound occurrence 约束出现bound variable 约束变量bounded lattice 有界格bridge 桥Bruijn theorem Bruijn 定理Burali-Forti paradox Burali-Forti 悖论Burnside lemma Burnside 引理Ccage 笼canonical epimorphism 标准满态射Cantor conjecture Cantor 猜想Cantor diagonal method Cantor 对⾓线法Cantor paradox Cantor 悖论cardinal number 基数Cartesion product of graph 图的笛卡⼉积Catalan number Catalan 数category 范畴Cayley graph Cayley 图Cayley theorem Cayley 定理center 中⼼characteristic function 特征函数characteristic of ring 环的特征characteristic polynomial 特征多项式check digits 校验位Chinese postman problem 中国邮递员问题chromatic number ⾊数chromatic polynomial ⾊多项式circuit 回路circulant graph 循环图circumference 周长class 类classical completeness 古典完全的classical consistent 古典相容的clique 团clique number 团数closed term 闭项closure 闭包closure of graph 图的闭包code 码code element 码元code length 码长code rate 码率code word 码字coefficient 系数coimage 上象co-kernal 上核coloring 着⾊coloring problem 着⾊问题combination number 组合数combination with repetation 可重组合common factor 公因⼦commutative diagram 交换图commutative ring 交换环commutative seimgroup 交换半群complement 补图(⼦图的余) complement element 补元complemented lattice 有补格complete bipartite graph 完全⼆分图complete graph 完全图complete k-partite graph 完全k-分图complete lattice 完全格composite 复合composite operation 复合运算composition (molecular proposition) 复合（分⼦）命题composition of graph (lexicographic product)图的合成（字典积）concatenation (juxtaposition) 邻接运算concatenation graph 连通图congruence relation 同余关系conjunctive normal form 正则合取范式connected component 连通分⽀connective 连接的connectivity 连通度consequence 推论（后承）consistent (non-contradiction) 相容性（⽆⽭盾性）continuum 连续统contraction of graph 图的收缩contradiction ⽭盾式（永假式）contravariant functor 反变函⼦coproduct 上积corank 余秩correct error 纠正错误corresponding universal map 对应的通⽤映射countably infinite set 可列⽆限集（可列集）covariant functor (共变)函⼦covering 覆盖covering number 覆盖数Coxeter graph Coxeter 图crossing number of graph 图的叉数cuset 陪集cotree 余树cut edge 割边cut vertex 割点cycle 圈cycle basis 圈基cycle matrix 圈矩阵cycle rank 圈秩cycle space 圈空间cycle vector 圈向量cyclic group 循环群cyclic index 循环（轮转）指标cyclic monoid 循环单元半群cyclic permutation 圆圈排列cyclic semigroup 循环半群DDe Morgan law De Morgan 律decision procedure 判决过程decoding table 译码表deduction theorem 演绎定理degree 次数,次(度)degree sequence 次(度)序列derivation algebra 微分代数Descartes product Descartes 积designated truth value 特指真值detect errer 检验错误deterministic 确定的diagonal functor 对⾓线函⼦diameter 直径digraph 有向图dilemma ⼆难推理direct consequence 直接推论（直接后承）direct limit 正向极限direct sum 直和directed by inclution 被包含关系定向discrete Fourier transform 离散 Fourier 变换disjunctive normal form 正则析取范式disjunctive syllogism 选⾔三段论distance 距离distance transitive graph 距离传递图distinguished element 特异元distributive lattice 分配格divisibility 整除division subring ⼦除环divison ring 除环divisor (factor) 因⼦domain 定义域Driac condition Dirac 条件dual category 对偶范畴dual form 对偶式dual graph 对偶图dual principle 对偶原则(对偶原理) dual statement 对偶命题dummy variable 哑变量（哑变元）Eeccentricity 离⼼率edge chromatic number 边⾊数edge coloring 边着⾊edge connectivity 边连通度edge covering 边覆盖edge covering number 边覆盖数edge cut 边割集edge set 边集edge-independence number 边独⽴数eigenvalue of graph 图的特征值elementary divisor ideal 初等因⼦理想elementary product 初等积elementary sum 初等和empty graph 空图empty relation 空关系empty set 空集endomorphism ⾃同态endpoint 端点enumeration function 计数函数epimorphism 满态射equipotent 等势equivalent