Jacobi Forms over Number Fields

OF THE DISSERTATION Jacobi Forms over Number Fields by Howard Skogman Doctor of Philosophy in Mathematics University of California San Diego, 1999 Professor Harold Stark, Chair We de ne Jacobi Forms over an algebraic number eld K and construct examples by rst embedding the group and the space into the symplectic group and the symplectic upper half space respectively. We then create symplectic modular forms and create Jacobi forms by taking the appropriate Fourier coe cients. We also prove some relations of these Jacobi forms over certain elds to other types of modular forms. vii Chapter 1 Introduction The functions considered in this paper are generalizations of the classical Jacobi forms. Classical Jacobi forms are functions satisfying two transformation properties a + b c + d; z c + d = (c + d)ke2 im cz2 c +d ( ; z) (1.1) and ( ; z+ + ) = e 2 im( 2 +2 z) ( ; z) (1.2) for all matrices 0@ a b c d 1A in Sl2(Z), and vectors [ ; ] in Z2: The original examples of such forms were created by Jacobi in [11]. These examples were types of theta functions of quadratic forms. For example, given a positive de nite l l matrix Q with rational integer entries and even diagonal entries, and given a xed vector ~b in Zl, de ne the function Q;b( ; z) = X ~r2Zl e i t~rQ~r +2 t~rQ~bz : These theta functions have the special transformation properties described above, however there was no investigation into general functions satisfying the transformation properties presented above until [6]. It is clear that these functions are generalizations of modular forms which are 1 2 functions satisfying f a + b c + d = (c + d)kf( ); 80@ a b c d 1A 2 Sl2(Z) because setting the variable z = 0 and considering the rst transformation formula 1.1 yields the modular form transformation formula. The most interesting feature of Jacobi forms is that although they are only slightly more complicated than classical modular forms, they provide information about much more complicated types of modular forms such as vector valued and symplectic modular forms. There are a number of isomorphisms between spaces of Jacobi forms and these other types of modular forms and therefore understanding the Jacobi forms gives information about the other spaces of modular forms. Many of these connections are presented in [6] but others are presented in [12],[13],[18], [19]. There have also been a large number of investigations into more general types of Jacobi forms. Some of these articles focus on replacing the variable z above with a general matrix which has produced more connections to symplectic modular forms and vector valued modular forms see for example [15], [22],[24],[25]. There are two principal constructions for most types of modular forms. One is through theta functions, and the other is through Eisenstein series. These Eisenstein series have also been created in the case of Jacobi forms and this has allowed the study of the space of Jacobi forms as an algebraic structure. These techniques have also led to more connections between Jacobi forms and other types of forms see [1],[2],[7]. Another type of generalization of Jacobi forms is functions satisfying the same sort of transformation properties with larger groups of matrices and translation vectors. This is the primary focus of this paper. There have been very few investigations into this area. Gritsenko in [9] studied functions which satis ed generalizations of the transformation formulas where the invariance properties were with respect to all 0@ a b c d 1A in Sl2(Z[p 1]) and [ ; ] in Z[p 1]2. Krieg in [14] studied similar forms with the ring Z[p 1] replaced by the ring of integers in any imaginary quadratic eld, and studied their connection to symplectic modular forms. Haverkamp in [10] studied forms where the rst transformation properties are with respect to Sl2(Z) but the second is with 3 respect to vectors in the ring of integers in an imaginary quadratic eld. The aim of this paper is a still further generalization where the transformation formulas are with respect to the ring of integers in any algebraic number eld, (i.e. nite degree extension of the rationalsQ). So the rst transformation formula is with respect to Sl2(OK) and the second is with respect to O2 K where OK is the ring of algebraic integers in a number eld K. The construction relies on extending the connection between Jacobi forms and symplectic modular forms to algebraic number elds. The actual construction produces the \natural" generalization of Jacobi's initial examples. One interesting di erence between this work and some of the above cited works [9],[10],[15], is the use of a di erent symplectic group. The cited works all used the Hermitian symplectic group which is all matrices M with entries in the ring of integers in the eld such that tMJM = J = 0@ 0 In In 0 1A where the bar denotes complex conjugation. The symplectic group employed here has a similar requirement without the bar on the rst matrix. Another interesting feature of the Jacobi forms constructed in this paper is the need to introduce the index vector. The m which appears in the classical transformation formulas is replaced by the inner product of a vector with itself. The vector only appears in the variables corresponding to complex conjugates of the eld and it causes a slight modi cation of the transformation formulas. The di erence is that instead of all of the factors having the inner product of the vector with itself, some appear with the inner product of the vector and its complex conjugate. This index vector does not appear in Jacobi forms of totally real elds and this allows (along with some other facts) to extend the connection to vector valued modular forms for totally real elds. 1.1 Some Notation Throughout the paper some standard notation will be used which is listed here. Q;Z;R;R+; C , will be used to denote the rational numbers, the rational integers, the real numbers, the positive real numbers and the complex numbers respectively. We will use K to denote an algebraic number eld and OK will be the ring of algebraic 4 integers in this number eld. A totally positive element 2 K will be denoted 0 and the notation 0 is used to denote an element that is totally positive or zero. We will denote by K ; 1 K the di erent and the inverse di erent of the eld K. The n n matrices with entries in F is denoted Mn(F). Given a matrix M 2Mn(F) the transpose of M will be denoted tM . The l l identity matrix will be Il. The imaginary part of is denoted Im( ). The n dimensional Z-module spanned by the faig1 i n is written [a1; a2; :::an]Z. Chapter 2 Jacobi Forms over Q All of the results presented in this chapter are presented in [6]. 2.1 The Jacobi Group Jacobi forms are functions satisfying transformation formulas related to a group acting on the domain of the function. In order to de ne Jacobi forms, the rst thing is to de ne is the Jacobi group and then introduce the space on which this group discretely acts. The Jacobi group is Sl2(Z)nZ2 J (Z) where the group Sl2(Z) consists of 2 2 matrices M = 0@a b c d1A 2 Sl2(Z): such that each entry is a rational integer a; b; c; d 2 Zand det(M) = ad bc = 1. Note this is not a direct product but a semi-direct product where the group operation is given by: for all M;M 0 in Sl2(Z), and [ ; ]; [ ; ] in Z2 (M; [ ; ]) (M 0; [ ; ]) = (MM 0; [ ; ]M 0+ [ ; ]): (2.1) The second entry is the product of the rst vector and the second matrix, added to the second vector. 