ON THETA FUNCTIONS OF ORDER FOUR

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Niveau: Supérieur, Licence, Bac+2
ON THETA FUNCTIONS OF ORDER FOUR YAACOV KOPELIOVICH, CHRISTIAN PAULY, AND OLIVIER SERMAN Abstract. We prove that the fourth powers of theta functions with even characteristics form a basis of the space H0(A,OA(4?))+ of even theta functions of order four on a principally polarized Abelian variety (A, ?) without vanishing theta-null. 1. Introduction Let (A, ?) be a principally polarized Abelian variety of dimension g defined over an alge- braically closed field k of characteristic different from two. By Mumford's algebraic theory of theta functions [M1] (see also the appendix of [B1]) there is a canonical bijection between the set of symmetric effective divisors ?? ? A representing the principal polarization and the set K of theta-characteristics ? of A. We denote by L the unique symmetric line bundle over A representing 2? and such that the linear system |L| contains the divisors 2?? for all ? ? K. The space H0(A,L2) of theta functions of order four decomposes under the natural involution of A into ±-eigenspaces H0(A,L2)+ and H0(A,L2)? of dimensions d+ = 2 g?1(2g + 1) and d? = 2 g?1(2g ? 1). We note that d+ and d? are equal to the number of even and odd theta-characteristics of A.

• section s? ?


• symmetric effective


• vanishing theta-null


• k?


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• all ? ?


• bundle over


• ??

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• Publié par: profil-vieg-2012
• Nombre de lectures: 9
• Langue: English

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SOME ABELIAN INVARIANTS OF 3-MANIFOLDS

LOUIS FUNAR

Someinvariantsforclosedorientable3-manifoldsaredenedusingaseriesofrepresentationsofthesymplec- tic groups and the theory of Heegaard splittings. They are natural extensions of theU(1) Chern-Simons- Witteninvariants.Theserepresentationscomefromthefunctionalequationsatisedbythethetafunctions of levelk analyze . Wethe values of these invariants for lens spaces.

Keywords functions, Heegaard splitting, tensor representation, symplectic groups.: Theta AMS Classic ation: 57 A 10, 14 K 25, 32 G 15.

0 . INTRODUCTION

The aim of this paper is to construct invariants of 3-manifolds using the endomorphisms of 1-homologies of surfaces determined by Heegaard splittings and representations of the symplectic group. This leads us to the study of actions of such endomorphisms on the space of theta functions on the Siegel space. The construction goes as follows. Any three manifold can be given an Heegaard decomposition, and hence can be written as the union of two handlebodies identied along a homeo- morphism of the surface boundary. After a choice of a basis of the 1-homology the homeomorphism induces an element of the symplectic group. The indeterminacy in the choice of this matrix can be analyzed to give invariants of the three manifold in question. We develop a particular invariant using actions on spaces of modular forms and analyze it in the case of lens spaces. Although the idea to consider theta functions is transparent from the notes of Oxford seminar [2] there is not an explicit treatment of Abelian Witten’s theory, from this perspective, on the author’s knowledge. Thus the goal of the present paper is to provide such a rigorous construction and a natural extension of it which leads us to some more general Abelian invariants. In ([12, 9]) the U(1) invariants are introduced as complex numbers modulo U(1) (or the group of roots of unity). In [26] some invariants are constructed in terms of the linking matrices of 3-manifolds, and their absolute value is the U(1) invariant. Our rst task will be to establish a family of invariants using representations of the symplectic group, and to check for the smallest group of roots of unity which have to appear as indeterminacy. Roughly speaking the usualU(1) gauge theory comes with a one dimensional vacuum vector associated to a handlebody and corresponding to a theta function with trivial characteristics. We generalize it to the case where the vacuum is degenerate, and is represented by a vector subspace of the space associated to the surface. Alternatively this amounts to consider a new representation of the symplectic group which is an exterior power of the former. Thiswaywederivenontrivialrenementf p,k of the usual Abelian invariantsf k,k which depend on the levelkand a divisorpofkconsidering such an extension is that, starting with interest in . The the standard U(1) TQFT we obtain other invariants (and furthermore also TQFT) which contain

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more topological information, as it can be deduced from the computations on lens spaces. It seems that this procedure could be carried on over some other TQFT. Another construction of 3-manifold invariants via representations of the mapping class groups was obtained by Kohno [20] for the SU(2) TQFT. Our invariants are certainly less sensitive than Kohno’s invariants: in particular the SU(2)-invariants can in some cases distinguish between a homology sphereandthestandardsphere,butourinvariantscannotdothat,becausetheyaredenedonthe symplectic group level rather than the mapping class group. However it is not at all clear whether all our invariantsf p,k can be deduced from the SU(2)-invariants. Some of the results of this paper have been announced in [9] and several related articles appeared (see [12, 26, 24, 7, 30, 31, 23]). Another semi-Abelian version was described in [11]. Aknowledgements.IambtedindenodhawensiosscuisedrohtrafuoneVrP.visoisadthestomy this subject, to V.Turaev and L.Guillou for their pertinent observations which improved the clarity of this paper. This paper is an expanded and updated version of the rst chapter of the author’s Ph.D. thesis at University of Paris-Sud.