category 等价范畴equivalent class 等价类equivalent matrix 等价矩阵equivalent object 等价对象equivalent relation 等价关系error function 错误函数error pattern 错误模式Euclid algorithm 欧⼏⾥德算法Euclid domain 欧⽒整环Euler characteristic Euler 特征Euler function Euler 函数Euler graph Euler 图Euler number Euler 数Euler polyhedron formula Euler 多⾯体公式Euler tour Euler 闭迹Euler trail Euler 迹existential generalization 存在推⼴规则existential quantifier 存在量词existential specification 存在特指规则extended Fibonacci number ⼴义 Fibonacci 数extended Lucas number ⼴义Lucas 数extension 扩充（扩张）extension field 扩域extension graph 扩图exterior algebra 外代数Fface ⾯factor 因⼦factorable 可因⼦化的factorization 因⼦分解faithful (full) functor 忠实（完满）函⼦Ferrers graph Ferrers 图Fibonacci number Fibonacci 数field 域filter 滤⼦finite extension 有限扩域finite field (Galois field ) 有限域（Galois 域）finite dimensional associative division algebra有限维结合可除代数finite set 有限（穷）集finitely generated module 有限⽣成模first order theory with equality 带符号的⼀阶系统five-color theorem 五⾊定理five-time-repetition 五倍重复码fixed point 不动点forest 森林forgetful functor 忘却函⼦four-color theorem(conjecture) 四⾊定理（猜想）F-reduced product F-归纳积free element ⾃由元free monoid ⾃由单元半群free occurrence ⾃由出现free R-module ⾃由R-模free variable ⾃由变元free-?-algebra ⾃由?代数function scheme 映射格式GGalileo paradox Galileo 悖论Gauss coefficient Gauss 系数GBN (G?del-Bernays-von Neumann system)GBN系统generalized petersen graph ⼴义 petersen 图generating function ⽣成函数generating procedure ⽣成过程generator ⽣成⼦（⽣成元）generator matrix ⽣成矩阵genus 亏格girth （腰）围长G?del completeness theorem G?del 完全性定理golden section number 黄⾦分割数（黄⾦分割率）graceful graph 优美图graceful tree conjecture 优美树猜想graph 图graph of first class for edge coloring 第⼀类边⾊图graph of second class for edge coloring 第⼆类边⾊图graph rank 图秩graph sequence 图序列greatest common factor 最⼤公因⼦greatest element 最⼤元（素）Grelling paradox Grelling 悖论Gr?tzsch graph Gr?tzsch 图group 群group code 群码group of graph 图的群HHajós conjecture Hajós 猜想Hamilton cycle Hamilton 圈Hamilton graph Hamilton 图Hamilton path Hamilton 路Harary graph Harary 图Hasse graph Hasse 图Heawood graph Heawood 图Herschel graph Herschel 图hom functor hom 函⼦homemorphism 图的同胚homomorphism 同态（同态映射）homomorphism of graph 图的同态hyperoctahedron 超⼋⾯体图hypothelical syllogism 假⾔三段论hypothese (premise) 假设(前提)Iideal 理想identity 单位元identity natural transformation 恒等⾃然变换imbedding 嵌⼊immediate predcessor 直接先⾏immediate successor 直接后继incident 关联incident axiom 关联公理incident matrix 关联矩阵inclusion and exclusion principle 包含与排斥原理inclusion relation 包含关系indegree ⼊次（⼊度）independent 独⽴的independent number 独⽴数independent set 独⽴集independent transcendental element 独⽴超越元素index 指数individual variable 个体变元induced subgraph 导出⼦图infinite extension ⽆限扩域infinite group ⽆限群infinite set ⽆限（穷）集initial endpoint 始端initial object 初始对象injection 单射injection functor 单射函⼦injective (one to one mapping) 单射（内射）inner face 内⾯inner neighbour set 内（⼊）邻集integral domain 整环integral subdomain ⼦整环internal direct sum 内直和intersection 交集intersection of graph 图的交intersection operation 交运算interval 区间invariant factor 不变因⼦invariant factor ideal 不变因⼦理想inverse limit 逆向极限inverse morphism 逆态射inverse natural transformation 逆⾃然变换inverse operation 逆运算inverse relation 逆关系inversion 反演isomorphic category 同构范畴isomorphism 同构态射isomorphism of graph 图的同构join of graph 图的联JJordan algebra Jordan 代数Jordan product (anti-commutator) Jordan乘积（反交换⼦）Jordan sieve formula Jordan 筛法公式j-skew j-斜元juxtaposition 邻接乘法Kk-chromatic graph k-⾊图k-connected graph k-连通图k-critical graph k-⾊临界图k-edge chromatic graph k-边⾊图k-edge-connected graph k-边连通图k-edge-critical graph k-边临界图kernel 核Kirkman schoolgirl problem Kirkman ⼥⽣问题Kuratowski theorem Kuratowski 定理Llabeled graph 有标号图Lah number Lah 数Latin rectangle Latin 矩形Latin square Latin ⽅lattice 格lattice homomorphism 格同态law 规律leader cuset 陪集头least element 最⼩元least upper bound 上确界（最⼩上界）left (right) identity 左（右）单位元left (right) invertible element 左（右）可逆元left (right) module 左（右）模left (right) zero 左（右）零元left (right) zero divisor 左（右）零因⼦left adjoint functor 左伴随函⼦left cancellable 左可消的left coset 左陪集length 长度Lie algebra Lie 代数line- group 图的线群logically equivanlent 逻辑等价logically implies 逻辑蕴涵logically valid 逻辑有效的（普效的）loop 环Lucas number Lucas 数Mmagic 幻⽅many valued proposition logic 多值命题逻辑matching 匹配mathematical structure 数学结构matrix representation 矩阵表⽰maximal element 极⼤元maximal ideal 极⼤理想maximal outerplanar graph 极⼤外平⾯图maximal planar graph 极⼤平⾯图maximum matching 最⼤匹配maxterm 极⼤项（基本析取式）maxterm normal form(conjunctive normal form) 极⼤项范式（合取范式）McGee graph McGee 图meet 交Menger theorem Menger 定理Meredith graph Meredith 图message word 信息字mini term 极⼩项minimal κ-connected graph 极⼩κ-连通图minimal polynomial 极⼩多项式Minimanoff paradox Minimanoff 悖论minimum distance 最⼩距离Minkowski sum Minkowski 和minterm (fundamental conjunctive form) 极⼩项（基本合取式）minterm normal form(disjunctive normal form)极⼩项范式（析取范式）M?bius function M?bius 函数M?bius ladder M?bius 梯M?bius transform (inversion) M?bius 变换（反演）modal logic 模态逻辑model 模型module homomorphism 模同态(R-同态)modus ponens 分离规则modus tollens 否定后件式module isomorphism 模同构monic morphism 单同态monoid 单元半群monomorphism 单态射morphism (arrow) 态射（箭）M?bius function M?bius 函数M?bius ladder M?bius 梯M?bius transform (inversion) M?bius 变换（反演）multigraph 多重图multinomial coefficient 多项式系数multinomial expansion theorem 多项式展开定理multiple-error-correcting code 纠多错码multiplication principle 乘法原理mutually orthogonal Latin square 相互正交拉丁⽅Nn-ary operation n-元运算n-ary product n-元积natural deduction system ⾃然推理系统natural isomorphism ⾃然同构natural transformation ⾃然变换neighbour set 邻集next state 下⼀个状态next state transition function 状态转移函数non-associative algebra ⾮结合代数non-standard logic ⾮标准逻辑Norlund formula Norlund 公式normal form 正规形normal model 标准模型normal subgroup (invariant subgroup) 正规⼦群（不变⼦群）n-relation n-元关系null object 零对象nullary operation 零元运算Oobject 对象orbit 轨道order 阶order ideal 阶理想Ore condition Ore 条件orientation 定向orthogonal Latin square 正交拉丁⽅orthogonal layout 正交表outarc 出弧outdegree 出次(出度)outer face 外⾯outer neighbour 外（出）邻集outerneighbour set 出(外)邻集outerplanar graph 外平⾯图Ppancycle graph 泛圈图parallelism 平⾏parallelism class 平⾏类parity-check code 奇偶校验码parity-check equation 奇偶校验⽅程parity-check machine 奇偶校验器parity-check matrix 奇偶校验矩阵partial function 偏函数partial ordering (partial relation) 偏序关系partial order relation 偏序关系partial order set (poset) 偏序集partition 划分,分划,分拆partition number of integer 整数的分拆数partition number of set 集合的划分数Pascal formula Pascal 公式path 路perfect code 完全码perfect t-error-correcting code 完全纠-错码perfect graph 完美图permutation 排列（置换）permutation group 置换群permutation with repetation 可重排列Petersen graph Petersen 图p-graph p-图Pierce arrow Pierce 箭pigeonhole