5 6 We also de ne, as usual, the congruence subgroup of level N , for N in Z+ as 0(N) := 8<:M = 0@a b c d1A ;M 2 and c 0 mod N9=; : and we may choose to look at forms transforming under 0(N)n (mZ)2 where m is in Z. The space on which this group acts is h C where h := fz 2 C ; Im z > 0g and C is the eld of complex numbers. The actions of J (Z) on h C are for M in Sl2(Z), M ( ; z) = 0@ a b c d 1A ( ; z) = a + b c + d; z c + d (2.2) and for [ ; ] in Z2, [ ; ] ( ; z) = ( ; z + + ): (2.3) The general element of this Jacobi group acts rst by the matrix and then by the vector which insures there is a group action on this space. To have a group action means that: 8g1; g2 2 J (Z); ( ; z) 2 h C g1 (g2 ( ; z)) = g1g2 ( ; z): This group action xes the space and can be shown to be discrete. Brie y, a discrete action is one with no sequences of distinct elements approaching a limit in the group. 2.2 Jacobi Forms De nition. A function ( ; z) : h C ! C that satis es a + b c + d; z c + d = (c + d)ke2 im cz2 c +d ( ; z) (2.4) ( ; z+ + ) = e 2 im( 2 +2 z) ( ; z) (2.5) for all 0@ a b c d 1A 2 Sl2(Z); [ ; ] 2 Z2, and is analytic in both variables is called a Jacobi Form of weight k and index m. 7 The rst transformation formula 2.4 is a generalization of the classical modular transformation formula, that is a function which satis es a formula like f 0@0@ a b c d 1A 1A = (c + d)kf( ); 80@ a b c d 1A 2 Sl2(Z): (2.6) A Jacobi form restricted to z = 0 is a modular form. The second transformation formula 2.5 is a type of elliptic transformation, which means it expresses an invariance under translations by the rational integer lattice spanned by f1; g. Because of these formulas, a Jacobi form satis es ( +1; z) = ( ; z+1) = ( ; z) and therefore possesses a Fourier expansion with respect to both ; z i.e. ( ; z) = 1 Xn=0 X r2Z; r2 4nm c(n; r)qn r where q = e2 i ; = e2 iz . Note that the r2 4nm is a condition to make the function analytic at in nity, i.e. analytic as ! i1. In the de nition of Jacobi forms one may also restrict to a subgroup of the Jacobi group, and one may allow a multiplier system in the rst transformation formula. Speci cally we may replace 2.4 by a + b c + d; z c + d = (M)(c + d)ke2 im cz2 c +d ( ; z) where (M) is a root of unity depending only on the matrix M . We will discuss the multiplier systems more in the following sections. 2.3 Relations to other types of modular forms Jacobi forms are related to other types of modular forms, notably symplectic and vector valued modular forms. We will only sketch some of the major results of the correspondences here. Symplectic modular forms of genus n are functions on h(n) = fZ = X + iY 2Mn(C ) j tZ = Z; Y > 0g; 8 i.e. n n symmetric matrices over the complex numbers with positive de nite imaginary part. The group Sp2n(Z) acts discretely on this space, where Sp2n(Z) = 8<:M = 0@ A B C D 1A 2M2n(Z) j tMJM = J = 0@ 0n In In 0n 1A9=; and 0n; In denote the n n zero and identity matrix respectively and all of A;B;C;D are n n matrices over the rational integers. This group may also be written as Sp2n(Z) = 8<:0@ A B C D 1A 2M2n(Z) j tAC = tCA; tDB = tBD; tAD tCB = In9=; : The action of Sp2n(Z) on h(n) is given by 0@ A B C D 1A Z = (AZ +B)(CZ +D) 1; 8 0@ A B C D 1A 2 Sp2n(Z); Z 2 h(n): De nition. A Symplectic modular form of weight k and genus n is a function f(Z) : h(n) ! C such that f(M Z) = det(CZ +D)kf(Z); 8 M = 0@ A B C D 1A 2 Sp2n(Z): (2.7) We may allow a multiplier system in the transformation formula. When the genus is two the variable Z may be written as Z = 0@ z z 2 1A ; ; 2 2 h; z 2 C ; where Im( )Im( 2) > Im(z)2: The correspondence between Jacobi forms and symplectic modular forms is given by Theorem. If f is a symplectic modular form of weight k and genus two, f(Z) = f( ; z; 2) has a Fourier expansion with respect to 2 of the form f( ; z; 2) = 1 X m=0 k;m( ; z)e2 im 2 where each of the k;m( ; z) are Jacobi forms of weight k and index m. 9 This theorem shows the close connection between Jacobi forms and symplectic modular forms. A generalization of this theorem will be used to create Jacobi forms over an algebraic number eld. It is interesting to note that this correspondence is somewhat reversible, in that symplectic modular forms may be created by taking certain Jacobi forms as the coe cients in a Fourier expansion when the genus is two. The relation to vector valued modular functions arises from a periodicity in the Fourier coe cients of a Jacobi form. To state this connection rst de ne for each congruence class modulo 2m m; ( ; z) = X r2Z; r mod2m qr2=4m r; q = e2 i ; = e2 iz which are Jacobi forms of weight 1 2 and index m on a subgroup of J (Z). If ( ; z) is a Jacobi form of weight k and index m, then there exist functions h ( ) such that ( ; z) = X mod2mh ( ) m; ( ; z): Where the h ( ) satisfy h ( + 1) = e2 im 2=4mh ( ) (2.8) h 1 = k p2 =i X mod2m e2 i =2mh ( ) (2.9) where the square root in 2.9 is given by the principal value. Theorem. The correspondence k;m( ; z) ! (h ) mod2m gives an isomorphism between the space of Jacobi forms of weight k and index m and vector valued modular forms satisfying 2.8, 2.9 and bounded as Im( ) goes to in nity. By vector valued modular forms of weight k 12 we mean functions ~h( ) = (h ( )) mod2m satisfying ~h(M ) = (c +d)k 12U(M)~h( ) where U(M) is some 2m 2m matrix and M = 0@ a b c d 1A in Sl2(Z). Both of these correspondences will be extended for certain types of number elds and then proofs of the results will be given. 10 2.4 Multiplier systems The more general de nitions of modular forms, Jacobi forms, and symplectic modular forms all include multiplier systems. This is especially important for those forms that have non-integer weight. The importance of the multiplier system can be seen most easily in the case of classical modular forms. We now state a de nition of modular forms on Sl2(Z) which incorporates a multiplier system and then give some justi cation for its necessity. De nition. A modular form of weight k and multiplier system (:::) is an analytic function f : h ! C such that 80@ a b c d 1A 2 Sl2(Z) f 0@0@ a b c d 1A 1A = f a + b c + d = 0@ a b c d 1A (c + d)kf( ); where (:::) is a root of unity (i.e. (:::)f = 1 for some f 2 Z) depending only on the matrix 0@ a b c d 1A. This may be seen as a generalization of the usual de nition of modular forms which is given with the multiplier system (M) 1 for all matrices M in Sl2(Z). In order to see how these multiplier systems arise let M = 0@ a b c d 1A be a matrix in Sl2(Z) and let (M; ) = (c + d). In order to have the group Sl2(Z) act on the space h it is necessary that for all M1;M2 in Sl2(Z); in h, M1 (M2 ) = M1M2 : So in order to be a modular form of weight 1 without a multiplier the function must have the property that (M1M2; ) = (M1;M2 ) (M2; ) and it is easy to check that has this property. However, if f is a modular form of weight 1 2 without a multiplier system then the requirement is (M1M2; ) 1 2 = (M1;M2 ) 1 2 (M2; ) 1 2 11 which is not necessarily true due to the ambiguity with how each of the square roots are taken. All that is known (M1M2; ) 1 2 = (M1;M2 ) 1 2 (M2; ) 1 2 : A multiplier system is introduced to handle the sign trouble in this case and is de ned as (:::) : Sl2(Z) ! Ul such that (M1M2) (M1M2; ) 1 2 = (M1) (M1;M2 ) 1 2 (M2) (M2; ) 1 2 : for all M1;M2 2 Sl2(Z) where Ul is the nite group of the l-th roots of unity. Note that it is not in general true that (M1M2) = (M1) (M2). In general it is very complicated to gure out the multiplier explicitly in terms of the matrix. Most important to the work presented here is the multiplier system of the symplectic theta function which is a symplectic modular form of weight 1 2 and satis es f 0@0@ A B C D 1A Z1A = f((AZ +B)(CZ +D) 1) = 0@ A B C D 1A det(CZ +D) 12 f(Z) for all matrices in a certain subgroup of Sp2n(Z). The multiplier system in this case is an eighth root of unity depending only on the matrix and the value of the square root that is taken. This is the most important case to this work because the symplectic theta function will be used to create the new Jacobi forms and then the transformation properties of the symplectic theta function are used to derive the transformation properties of the Jacobi forms. Stark in [21] determined the explicit multiplier system for the symplectic theta function in some important special cases and this could be used to determine the exact root of unity in the Jacobi transformation properties. Chapter 3 Number elds In this chapter we present some notation and facts about algebraic number elds. The notation will be used throughout the rest of the paper. All of the results presented here are standard and contained in most books about algebraic number theory for example [5], [20]. 