1 . STATEMENT OF THE MAIN RESULT LetM 3  a Heegaard splitting of Considerbe a closed connected and oriented 3-manifold.M 3 = T g ∪ ϕ T g into two handlebodies of genusgglued together along their common surface  g using the homeomorphismϕ:  g  → g . Notice thatϕis not uniquely determined by the Heegaard splitting. In fact it can be composed (to the left and to the right) by any homeomorphism which extends to the whole handlebodyT g  anbounding the surface (i.e. extendable homeomorphism as considered by Suzuki [28] and Kohno [20]). SetM g for the mapping class group of the genusgsurface andM g + for the image inM g  We have a canonical surjectionof the subgroup of extendable homeomorphisms. s:M g  →Sp(2gZ that a symplectic basis in the homology Assume) onto the symplectic group. of the surface  g is chosen. Therefores(M g + ) = Sp + (2gZ) can be easily described as the set of symplectic matrices having the form " A 0D B # with respect to the usual splitting intoggmatrices. Remark that the tower of groups Sp(2gZ) has an exterior multiplication law, namely the symplectic sum  s : Sp(2gZ)Sp(2hZ) →Sp(2(g+h)Z) given by the formula: " DC BA #  s " A C 0 0 B D 0 0 # = " A C   A C 0 0 B D   B D 0 0 # .

LetAdenote an arbitrary set.

Denition set of functions1.1. TheF g : Sp(2gZ) →A,g∈Z + , is called an Abelian invariant ifthefollowingtwoconditionsarefullled: 1.F g (axb) =F g (x), for allx∈Sp(2gZ),a b∈Sp + (2gZ),g∈Z + , 2.F g+1 (x s ) =F g (x), for allx∈Sp(2gZ),g∈Z + , where= " 0 1   0 1 # ∈SL(2Z).

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Observethatan(Abelian)invariantdenesatopologicalinvariantforclosed3-manifoldsbymeans of the formula: F(M 3 ) =F g (s(ϕ)). The two conditions stated above forF  and the Reidemester-Singer theorem (see [6, 27]) prove the independence on the various choices which can be made on the right hand term. The natural way to get Abelian invariants is to use the representations of Sp(2gZ).

Denition1.2. A tensor representation of the symplectic group consists in the following data: 1. The hermitian vector spacesW g V g satisfyingW g =W 1 g ,V g =V 1 g . 2. A sequence of unitary representations g : Sp(2gZ) →U(V g hcihw)luftsllocehtidnnsio  g+h (x s y) = g (x) h (y) for allx y g happropriately chosen. W g is g (Sp + (2gZ))-invariant. Let W g denote the projection ofV g ontoW g . Forx∈End(V g ) set det W g (x) = det( W g x). We will assume that –det W 1  1 ()6= 0, –0 does not belong to S g>0 det W g ( g (Sp + (2gZ))),

hold. We denote byR(   V   W  ) the (multiplicative) group generated by S g>0 det W g ( g (Sp + (2gZ))) C  .

Fromsuchdatawecanndaninvariantbymeansof

LEMMA 1.3.To each tensor representation= (   V   W  )of the symplectic group we can associate an Abelian invariantF g () : Sp(2gZ) →C/R(), by means of the following formula: F g (;x) = [det W 1  1 ())]  g h det W g ( g (x)) i m(g)  wherem(g) = (dim(W 1 )) 1 g . Proof.The following equality

det W g ( g (cx)) = det W g ( g (c))det W g ( g (x)) holds wheneverc∈Sp + (2gZ), becauseW g is g (Sp + (2gZ derive that))-invariant. Next we det W g+1 ( g+1 (x s )) = det W g W 1 ( g (x) 1 ()) = h det W g ( g (x)) i dimW 1 [det W 1 ( 1 ())] dimW g  so our claim follows.✷ Our main result consists in the construction of a tensor representation of the symplectic group. Weneedrstsomenotations: V g (k) =C<  m ;m∈(Z/kZ) g >, for evenk, W g (p k) =C<  pm ;m∈(Z/kZ) g >, wherepdividesk.

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