principle 鸽⼦笼原理planar graph （可）平⾯图plane graph 平⾯图Pólya theorem Pólya 定理polynomail 多项式polynomial code 多项式码polynomial representation 多项式表⽰法polynomial ring 多项式环possible world 可能世界power functor 幂函⼦power of graph 图的幂power set 幂集predicate 谓词prenex normal form 前束范式pre-ordered set 拟序集primary cycle module 准素循环模prime field 素域prime to each other 互素primitive connective 初始联结词primitive element 本原元primitive polynomial 本原多项式principal ideal 主理想principal ideal domain 主理想整环principal of duality 对偶原理principal of redundancy 冗余性原则product 积product category 积范畴product-sum form 积和式proof (deduction) 证明（演绎）proper coloring 正常着⾊proper factor 真正因⼦proper filter 真滤⼦proper subgroup 真⼦群properly inclusive relation 真包含关系proposition 命题propositional constant 命题常量propositional formula(well-formed formula,wff)命题形式（合式公式）propositional function 命题函数propositional variable 命题变量pullback 拉回(回拖) pushout 推出Qquantification theory 量词理论quantifier 量词quasi order relation 拟序关系quaternion 四元数quotient (difference) algebra 商（差）代数quotient algebra 商代数quotient field (field of fraction) 商域（分式域）quotient group 商群quotient module 商模quotient ring (difference ring , residue ring) 商环（差环，同余类环）quotient set 商集RRamsey graph Ramsey 图Ramsey number Ramsey 数Ramsey theorem Ramsey 定理range 值域rank 秩reconstruction conjecture 重构猜想redundant digits 冗余位reflexive ⾃反的regular graph 正则图regular representation 正则表⽰relation matrix 关系矩阵replacement theorem 替换定理representation 表⽰representation functor 可表⽰函⼦restricted proposition form 受限命题形式restriction 限制retraction 收缩Richard paradox Richard 悖论right adjoint functor 右伴随函⼦right cancellable 右可消的right factor 右因⼦right zero divison 右零因⼦ring 环ring of endomorphism ⾃同态环ring with unity element 有单元的环R-linear independence R-线性⽆关root field 根域rule of inference 推理规则Russell paradox Russell 悖论Ssatisfiable 可满⾜的saturated 饱和的scope 辖域section 截⼝self-complement graph ⾃补图semantical completeness 语义完全的（弱完全的）semantical consistent 语义相容semigroup 半群separable element 可分元separable extension 可分扩域sequent ⽮列式sequential 序列的Sheffer stroke Sheffer 竖（谢弗竖）simple algebraic extension 单代数扩域simple extension 单扩域simple graph 简单图simple proposition (atomic proposition) 简单（原⼦）命题simple transcental extension 单超越扩域simplication 简化规则slope 斜率small category ⼩范畴smallest element 最⼩元（素）Socrates argument Socrates 论断（苏格拉底论断）soundness (validity) theorem 可靠性（有效性）定理spanning subgraph ⽣成⼦图spanning tree ⽣成树spectra of graph 图的谱spetral radius 谱半径splitting field 分裂域standard model 标准模型standard monomil 标准单项式Steiner triple Steiner 三元系⼤集Stirling number Stirling 数Stirling transform Stirling 变换subalgebra ⼦代数subcategory ⼦范畴subdirect product ⼦直积subdivison of graph 图的细分subfield ⼦域subformula ⼦公式subdivision of graph 图的细分subgraph ⼦图subgroup ⼦群sub-module ⼦模subrelation ⼦关系subring ⼦环sub-semigroup ⼦半群subset ⼦集substitution theorem 代⼊定理substraction 差集substraction operation 差运算succedent 后件surjection (surjective) 满射switching-network 开关⽹络Sylvester formula Sylvester公式symmetric 对称的symmetric difference 对称差symmetric graph 对称图symmetric group 对称群syndrome 校验⼦syntactical completeness 语法完全的（强完全的）Syntactical consistent 语法相容system ?3 , ?n , ??0 , ??系统?3 , ?n , ??0 , ??system L 公理系统 Lsystem ?