3.1 The conjugates of a number eld LetK be an algebraic number eld, that isK C , K is a nite degree extension of Q. It is well known that K may be written as K = Q( ) where satis es an irreducible nth degree polynomial whose coe cients are rational integers, and then n is the degree of the extension. Given , every in K may be expressed as a rational combination of powers of , for example = Pn 1 j=0 bj j where each bi is in Q. If the other roots of the polynomial satis ed by are denoted (1); (2); :::; (n), assume the roots are ordered so that = (1), then the conjugates of the eld K are de ned as the elds K(i) = Q( (i)). Therefore, if an element is in K; and = Pn 1 j=0 bj j then the conjugates of are denoted (i) = Pn 1 j=0 bj (i)j in K(i). If (i) is a real number, K(i) is called a real conjugate of the eld K and if (i) is in C but not in R, K(i) is called a complex conjugate of K. Note that complex conjugates of always come in pairs, if (i) = a + bi is one root of the polynomial satis ed by then (i) = a bi is also a root. If K(i) = Q( (i)) is a complex conjugate then denote the corresponding complex 12 13 conjugate eld K(i) = Q( (i)): Assume K is an nth degree extension, denoted [K : Q] = n; n = r1+2r2 where r1 is the number of real conjugates of K and 2r2 is the number of complex conjugates of K. De ne OK as the ring of integers in K, which is all elements of K satisfying a monic polynomial with coe cients in Z. It is known that OK is an n-dimensional Zmodule written OK = [!1; !2; :::!n]Z, so for all in OK ; there exists l1; l2; :::; ln 2 Zsuch that =Pnj=1 li!i. For the rest of the paper the conjugates are labelled so that the rst r1 are the real conjugates and the rest are such that Kr1+j = Kr1+r2+j ; for all 1 j r2. The invertible elements of OK are called the units of K and denoted UK , UK = f" 2 OK j " 6= 0; " 1 2 OKg: An element a in K is called totally positive, denoted by a 0, if all the real conjugates of a are positive. It will also be useful to use the notation a 0 to mean that a is totally positive or zero. The trace and the norm of an element a in K are respectively given by TrK=Q(a) := n Xj=1 a(j); and NK=Q(a) := N (a) := n Y j=1a(j): For any a in K both TrK=Q(a);Nk=Q(a) are in Q; in particular if a is in OK then both TrK=Q(a);Nk=Q(a) are in Z. The discriminant of K denoted K = det(W )2 where W = 0BBB@ !(1) 1 : : : !(1) n ... ... !(n) 1 : : : !(n) n 1CCCA : Where the !j were de ned above as the rational integer basis elements of OK . The discriminant of a number eld is a rational integer. 14 3.2 Ideals in number elds, the Di erent There are certain subsets of the eld K which are known as ideals. Ideals play an important role in a number eld because the arithmetic of numbers is replaced by the arithmetic of ideals. De nition. An ideal of K is a subset a OK such that (1) 8a; b 2 a; a+ b 2 a (2) 8b 2 OK ; ba a. However, zero is not considered to be an ideal even though it does satisfy all of the hypotheses. The ideals of a number eld possess an arithmetic structure in that ideals may be added or multiplied together, or divided one by the other. De ne for two ideals a; b K; a+ b = (a[ b) that is the ideal generated by the union of the two ideals, and de ne ab as the ideal generated by all elements of the form for in a; in b. In order to de ne division of ideals it is necessary to de ne the inverse of an ideal. The inverse of an ideal a is a 1 = fb 2 K j 8a 2 a; ba 2 OKg: It is not di cult to check that the above de nition is an ideal. There are also notions of prime ideals and the unique factorization of ideals into the product of prime ideals but this will not be needed. Now de ne the inverse di erent of K to be 1 K where 1 K = fa 2 K j TrK=Q(ab) 2Z; 8b 2 OKg: The Di erent of K is K , i.e. the inverse of the ideal 1 K . All ideals in a number eld of degree n have a n-dimensional rational integer basis. So if a is an ideal in K and is in a then = Pnj=1 cjaj where the aj are xed elements of K and the cj are in Z. This representation is denoted a = [a1; a2; :::an]Z. The conjugate ideals are then given by a(j) = [a(j) 1 ; a(j) 2 ; :::a(j) n ]Z. This type of basis may be used to turn a sum over elements of the ideal and the conjugates of the element, into a sum over the rational integers. For example if 2 a 15 and =Pnj=1 cjaj as above then 0BBB@ (1) ... (n) 1CCCA = 0BBB@ a(1) 1 ::: a(1) n ... ... a(n) 1 ::: a(n) n 1CCCA0BBB@ c1 ... cn 1CCCA = A0BBB@ c1 ... cn 1CCCA : The matrix representation of the ideal a is invertible and the inverse of the matrix A gives the basis for the ideal a 1 1 K . The inverse matrix has the same structure as A, for example there exist b1; b2; :::bn 2 K such that A 1 = 0BBB@ b(1) 1 ::: b(1) n ... ... b(n) 1 ::: b(n) n 1CCCA then the ideal a 1 1 K has a rational integer basis a 1 1 K = [b1; b2; :::bn]Z. Chapter 4 Jacobi forms over number elds In this chapter the Jacobi group over a number eld K is de ned as well as the space on which it acts, and the de nition of a Jacobi form over a number eld is given. All of the notation from the previous chapter carries over into the rest of this paper including the de nitions of K; r1; r2; K(j);OK;W; etc. 4.1 The Jacobi group The Jacobi group of the number eld K will be denoted J (K) where J (K) = Sl2(OK)nO2 K : The group operation is the same as in the classical case, see 2.1. This group acts on H = hr1 h(Q)r2 C r1 Qr2 where Q is used to denote Q = fx+ y jx; y 2 C ; 2 = 1; a = a; 8a 2 C g which is known as the full ring of quaternions. It is sometimes written as Q = fa+ bi+ cj + d j a; b; c; d2 R; i2 = j2 = 2 = 1; ij = ; j = i; ji= g: 16 17 h(Q) is used to denote the upper half plane of quaternions h(Q) = fx+ y 2 Q j y 2 R+g which may also be written as all quaternions that have no j component and positive component. It will be crucial in all of the following calculations to be careful about the ordering of elements of the quaternions since elements of this ring do not commute. This space H where this Jacobi group acts is composed of one copy of h C for each real conjugate of the eld, and one copy of h(Q) Q for every pair of complex conjugates. Variables of this space will be listed as ( 1; 2; :::; r1+r2 ; z1; :::; zr1+r2) where each of the variables is in the appropriate upper half space and each of the z variables is in the appropriate eld. Note the ordering of the conjugates is retained from chapter 4. So the rst r1 of the upper half plane j (or full eld zj) variables are in h (or C ) and the next r2 of the variables are in h(Q) (or Q). The quaternionic variables will be represented as j = xj + yj ; 8 j 2 h(Q) and zj = uj + vj ; 8zj 2 Q: The actions of J (OK) on the space H are given by 80@ 1A 2 Sl2(OK); 0@ 1A ( 1; ::: r1+r2 ; z1; :::zr1+r2) = ( (1) 1 + (1) (1) 1 + (1) ; ::: (r1) r1 + (r1) (r1) r1 + (r1) ; ( (r1+1) r1+1 + (r1+1))( (r1+1) r1+1 + (r1+1)) 1; ::: z1 (1) 1 + (1) ; ::: zr1 (r1) r1 + (r1) ; ( r1+1 (r1+1) + (r1+1)) 1zr1+1:::); (4.1) 8[ ; ] 2 O2 K ; [ ; ] ( 1; ::: r1+r2 ; z1; :::zr1+r2) = ( 1; ::: r1+r2 ; z1 + 1 (1) + (1); :::; zr1 + r1 (r1) + (r1); zr1+1 + r1+1 (r1+1) + (r1+1); :::): (4.2) 18 As before, a general element of the Jacobi group acts rst by the matrix and then by the vector. These actions are just the di erent conjugates of the group elements acting on the di erent copies of the upper half spaces and the full elds by the same actions as in the classical case. The de nition of congruence subgroups as above may be extended to K and thereby de ne subgroups of the Jacobi group using subgroups of Sl2(OK) and sublattices in the ring of integers. For example, extending the notation from earlier, given an ideal N OK de ne 0(N) = 8<:M = 0@a b c d1A ; M 2 Sl2(OK) and c 2 N9=; : and so a subgroup of J (OK) of the form 0(N)n a2 for ideals a;N OK may be used in place of J (OK). 