公理系统?system L1 公理系统 L1system L2 公理系统 L2system L3 公理系统 L3system L4 公理系统 L4system L5 公理系统 L5system L6 公理系统 L6system ?n 公理系统?nsystem of modal prepositional logic 模态命题逻辑系统system Pm 系统 Pmsystem S1 公理系统 S1system T (system M) 公理系统 T(系统M)Ttautology 重⾔式（永真公式）technique of truth table 真值表技术term 项terminal endpoint 终端terminal object 终结对象t-error-correcing BCH code 纠 t -错BCH码theorem (provable formal) 定理（可证公式）thickess 厚度timed sequence 时间序列torsion 扭元torsion module 扭模total chromatic number 全⾊数total chromatic number conjecture 全⾊数猜想total coloring 全着⾊total graph 全图total matrix ring 全⽅阵环total order set 全序集total permutation 全排列total relation 全关系tournament 竞赛图trace (trail) 迹tranformation group 变换群transcendental element 超越元素transitive 传递的tranverse design 横截设计traveling saleman problem 旅⾏商问题tree 树triple system 三元系triple-repetition code 三倍重复码trivial graph 平凡图trivial subgroup 平凡⼦群true in an interpretation 解释真truth table 真值表truth value function 真值函数Turán graph Turán 图Turán theorem Turán 定理Tutte graph Tutte 图Tutte theorem Tutte 定理Tutte-coxeter graph Tutte-coxeter 图UUlam conjecture Ulam 猜想ultrafilter 超滤⼦ultrapower 超幂ultraproduct 超积unary operation ⼀元运算unary relation ⼀元关系underlying graph 基础图undesignated truth value ⾮特指值undirected graph ⽆向图union 并(并集)union of graph 图的并union operation 并运算unique factorization 唯⼀分解unique factorization domain (Gauss domain) 唯⼀分解整域unique k-colorable graph 唯⼀k着⾊unit ideal 单位理想unity element 单元universal 全集universal algebra 泛代数（Ω代数）universal closure 全称闭包universal construction 通⽤结构universal enveloping algebra 通⽤包络代数universal generalization 全称推⼴规则universal quantifier 全称量词universal specification 全称特指规则universal upper bound 泛上界unlabeled graph ⽆标号图untorsion ⽆扭模upper (lower) bound 上（下）界useful equivalent 常⽤等值式useless code 废码字Vvalence 价valuation 赋值Vandermonde formula Vandermonde 公式variery 簇Venn graph Venn 图vertex cover 点覆盖vertex set 点割集vertex transitive graph 点传递图Vizing theorem Vizing 定理Wwalk 通道weakly antisymmetric 弱反对称的weight 重（权）weighted form for Burnside lemma 带权形式的Burnside引理well-formed formula (wff) 合式公式(wff) word 字Zzero divison 零因⼦zero element (universal lower bound) 零元（泛下界）ZFC (Zermelo-Fraenkel-Cohen) system ZFC系统form)normal(Skolemformnormalprenex-存在正则前束范式(Skolem 正则范式)3-value proposition logic 三值命题逻辑。

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a r X i v :h e p -p h /9907543v 1 29 J u l 1999DESY 99–103hep-ph/9907543July 1999Relations Between PolarizedStructure Functions1Johannes Bl¨u mleinDESY Zeuthen,Platanenallee 6,D-15735Zeuthen,GermanyAbstractThe status of the twist–2and the twist–3integral relations between polarized structurefunctions in deep inelastic scattering is discussed.The relations can be tested in theupcoming experiments in the range Q 2∼>M 2p .Relations Between Polarized Structure FunctionsJohannes Bl¨u mleinDeutsches Elektronen-Synchrotron DESY,Platanenallee 6,D-15735Zeuthen,Germany Abstract:The status of the twist–2and the twist–3integral relations betweenpolarized structure functions in deep inelastic scattering is discussed.The relationscan be tested in the upcoming experiments in the range Q 2∼>M 2p .1IntroductionThe measurement of the nucleon structure functions in polarized deeply inelastic scattering reveals the behaviour of quarks and gluons in outer magnetic ﬁelds and allows to test basicpredictions of Quantum Chromodynamics (QCD)in the short–distance range Q 2≫M 2p .In this domain the structure of the hadronic tensor W µνwhich describes the