4.2 Jacobi forms In order to de ne Jacobi forms over K it is necessary to de ne the transformation formulas and transformation factors for the Jacobi group. It will be useful to de ne an exponential of a quaternion as e[:::] = e2 i[:::]; e[a+ b ] = e[a+ a+ i(b+ b)] = e2 i(a+a+i[b+b]); 8a+ b 2 Q: To reduce some notation in the formulas the factorN ( + ) is used to denote the factor that replaces the (c + d) in the classical transformation formula 2.4, so denote for in hr1 h(Q)r2; ; in OK N ( + ) = r1 Y j=1( (j) j + (j)) r1+r2 Y j=r1+1(j (j)xj + (j)j2 + y2 j j (j)j2) (4.3) where the notation j = xj + yj ; for all j in h(Q) is used. The usual complex norm is denoted by j:::j, ja+ bij2 = a2 + b2. In the case of number elds, especially elds with complex conjugates, it is necessary to de ne the index vector associated with the index m. The index vector will be a certain complex vector, denoted ~ m of length equal to twice the weight k of 19 the form, such that t~ m(j) ~ m(j) = m(j); for all 1 j n. The explicit nature and the appearance of the index vector as well as what is meant by its conjugates will be made clear in the construction. De nition. A Jacobi form of weight k and index m and index vector ~ m for the number eld K is a function (~ ; ~z) : hr1 h(Q)r2 C r1 Qr2 ! C satisfying 0@0@ 1A (~ ; ~z)1A = 0@ 1AN ( + )k0@ r1 Y j=1 e[m(j) (j)z2 j (j) j + (j) ]1A 0@ r1+r2 Y j=r1+1 e[ t~ m(j)(uj + vj )( (j) j + (j)) 1 (j)(uj + vj )~ m(j)]1A (~ ; ~z) (4.4) and ([ ; ] (~ ; ~z)) = 0@ r1 Y j=1 e[ m(j)( (j)2 j + 2 (j)zj)]1A 0@ r1+r2 Y j=r1+1 e[ t~ m(j)( (j) j (j) + 2 (j)zj)~ m(j)]1A (~ ; ~z) (4.5) for all 0@ 1A 2 Sl2(OK); [ ; ] 2 O2 K , where is a root of unity depending only on the matrix. It is important to note that the forms which will be created have weight k in Zor k in Z+ 12 , however the index m is allowed to be any integer in the eld. These transformation formulas are easily seen to be generalizations of the classical case. The transformation factors are essentially one conjugate of the factor from the classical action for each conjugate of K, where one must be careful in the ordering of the terms for the quaternionic variables. This de nition may be restricted to subgroups of the Jacobi group. The root of unity depending on the matrix will be an eighth root of unity, i.e. (M)8 = 1 for all M in Sl2(OK) which may be determined using work of Stark in [21]. 20 Note there is no multiplier system in the second transformation formula 4.5. The group action by elements of the form [ ; ] along with the fact that theta functions will be used in the construction, force the multiplier system to be trivial on these Jacobi group elements. The only di erences from the exact form of the classical formulas and factors results from the noncommutative nature of the quaternionic variables. It is also useful to note that the index vector is only necessary for elds with complex conjugates. So for totally real elds it is only required to specify the index and not the index vector. Also for totally real elds, it is possible to require that the Jacobi forms be analytic and in fact this will be necessary in order to prove the connection to vector valued modular forms. However, for the general algebraic number eld there is no notion of an analytic function so it is left out of the de nition. Chapter 5 Construction In this chapter a construction will be outlined that produces Jacobi forms of weight 1 2 , indexes of the form 2, and index vectors of the form ( ), for in OK . There is a method of producing arbitrary weights using quadratic forms which we will delay until chapter 7 and suggest [17] for more information. The idea of the construction is to use the relationship between Jacobi forms and symplectic modular forms. Speci cally, rst create a symplectic modular form which transforms over a number eld, then take a Fourier expansion with respect to the appropriate variables and the Fourier coe cients will be Jacobi forms. It will be necessary to explicity show this since it is not obvious from the previous result how to generalize this property to number elds. The speci c function we will create as a rst example is a generalization of the classical Jacobi theta function 1( ; z) =X n2Ze i(n2 +2nz): which is a Jacobi form of weight 1 2 and index 1 on a subgroup of the Jacobi group J (Z). 5.1 The index vector To introduce the index vector, it is easiest to describe how it appears. First, one creates a more general type of Jacobi theta function for any symmetric, positive de nite quadratic form Q. That is, if Q is an l dimensional symmetric quadratic form 21 22 over the rational integers with even diagonal entries, i.e. for a in Zl; Q(a) = taQa and Q(a) > 0; for all a in Zl; a 6= 0, then x a vector b in Zl and de ne Q( ; z) = X a2Zl e i( taQa +2 taQbz) which is a Jacobi form of weight l 2 and index Q(b). The natural generalization of this form may be given for an l dimensional symmetric quadratic form QK whose entries are in OK . It is known that such a form decomposes as Q(j) K = tL[j]L[j], and for a xed b in Ol K the general Jacobi theta fuction of a quadratic form may be written as QK ;b(~ ; ~z) = X a2Ol K exp( i(r1+r2 Xj=1 ta(j) tL[j] jL[j]a(j) + 2 ta(j) tL[j]zjL[j]b(j))); where we are using the previously de ned exponential for quaternions. Such a form will be constructed and shown to be a Jacobi form over K of weight l 2 and the index tbQKb = tb tLLb, except that the non complex parts of the quaternionic transformation factors will have an index of tb tLLb. So if the index m =Pli=1 c2i the non complex parts of the quaternionic transformation factors have an index of Pli=1 ci ci. This creates the need to specify the index vector which is ~ m(j) = L[j]b(j) for all of the conjugates in order to completely specify the transformation factors. Actually knowing the index vector is overkill. The transformation formulas are completely detemined by the index pair Q(b) and the index for the non complex part of the quaternionic factors tb tLLb. So if another index vector has the same index pair as Lb, the Jacobi form with this index vector will transform identically to the Jacobi form with the index vector Lb. This means that instead of requiring that forms have the same index vector, the de nition of Jacobi forms over K actually requires that the index vector for the form be in the same class as the index vector in the de nition. Where the class of an index vector is determined by the index pair. The rst construction will follow this idea in the case where QK = L = (1) the 1 1 identity matrix. The problems with the class of the index vector will not arise until the general construction in chapter 7. 