process can be investigated in terms of the light–cone expansion [1].We restrict our consideration to those contributions to W µνwhich contribute to the scattering cross sections in the massless quark case.There W µνis a current conserving quantity for the electro–weak interactions,W µν=−g µν+q µq νP ·qF 2(x,Q 2)−iεµνλσq λP σP ·q g 1(x,Q 2)+iεµνλσq λ(P ·qS σ−S ·qP σ)2−S ·qˆPµˆP νP ·q +S ·qˆPµˆP νq 2)(S ·q )q 2q µ,ˆSµ=S µ−S ·q2Twist DecompositionThe Fourier transform of the Compton amplitude T NCµν(x)for neutral current interactions readsi d4xe iqx T NCµν=− d4k U(k)γµ(g V1+g A1γ5)ˆk+ˆq(2π)4(k−q)2γν(g V2+g A2γ5)U(k),(2)=−i(g V1g V2+g A1g A2)εµανβqαuβ++(g V1g A2+g A1g V2)Sµανβ[qαuβ−+uαβ],where U(k)= d4k/(2π)4e−ikxψ(x),Sµανβ=gµαgµβ+gµβgνα−gµνgαβ,anduβ±=− d4k U(k)γβγ5(2π)4(k+q)2U(k)−(q↔−q).(4) The expansion of the denominators(k±q)2results into the operator–representationuβ±= n even,odd 2Q2 n+1qµ1...qµnΘ+β{αµ1···µn}.(6) The operatorsΘ+β{µ1···µn}are given byΘ+β{µ1···µn}= d4k U(k)γβγ5kµ1...kµn U(k)=Θ+β{µ1···µn}S +Θ+β{µ1···µn}R(7)and may be decomposed into a fully symmetric and a remainder part with respect to their indices.The former contribution is of twist–2while the latter is a twist–3operator.In general this decomposition has to be performed accounting for target mass eﬀects,see Ref.[2]for details.In the massless case the corresponding representations were given in Ref.[3]2.Whereas the above decomposition of the Compton amplitude yields contributions to all the eight Lorentz tensors of Eq.(1)due toΘ+β{µ1···µn}Sthis is the case for the polarized partforΘ+β{µ1···µn}R only keeping all the nucleon mass terms[2].Taking the limit M p→0onlycontributions∝g2and g3are obtained,which were studied previously in Ref.[3].The deep inelastic structure functions contain contributions of diﬀerent twist.Since the individual twist terms obey independent renormalization group equations their scaling viola-tions are diﬀerent.Moreover the diﬀerent twist operators aquire separate expectation values which are related to diﬀerent target mass corrections in general.Because of this the structure functions F i and g i have to be represented as linear superpositions of their twist contributions, the scaling violations of which are calculated individually.Henceforth we will discuss the twist contributions separately.3Twist–2RelationsIn lowest order in αs the twist–2contributions to the structure functions g i |5i =1are determined by a single quarkonic operator matrix element a n for each moment n in the massless caseg i (n )=10dxx n −1g i (x ).(8)These non–perturbative functions are diﬀerent for the sets of structure functions g 1,2and g 3,4,5due to the corresponding quark contents.At the level of twist–2the former ones are ∝∆q (x )+∆q (x ).The nucleon mass dependence may induce involved expressions for the diﬀerent structurefunctions which are typically of the form,cf.[2],3g±3τ=2(x)=qg q V g qA2x 2ξ1F ±q (ξ1)+12M 2x 3/Q 2ξ11ξ1dξ22(4M 2x 2/Q 2)2ξ21ξ2dξ3yg II1(y,Q 2)(10)g II3(x,Q 2)=2x1xdy3Numerical results are presented in Ref.[5].4The validity of Eq.