23 5.2 Symplectic theta function We now introduce the function which will be used to create a symplectic modular form, namely the symplectic theta function Sp0@Z;0@ U V 1A1A = X m2Zn e i( t(m+V )Z(m+V ) 2 tmU tV U) (5.1) where Z is in h(n), and U; V in C n are xed. Sp(Z) is a symplectic modular form of weight 12 with a multiplier system for (n) where (n) = 8<:M = 0@ A B C D 1A 2 Sp2n(Z) j A tB; C tD have even diagonal entries9=; : Speci cally, this means 8 M = 0@ A B C D 1A 2 (n) Sp0@0@ A B C D 1A Z;M 0@ UV 1A1A = (M)det(CZ +D) 1 2 Sp0@Z;0@ UV 1A1A (5.2) where (M) is an eighth root of unity depending only on the matrix M and the value of the square root that is taken. This root of unity can be explicitly determined in certain cases using results of [21]. There is an alternate presentation of this theta function which changes the group under which this function transforms. Namely by replacing the i that appears in 5.1 by a 2 i produces a function that does not transform properly under (n) but under the group (n) 0 (4) = 8<:0@ A B C D 1A 2 Sp2n(Z) j C has entries divisible by 49=; : This type of group is more common in the literature, and make calculations simpler to present however it is quite a bit smaller than (n) so the simpli cation comes at the price of a large number of transformation formulas. In this paper the \2 i" version will be used to simplify the statements of results and to simplify the construction however the construction with the \ i" version works just as well. 24 5.3 Embeddings The method of creating a symplectic theta function that transforms appropriately is to embed J (OK) into Sp4n(Z) and the space hr1 h(Q)r2 C r1 Qr2 into h(2n). We will introduce notation to make this easier to state. We set for (~ ; ~z) in hr1 h(Q)r2 C r1 Qr2 d1(~ ) = 0BBBBB@ 1 0 ::: 0 2 ... . . . r1 1CCCCCA ; d1(~z) = 0BBBBB@ z1 0 ::: 0 z2 ... . . . zr1 1CCCCCA d2(~x) = 0BBBBB@ xr1+1 0 ::: 0 xr1+2 ... . . . xr1+r2 1CCCCCA ; d2(~u) = 0BBBBB@ ur1+1 0 ::: 0 ur1+2 ... . . . ur1+r2 1CCCCCA d3(~x) = 0BBBBB@ xr1+1 0 ::: 0 xr1+2 ... . . . xr1+r2 1CCCCCA ; d3(~u) = 0BBBBB@ ur1+1 0 ::: 0 ur1+2 ... . . . ur1+r2 1CCCCCA : As before, set j = xj + yj ; zj = uj + vj for r1 < j r1 + r2 and similarly de ne d2(~y); d2(~v); d3(~y); d3(~v). It should be noted that all of the following quaternionic embeddings are based on the same representation, i.e. a+ b 2 Q ,! 0@ a ib i b a 1A 2M2(C ): (5.3) Now embed the space H into h(2n) by sending (~ ; ~z) to Z = 0BBBBBBBBBBB@ d1(~ ) 0 0 d1(~z) 0 0 0 d2(~x) id2(~y) 0 d2(~u) id2(~v) 0 id3(~y) d3(~x) 0 id3(~v) d3(~u) d1(~z) 0 0 d1(~ 0) 0 0 0 d2(~u) id3(~v) 0 d2(~x0) id2(~y0) 0 id2(~v) d3(~u) 0 id3(~y0) d3(~x0) 1CCCCCCCCCCCA : 25 There are an extra set of upper half plane variables ~ 0 (where the quaternionic variables are denoted 0 j = x0j + y0 j ), which were embedded in the lower right n n corner of the matrix. The Fourier expansion will be formed with respect to these extra variables, after creating the symplectic modular form. Another embedding as in chapter 4 allows a sum over rational integers instead of integers in OK , recall the de nition from Chapter 3, W = 0BBB@ !(1) 1 : : : !(1) n ... ... !(n) 1 : : : !(n) n 1CCCA where OK = [!1; :::!n]Z: This matrix W converts rational integers into elements of OK , i.e. given in OK ; = Pnj=1 ai!i with aj in Z; 1 j n, then 0BBBBB@ (1) (2) ... (n) 1CCCCCA = 0BBB@ !(1) 1 : : : !(1) n ... ... !(n) 1 : : : !(n) n 1CCCA0BBBBB@ a1 a2 ... an 1CCCCCA : De ne c W = 0@ W 0 0 W 1A and create the symplectic theta function 0@Z;0@ 00 1A1A = X m2Z2n e2 i tm tc WZc Wm = X m2Z2n e2 i tm ~ Zm where e Z = tc WZc W: In order for this to be a symplectic theta function it is necessary to know that e Z is in the space h(2n). For this we need the following lemma. Lemma. Assume V and T are invertible complex matrices, then the imaginary part of Z = tTV T is positive de nite if 1 2i(V tV ) is positive de nite. Proof. The imaginary part of tTV T is 1 2i [ tTV T tTV T ] = 1 2i [ tTV T tT tV T ] since Z is symmetric. Simplifying we have this is tT [ 1 2i(V tV )]T and the expression in the parentheses is positive de nite by assumption. Multiplying by tT and T does not a ect the positive de nite nature of the expression in parentheses. 26 The converse of the lemma is also true but will be unnecessary in the construction. To see that e Z is in h(2n), de ne S = 0BBBBBBBBBBB@ Ir1 0 ::: 0 0 Ir2 ... Ir2 0 Ir1 0 Ir2 Ir2 0 1CCCCCCCCCCCA then SZS is the same matrix as Z with the complex conjugate pairs switched in all of the variables. Similarly Sc W is the same as the matrix c W with the complex conjugate pairs of rows switched which will be written as W . Note S2 = I2n and write tc WZc W = tc WSSZc W = tWSZc W and the imaginary part of SZ is (using the notation from above) 0BBBBBBBBBBB@ d1(y) d1(v) d3(y1) d3(v1) d3(u2) d2(y1) d2(u2) d2(v1) d1(v) d1(y0) d2(v1) d3(u2) d3(y0 1) d2(u2) d3(v1) d2(y0 1) 1CCCCCCCCCCCA : (5.4) This matrix can be forced to be positive de nite simply by picking the y0 and the y0 1 variables to be large enough based on the other variables in the matrix. So some conditions on the extra upper half plane variables forces the imaginary part of the matrix in 5.4 to be positive de nite and therefore the matrix tc WZc W = tWSZc W has positive de nite imaginary part and since it is obviously symmetric it is in h(2n). Expanded out (Z) = (~ ; ~z; ~ 0) looks like (~ ; ~z; ~ 0) = X 1; 22OK e[ r1 Xj=1 (j)2 1 j + 2 (j) 1 (j) 2 zj ]e[ r1+r2 X j=r1+1 (j) 1 j (j) 1 + 2 (j) 1 zj (j) 2 ] e[ r1 Xj=1 (j)2 2 0 j ]e[ r1+r2 X j=r1+1 (j) 2 0 j (j) 2 ]: (5.5) 27 This rst line mimics the classical Jacobi theta functions where one thinks of the 2 as xed and the quadratic form Q = (1). In fact the rst line is the Fourier expansion with respect to the 0 j , so the Fourier expansion with respect to ~ 0 may be written (~ ; ~z; ~ 0) = X 22OK 1 2 ; 2(~ ; ~z)e[ r1 Xj=1 (j)2 2 0 j ]e[ r1+r2 X j=r1+1 (j) 2 0 j (j) 2 ] where 12 ; 2(~ ; ~z) = X 12OK e[ r1 Xj=1 (j)2 1 j + 2 (j) 1 (j) 2 zj ]e[ r1+r2 X j=r1+1 (j) 1 j (j) 1 + 2 (j) 1 zj (j) 2 ]: To write this expansion properly we should group the coe cients corresponding to 2 and 2 should be grouped together, however since these terms are identical we leave them separate for now. The next step in the construction is to embed the Jacobi group into Sp4n(Z) and show that the action on the embedded variables (~ ; ~z) as an element of h(2n) is the claimed group action on the space. First de ne the notation d( ); for in K and extend the notation from above for d1; d2; d3 to elements of K d( ) = 0BBBBB@ (1) 0 : : : 0 0 (2) ... . . . 0 (n) 1CCCCCA = 0BB@ d1( ) d2( ) d3( ) 1CCA : The embeddings of the elements of the Jacobi group are: 0@ 1A 2 Sl2(OK) ,! 0BBBBB@ d( ) 0 d( ) 0 0 In 0 0 d( ) 0 d( ) 0 0 0 0 In 1CCCCCA = 0@ A B C D 1A ; [ ; ] 2 O2 K ,! 0BBBBB@ In 0 0 d( ) d( ) In d( ) d( ) 0 0 In d( ) 0 0 0 In 1CCCCCA = 0@ A0 B0 C 0 D0 1A : 28 With these embeddings 0@ 1A (~ ; ~z; ~ 0) ,! 0BBBBBBBBBBB@ d1( ~ + ) 0 0 d1( ~z) 0 0 0 d2( ~ x+ ) id2( ~ y) 0 d2( ~ u) id2( ~v) 0 id3( ~ y) d3( ~ x+ ) 0 id3( ~v) d3( ~ u) d1(~z) 0 0 d1(~ 0) 0 0 0 d2(~u) id3(~v) 0 d2(~x0) id2(~y0) 0 id2(~v) d3(~u) 0 id3(~y0) d3(~x0) 1CCCCCCCCCCCA 0BBBBBBBBBBB@ d1( ~ + ) 0 0 d1( ~z) 0 0 0 d2( ~x+ ) id2( ~y) 0 d2( ~u) id2( ~v) 0 id3( ~y) d3( ~x+ ) 0 id3( ~v) d3( ~u) 0 0 0 In 0 0 0 0 0 0 In 0 0 0 0 0 0 In 1CCCCCCCCCCCA 1 : It may be veri ed that this action with these embeddings takes the variables j ! (j) j + (j) (j) j + (j) ; zj ! zj (j) j + (j) ; 8 1 j r1 (5.6) j ! ( (j) j + (j))( (j) j + (j)) 1; zj ! ( j (j) + (j)) 1zj ; 8 r1 j r1 + r2 (5.7) 0 j ! 0 j (j)z2 j (j) j + (j) ; 8 1 j r1 (5.8) 0 j ! 0 j (uj + vj )( (j) j + (j)) 1 (j)(uj + vj ); 8 r1 j r1 + r2: (5.9) Where the quaternionic factors are still embedded using 5.3. These are exactly the proscibed actions of the matrix 0@ 1A on the space H, and the extra variables 0 j are transformed by subtracting the exponential transformation factor from formula 4.4 for the appropriate conjugate to the 0 j . 