(10)in the presence of target mass corrections was shown in [6]before.Expansions in terms of M 2/Q 2as considered in this paper,however,turn out to introduce artiﬁcial singularities in the range x →1,which are not present in the resummed expressions [2].3the consideration of mass corrections appears to be necessary also for consistency reasons since the structure functions containing twist–3terms in the case of longitudinal nucleon polarization are weighted by a factor of M2/S in the scattering cross section.The explicit relations for the target mass corrections to the twist–3contributions to g i|5i=1are given in[2].They obey the following relations:g III1(x,Q2)=4M2x2yg III2(y,Q2) ,(13)4M2x2Q2 +3 1x dyyg III4(y,Q2),(15)which hold after the inclusion of the target mass corrections.Whereas the relations(14,15)are relevant in the presence of weak interactions only,Eq.(13)can be tested already for purely electromagnetic interactions in the domain of lower values of Q2.For an experimental determination of the twist–3contributions to the structure functions g1(x,Q2)and g2(x,Q2)one can proceed as follows.From Eq.(13)it is evident,that for Q2≫M2g1(x,Q2)receives only twist–2contributions.g1can be measuredﬁrstly in this range.Its twist–2contribution at lower values of Q2can be obtained solving the twist–2 evolution equations for g1(x,Q2).Then one can determine the twist–2contribution to g2(x,Q2) by the Wandzura–Wilzcek relation,Eq.(10).Assuming that the contributions of twist–4and higher are suppressed the twist–3contributions to g1,2(x,Q2)can be extracted form the data by subtraction of the twist–2pieces.Relation(13)[2]may now be tested in calculating the twist–3 contribution from the twist–3contribution to g2(x,Q2)and comparing with the measurement.Finally we would like to comment on two other relations.The Burkhardt–Cottingham sum rule[11]10dxg2(x,Q2)=0(16)is consistent with the results of the local light cone expansion.Both the corresponding expec-tation values for twist–2and twist–3are absent in the respective series expansion also in the presence of target mass corrections[2].A second relation[12]10dxx[g1(x)+2g2(x)]=0(17)also holds in the presence of target mass corrections as long as Q2>M2p.One may cast the respective relations into the following form,cf.[2]:10dxx[g1(x)+2g2(x)]=e2q M p 10dx h1(x)−1−M2p x25SummaryIn the range of low values of Q2∼>M2p nucleon mass corrections to the polarized structure functions in deep inelastic scattering are essential.After the inclusion of these corrections a symmetric picture is obtained comparing the respective twist–2and twist–3terms,unlike the massless case.In lowest order inαs the twist–2and the twist–3contributions of the ﬁve polarized structure functions are connected by three relations.These are the Wandzura–Wilczek relation,a relation by the author and Kochelev and the Dicus relation for the twist–2 terms,and three relations by the author and Tkabladze for the twist–3terms.While the Dicus relation receives aﬁnite target mass correction,the other relations do not,or they doﬁrstly result after the inclusion of the target mass corrections at all.The relations being present for purely electromagnetic interactions can be tested through precision measurements of the structure functions g1(x,Q2)and g2(x,Q2)in the near future.Acknowledgement.For discussions I would like to thank N.Kochelev and abladze. 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