29 It should be noted that the action 0@ A B C D 1A Z is the same as the action 0@ tWA tW 1 tWBW W 1C tW 1 W 1DW 1A ~ Z (recall ~ Z = tWZW ). In order for this embedding of J (OK) to act on the symplectic theta function it is necessary that these embedded matrices be in Sp4n(Z) or to be more precise (2n) 0 (4). The next step is to calculate the conditions on ; ; ; that will force the matrix 0@ A B C D 1A = 0@ tWA tW 1 tWBW W 1C tW 1 W 1DW 1A 2 (2n) 0 (4): Since W is a matrix of integers in the eld and contains a basis for the ring of integers OK the inverse matrix W 1 is a basis for 1 K the inverse di erent of the eld K. It is easy to check that each of the entries in the matrices A; B; C; D is the trace of an element of K. Since W 1 is a basis for the inverse di erent, as long as ; ; are in OK the entries of A; B; D will be traces of elements of 1 K and therefore rational integers. In order for C to be integral it is required that in K , i.e. must be in the di erent. This is because the entries of C are traces of elements in 2 K ( ) and by the de nition of the inverse di erent, an element of the inverse di erent has integral trace. So as long as is in K , C will have entries that are traces of elements in 1 K . In order for this matrix to be in (2n) 0 (4) it is su cient to require that 4 divide . Therefore as long as 0@ 1A 2 0(4 K) the matrix 0@ A B C D 1A 2 (2n) 0 (4). It is easy to verify that det( C ~ Z + D) = det(CZ +D) = N ( + ): Therefore matching the Fourier coe cients of the symplectic theta function and the transformed version by 5.2 and using the fact that (Z) is a symplectic modular form, the coe cients satisfy 1 2 ; 2 0@0@ 1A ~ ; ~z1A = (M)N ( + ) 12 e[ r1 Xj=1 (j)2 2 (j)z2 j (j) j + (j) ] 30 e[ r1+r2 X j=r1+1 (j) 2 (uj + vj )( (j) j + (j)) 1 (j)(uj + vj ) (j) 2 ] 1 2 ; 2(~ ; ~z): This agrees with the formula 4.4 given in the de nition of a Jacobi form overK of weight 1 2 , index 22 and index vector ( 2). The second transformation law is proved by looking at the actions of [ ; ] in O2 K embedded in Sp4n(Z) and its action on the variables (~ ; ~z; ~ 0) and comparing Fourier coe cients. The embedded [ ; ] matrix takes j ! j ; zj ! zj + j (j) + (j); 8 1 j r1 + r2 0 j ! 0 j + (j)2 j + 2 (j)zj + (j) (j) 8 1 j r1 0 j ! 0 j + (j) j (j) + 2 (j)zj + (j) (j); 8 r1 + 1 j r1 + r2: It is trivial to see that det(C 0 e Z+D0) = 1, and therefore by the same arguement as above that by matching the Fourier coe cients on both sides of the symplectic theta function transformation formula 5.2 one extracts the transformation formulas for these Fourier coe cients. These formulas are exactly those given in 4.5 of the de nition for a Jacobi form of weight 1 2 index 22 and index vector ( 2). It is also easy to check that in order for 0@ tWA0 tW 1 tWB0W W 1C 0 tW 1 W 1D0W 1A 2 (2n) 0 (4) the only requirement is that [ ; ] be in O2 K since W 1C0 tW 1 = 0. Therefore by this construction, one is able to produce Jacobi forms on 0(4 K)nO2 K of weight 12 and index 22 and index vector ( 2) for all 2 in OK . Chapter 6 Relation to vector valued modular forms In this chapter the algebraic number eld K is required to be totally real, i.e. K R, though K is still assumed to be an nth degree extension. This restriction is necessary because it is only for the totally real elds that one may require the forms to be analytic, (it is only in the case of real elds and complex variables that analytic functions make sense). In order for the forms to be analytic the construction in chapter 7 would need to be limited to quadratic forms all of whose conjugates are positive de nite (since K is totally real this makes sense). The analyticity of the Jacobi forms overK plus the fact that the Jacobi forms are periodic in both variables produces a \nice" Fourier expansion. The periodicity may be seen from letting (~ ; ~z) be a Jacobi form over K, then because of the transformation formulas 4.4,4.5 (~ ; ~z) = (~ + 1; ~z) = (~ ; ~z + 1) where for example ~ + 1 means to add 1 to each element of the vector. Therefore the Fourier expansion of has the form (note r2 = 0; so n = r1) (~ ; ~z) = X ; 2 1 K c( ; )e[ n Xj=1 (j) j + (j)zj ] (6.1) where each of the c( ; ) is constant and 1 K is the inverse di erent as introduced earlier. In fact, for the forms that will be constructed in chapter 7 the Fourier expansion is only 31 32 over OK 1 K however in the general case this may not be assumed. The \nice" property of these Fourier expansions is that the Fourier expansion is with respect to ; z instead of just x, and u. For a general algebraic number eld, we do not have enough information about the Fourier expansions of our Jacobi forms to prove many of these results. We introduce notation which will be helpful. Earlier the trace TrK=Q of an element of the eld K was de ned, now extend this de nition to include the variables in the space H. For example, let ~ be an n dimensional upper half plane variable as usual and let 2 K then set TrK=Q( ~ ) = n Xj=1 (j) j : We similarly de ne the trace of the n dimensional full eld variable which has been written as ~z. 6.1 Restriction in the Fourier expansion For Jacobi forms of index m over Q, the functions are required to be analytic and this forces an extra condition on the Fourier expansion which has the form ( ; z) = 1 X n=0 X r2Z; r2 4nm c(n; r)qn r; q = e2 i ; = e2 iz : The condition that r2 4nm is a condition to make the function analytic as ! i1. A similar condition arises with analytic Jacobi forms over totally real number elds. Theorem. If (~ ; ~z) is an analytic Jacobi form of weight k and index m over a totally real number eld K of degree n then m 0 and (~ ; ~z) = X 2 1 K ; 0 X ( 2OK j 2 4 m) c( ; )e2 iTrK=Q ( + z): (6.2) The condition on the second sum is that for all 1 j n; (j)2 4 (j)m(j): Proof. Assume that (~ ; ~z) is a Jacobi form over K and has a Fourier expansion of the form 6.1. Since the Jacobi form is required to be analytic, the form restricted to (~ ; 0) 33 is also an analytic function of ~ . Therefore (~ ; 0) will have a Fourier expansion with respect to only totally positive elements of 1 K and zero, because if there was an in the Fourier expansion such that (j) < 0 for some 1 j n then by letting j ! i1, the function would diverge and therefore not be analytic. Similarly for any 2 OK the function (~ ; 0) = f(~ ) = e2 iTrK=Q (m 2 ) ([ ; 0] (~ ; 0)) is analytic in ~ . The Fourier expansion of f(~ ) is f(~ ) = X ; 2 1 K c( ; )e[TrK=Q( + + 2 m)]: In order for this function to be analytic in all of the j it is necessary that (j) + (j) (j) + (j)2m(j) > 0; 81 j n; 2 OK : This is a quadratic polynomial in (j) so it will always be positive provided the discriminant is negative or zero, i.e. (j)2 4m(j) (j) 0 and since (j) > 0, m(j) > 0 and this was what was claimed in the theorem. Therefore for totally real number elds the index is always totally positive as well as the condition on the Fourier expansion 2 4 m 0 or 2 4 m = 0. 6.2 Periodicity in the Fourier coe cients Given (~ ; ~z) is a Jacobi form over a totally real number eld K, the Fourier expansion may be written as (~ ; ~z) = X 2 1 K ; 0 X ( 2OK j 2 4 m) c( ; )e2 iTrK=Q ( + z): Since there is no need of an index vector for this case we will just specify that the index of the form is m 2 OK . Theorem. Given that (~ ; ~z) is an analytic Jacobi form of index m over a totally real number eld K, has a Fourier expansion of the form given above and the Fourier coe ecients c( ; ) depend only on mod 2m and on 4 m 2. 34 Proof. The second transformation formula 4.5 for Jacobi forms over K yields, for any ; 2 OK ([ ; ] (~ ; ~z)) = X ; 2 1 K c( ; )e[TrK=Q( + (z + + ))] = e[ TrK=Q(m 2 + 2m z)] (~ ; ~z): (6.3) It follows that (~ ; ~z) = e[TrK=Q(m 2 + 2m z)] X ; 2 1 K c( ; )e[TrK=Q( + (z + + ))] = X ; 2 1 K c( ; )e[TrK=Q([ + 2m+ ] + [ + 2 m]z)]: (6.4) Therefore c( ; ) = c( + 2m+ ; + 2 m) = c( 0; 0): (6.5) Then since 4 m 2 = 4 0m 02, the Fourier coe cients depend only mod 2m and on 4 m 2. 6.3 Connection to vector valued modular forms Assume we are given a Jacobi form over a totally real number eld K with a Fourier expansion as above. The periodicity of the Fourier coe cients described in the previous section leads to a connection between Jacobi forms and vector valued modular forms just as in the classical case. Note that the Fourier coe cients of a Jacobi form for the real number eld K may be expressed as c( ; ) = c (N) = c N + 2 4m ; for mod 2m; = N + 2 4m : Note that we are allowed to assume 2 4 m = N 0 or N = 0 by the earlier theorem. Extend the de nition to all totally positive numbers (or zero) N by setting c (N) = 0 if N 6 2 mod 2m. Now de ne the following functions h (~ ) = X N 0 c (N)e2 iTrk=Q (N =4m) (6.6) 35 #m; (~ ; ~z) = X 2 1 K mod2m e2 iTrK=Q ( 2 =4m+ z): (6.7) Then we have (~ ; ~z) = X 2 1 K ; 0 X ( 2OK j 2 4 m) c( ; )e2 iTrK=Q ( ~ + ~z) = X mod2m X ( mod2m) X N 0 c (N)e TrK=Q N + 2 4m + z = X mod2mh (~ )#m; (~ ; ~z): (6.8) Now it is easy to check that #m; (~ + 1; ~z) = e2 iTrK=Q ( 2 4m )#m; (~ ; ~z) or more generally#m; (~ + ; ~z) = e2 iTrK=Q ( 2 4m )#m; (~ ; ~z); 8 2 OK (6.9) and therefore since (~ + ; ~z) = (~ ; ~z), we get h (~ + ) = e 2 iTrK=Q ( 2 4m )h (~ ); 8 2 OK : (6.10) Now it remains to nd a formula for h ( ~ 1 ), (note that ~ 1 = ( 1 1 ; ::: 1 n )) because results of Vaserstein, Cooke, and Liehl prove that the matrices 0@ 0 1 1 0 1A ;0@ 1 0 1 1A ; 2 OK generate Sl2(OK), see [3],[4],[16],[23]. So knowing h (~ + ) and h ~ 1 gives the transformation properties of h (~ ) for all 0@ 1A 2 Sl2(OK). In order to nd h ( ~ 1 ) we will create the theta functions #m; (~ ; ~z) as the coe cients of a symplectic theta function and then specify the transformation formula for #m; ( ~ 1 ; ~z ) where the variables have the obvious meaning. De ne the following matrices Z = 0@ d1( ~ 2m) d1( ~z 2m) d1( ~z 2m) d1( ~ 0 2m) 1A 36 M1 = 0BBB@ 2m(1) 0 ::: 0 . .. ... 2m(n) 1CCCA ; M = 0@ M1 0 0 M1 1A : Note that m 0 because of the earlier theorem and therefore the matrix M1 is positive de nite. Therefore by selecting the imaginary parts of the 0 variables such that yjy0 j > v2 j for all 1 j n using the notation introduced earlier, the variable Z is in the symplectic upper half space. Recall the matrix c W = 0@ W 0 0 W 1A ; W = (!j i )1 i;j n; OK = [!1; :::!n]Z as de ned earlier. Also let ~ V = 0@ v1 v2 1A 2 C 2n where v1; v2 2 C n . Let 2 OK be a representative of a congruence class modulo m and let 2m = Pnj=1 bj!j ; where the bj 2 Q and let v1 = (bj)1 j n; v2 = (0). Then set m; 0@Z;0@ 0 ~ V 1A1A = X r2Z2n e i t(~ V+r) tc WMZMc W (~ V+r): Note that the presence of the matrix M changes the sum over rational integer vectors into a sum over the elements of the ideal (2m) instead of OK as in the rst construction. So the sum of elements of the form t(~ V + r) is actually a sum over the elements of O2 K of the form 0@ 1 2 1A where 2 2 OK and 1 is equivalent to mod 2m. Expanded out this looks like m; (~ ; ~z; ~ 0;0@ 0 ~ V 1A) = X 1 2 OK ; 2 2 (2m); 1 mod 2m e[TrK=Q( 21 4m + 2( 1 2) z 4m)]e[TrK=Q( 22 0 4m)] (6.11) which again yields the Fourier expansion with respect to the 0. The 2 = 2m Fourier coe cient is then #m; (~ ; ~z). By using the matrix 0@ 0 1 1 0 1A in Sl2(Z) embedded as 37 in chapter 5 and considering the expansion of m; 0@0@ 0 1 1 0 1A Z;0@ 0 1 1 0 1A 0@ 0 ~ V 1A1A and the transformation formula for the symplectic theta function 5.2 (note that the vector notation~ has been dropped to ease the notation, however all of the ; z; 0 should be considered as vectors), m; 0@0@ 0 1 1 0 1A Z;0@ 0 1 1 0 1A 0@ 0 ~ V 1A1A = m; 0@ 1; z ; 0 z2 ;0@ ~ V 0 1A1A = X 1; 22OK exp[ iTrK=Q( 21 1 2m + 2( 1 2) z 2m 1 2m)]exp[ iTrK=Q( 22 0 z 2m )] = 0@ 0 1 1 0 1A ( n Y j=1 j 2m(j) ) 1 2 0@Z;0@ 0 ~ V 1A1A : For ease of notation, let (S) = 0@ 0 1 1 0 1A. By again examining the 2 = 2m coe cient and equating the di erent 2 = 2m coe cients, we get that #m; 1 ; z = (S)( n Y j=1 j 2m(j) ) 1 2 e2 iTrK=Q (mz2= ) X mod2m e[TrK=Q( 2m)]#m; (~ ; ~z): (6.12) Therefore since m; 0@0@ 0 1 1 0 1A (Z;0@ 0 ~ V 1A)1A = (S)N ( )ke2 iTrK=Q (mz2= ) m; 0@Z;0@ 0 ~ V 1A1A ; we have h 1 = (S) 1N ( )k( n Y j=1 j 2m(j) ) 12 X mod2m exp(2 iTrK=Q( 2m))h ( ) (6.13) and we have now essentially proved the following correspondence Theorem. For a totally real number eld K the correspondence between analytic Jacobi forms of weight k and index m and vector valued modular forms satisfying the transformation formulas 6.10,6.13 over K and bounded as j !1; 8 1 j n, of weight k 12 38 given by ( ; z) = X mod2mh ( )#m; ( ; z) ! (h ( )) mod2m is an isomorphism between the spaces. There is a small ambiguity which arises due to the fact that there is a choice of which square roots are taken, so in fact we need a metaplectic covering of the group Sl2(OK) to account for this. The metaplectic covering will not be pursued here except to note that a metaplectic covering of the group Sl2(OK) is a group that in essence contains group elements combined with a multiplier system evaluated at the group element. This functions much in the same way as the multiplier systems discussed earlier. Chapter 7 General construction with quadratic forms The initial example given in chapter 5 created Jacobi forms of weight 12 , index 2, and index vector ( ) for all in OK . We now present a more general construction which will create forms of arbitrary half integral weight, more general indexes in OK and much more general index vectors. The forms which will be created here are going to be generalizations of the Jacobi theta functions for a positive de nite quadratic form over the rational integers of the type that appear in [11] and more recently in section 5.1. For example, let Q be a positive de nite quadratic form with entries in Zso for a in Zn; a 6= 0; Q(a) = taQa > 0, and x b in Zn then de ne Q;b( ; z) = X a2Zn e2 i( taQa +2 taQbz): The quadratic forms used in this construction will be required to be positive de nite in all of the real conjugates and this is a signi cant restriction. For this general construction we are using the \2 i" version of the symplectic theta function which will force all of the transformation formulas onto a 0 congruence subgroup instead of the type congruence subgroups. The use of the \ i" version is almost identical with the only di erence being the subgroup of the symplectic group under which the functions transform. There is a more general construction for theta functions of inde nite quadratic forms (i.e. not positive de nite) over number elds but for this we simply refer the reader 39 40 to [17], and [8] which shows how to create theta functions of inde nite quadratic forms. The techniques used in this paper may also be generalized to the inde nite quadratic forms with majorants by combining the methods of this paper and [17]. 7.1 Quadratic forms There are some facts about quadratic forms which will be useful in order to create Jacobi forms. A quadratic form is a symmetric function Q : C n C n ! C which will be represented as a matrix so Q(~a;~b) = t~aQ~b; Q 2Mn(C ); ~a;~b 2 C n : A symmetric quadratic form is such that Q(~a;~b) = Q(~b;~a) so it is necessary that the matrix Q be symmetric, tQ = Q. When Q is a matrix of real numbers, we call Q positive de nite if Q(~a) Q(~a;~a) > 0; 8 ~a 2 C n : Note that it is meaningless to talk about a quadratic form over the complex numbers being positive de nite, however in the case of complex entries, a quadratic form will be required to have non-zero determinant. Given such a matrix there will always be an upper triangular matrix L 2 Mn(C ) such that Q = tLL; L = (li;j)1 i;j n with lij = 0 for j < i.For this construction, the quadratic form Q is in Mn(OK); Q = (qs;t)1 s;t n so all of the entries are integers in the number eld. Since Q has entries in K, it is reasonable to de ne the conjugates Q(j) = (q(j) s;t )1 s;t n and denote the decomposition of each conjugate of the form L[j] so that Q(j) = tL[j]L[j], also denote L[r1+r2+j] = L[r1+j] to continue the notation and ordering from earlier. There is a signi cant restriction to the quadratic forms for this construction which is that the real conjugates of Q which 41 are Q(j); 1 j r1 must be positive de nite. It is possible to do the construction without this restriction but this would force the introduction of majorants and even more notation. 7.2 The general theta functions Given a quadratic form over OK as in the previous section, we will create a theta function of the form for b in Ol K Q;b(~ ; ~z) = X a2Ol K e[(r1+r2 Xj=1 ta(j) tL[j] jL[j]a(j) + 2 ta(j) tL[j]zjL[j]b(j))]: (7.1) These functions will be produced as before by creating a symplectic theta function and choosing the appropriate Fourier coe cients. This construction and the veri cation of the construction will prove the following theorem Theorem. Let K be an algebraic number eld of degree n = r1 + 2r2 as usual. Let Q be a l l quadratic form with entries in OK , such that Q(j) > 0 for 1 j r1 (i.e. all of the real conjugates are positive de nite), and let g be the greatest common divisor of the principal minors of Q, and let Q(j) = tL[j]L[j] for 1 j n then the function l 2 ;~ m;m(~ ; ~z) = X~ m;m Q;b(~ ; ~z) is a Jacobi form of weight l 2 , index Q(b), and index vector Lb for the subgroup 0(4 Kg 2det(Q)2)nO2 K J (OK). The sum over ~ m;m means to sum over all vectors b 2 Ol K such that Q(b) =m and tb tL[j]L[j]b = tm(j)m(j) for all r1 < j r1 + r2. In the case where there is only one b in Ol K in the sum then this is the natural generalization of the Jacobi theta function. However, for a general index pair fm; ~ mg all that can be shown is that the sum is over a nite number of b. Note K is the di erent of K as in the earlier notation. 42 We begin by introducing notation let dl(a) = 0BBBBB@ a 0 ::: 0 a ... . . . a 1CCCCCA which is an l l matrix. Then for ~ in hr1 h(Q)r2 and other similar variables dl;1(~ ) = 0BBBBB@ dl( 1) 0 ::: 0 dl( 2) ... . . . dl( r1) 1CCCCCA dl;2(~x) = 0BBBBB@ dl(xr1+1) 0 ::: 0 dl(xr1+2) ... . . . dl(xr1+r2) 1CCCCCA dl;3(~x) = 0BBBBB@ dl( xr1+1) 0 ::: 0 dl( xr1+2) ... . . . dl( xr1+r2) 1CCCCCA where the notation j = xj + yj for j in h(Q) is used. Similarly de ne these matrices for the other variables yj ; zj; uj ; vj; 0 j ; x0j; y0 j . Now de ne the l nl matrix W (j) = 0BBBBB@ !(j) 1 ::: !(j) n 0 ::: 0 ::: 0 !(j) 1 ::: !(j) n 0 ::: . . . !(j) 1 ::: !(j) n 1CCCCCA and de ne the matrix c W and b L to be f W = 0BBBBB@ W (1) W (2) ::: W (n) 1CCCCCA ;c W = 0@ f W 0 0 f W 1A ; e L = 0BBBBB@ L[1] 0 ::: 0 L[2] ... . . . L[n] 1CCCCCA ; b L = 0@ e L 0 0 e L 1A ; 43 where the L[j] were de ned in the previous section. The variable for the symplectic theta function is de ned similarly as e Z = 0BBBBBBBBBBB@ dl;1(~ ) 0 0 dl;1(~z) 0 0 0 dl;2(~x) idl;2(~y) 0 dl;2(~u) idl;2(~v) 0 idl;3(~y) dl;3(~x) 0 idl;3(~v) dl;3(~u) dl;1(~z) 0 0 dl;1(~ 0) 0 0 0 dl;2(~u) idl;3(~v) 0 dl;2(~x0) idl;2(~y0) 0 idl;2(~v) dl;3(~u) 0 idl;3(~y0) dl;3(~x0) 1CCCCCCCCCCCA : Now create the symplectic theta function Sp;Q( e Z) = X ~a2Z2nl e2 i( t~a tc W tb Le Zb Lc W~a); (7.2) but for this to be a symplectic theta function it must be veri ed that tc W tb L e Zb Lc W is in h(2nl) or at least that one can pick the extra upper half plane variables in order to put this matrix into the symplectic upper half space. The matrix is obviously symmetric by construction therefore it remains to show it has positive de nite imaginary part. By the same reasoning as earlier consider the matrix e S e S = 0BBBBBBBBBBB@ Ilr1 0 ::: 0 0 Ilr2 ... Ilr2 0 Ilr1 0 Ilr2 Ilr2 0 1CCCCCCCCCCCA which has the same e ect as before, it switches the complex conjugate rows of the matrix Lc W which is denoted as LW = e SLW . Therefore in order to show this matrix tc W tb L e Zb Lc W (7.3) has positive de nite imaginary part, note e S2 = I2nl and consider tc W tb L e Zb Lc W = tc W tb Le S e S e Zb Lc W = tW tLe S e ZLW 44 and if e S e Z has positive de nite imaginary part then so does 7.3 by the earlier lemma in chapter 5. The imaginary part of e S e Z is 0BBBBBBBBBBB@ dl;1(y) dl;1(v) dl;3(y1) dl;3(v1) dl;3(u2) dl;2(y1) dl;2(u2) dl;2(v1) dl;1(v) dl;1(y0) dl;2(v1) dl;3(u2) dl;3(y0 1) dl;2(u2) dl;3(v1) dl;2(y0 1) 1CCCCCCCCCCCA : (7.4) This matrix is just l copies of the matrix in 5.4 and since we can pick the y0 and the y0 1 variables so that 5.4 is positive de nite, the same choices of y0 and y0 1 will force the above matrix to be positive de nite. Expanded out this symplectic theta function 7.2 looks like Sp;Q( e Z) = Sp;Q(~ ; ~z; ~ 0) and Sp;Q(~ ; ~z; ~ 0) = X a;b2Ol K e[(r1+r2 Xj=1 ta(j) tL[j] jL[j]a(j) + 2 ta(j) tL[j]zjL[j]b(j))] e[( r1 Xj=1 tb(j) tL[j] 0 jL[j]b(j))]e[( r1+r2 X j=r1+1 tb(j) tL[j] 0 jL[j]b(j))] = X b2Ol K Q;b(~ ; ~z)e[( r1 Xj=1 tb(j) tL[j] 0 jL[j]b(j))]e[( r1+r2 X j=r1+1 tb(j) tL[j] 0 jL[j]b(j))] (7.5) where Q;b(~ ; ~z) = X a2Ol K e[(r1+r2 Xj=1 ta(j) tL[j] jL[j]a(j) + 2 ta(j) tL[j]zjL[j]b(j))]: This last representation shows how the Jacobi theta functions Q;b arise as the Fourier coe cients with respect to the 0 variables. However, there may be a number of vectors b in Ol K which are all part of the same Fourier coe cient because they all produce the same index Q(b). In particular this last representation should be written as X m2OK X (b2Ol K j Q(b)=m) Q;b(~ ; ~z)e[ r1 Xj=1m(j) 0 j ]e[ r2 X j=r1+1 tb(j) tL[j] 0 jL[j]b(j)]: (7.6) 45 Some of these Jacobi theta functions will be separated from each other so that they will be shown to transform as Jacobi forms independent of the other Jacobi theta functions in the coe ceint. Some of these Q;b will still be grouped together in the Fourier coe cient but it will be shown that they as a group transform correctly according to the de nition of Jacobi forms over K with a given index vector. Before we can fully state the transformation formulas of these Fourier coe cients it is necessary to examine the structure of the coe cient. The formula above 7.6 is a Fourier expansion with respect to the 0 j , for 1 j r1, and x0j ; for r1+1 j r1+r2 however the y0 j , with r1+1 j r1+r2 are not part of the expansion (because the form is not analytic). They are grouped in simply as a matter of notational convenience. However, it is possible to assume that the non-complex parts of the quaternionic variables, the y0 j are part of the Fourier expansion at least in order to show the transformation formulas. It was already noted that all of the Jacobi theta functions with index vectors b such that Q(b) =m are summed to make themth coe cient. It is possible to separate these into classes based on the associated index pair (Q(b); tb tLLb). Since there are only a nite number of vectors b which produce the same index there is a smallest tb tLLb because these are all real numbers. The sum of these Jacobi theta functions with the same index pair can then be separated from the rest of the Fourier coe cient by letting the y0 j for r1+1 j r1+r2 goto in nity. This will force the theta functions associated to the larger tb tLLb to goto zero faster. Now that this portion of the Fourier coe cient is separated it may be shown to be a Jacobi form over K of weight l 2 , index index Q(b) and index vector Lb where b is the vector associated to any one of the theta functions in the class. Similarly once this class of Jacobi theta functions is shown to be a Jacobi form one can take the next smallest tb tLLb and repeat the process thus separating out all of the di erent classes of theta functions. 7.3 Transformation formulas Now it is necessary to show this sum over a class of Jacobi theta functions transforms correctly. As before, it is necessary to extend the group embeddings from the rst example so that the embedded group will act on e Z . We generalize the earlier 46 de nition of d( ) to dl( ), for in OK dl( ) = 0BBBBB@ dl( (1)) 0 : : : 0 0 dl( (2)) ... . . . 0 dl( (n)) 1CCCCCA = 0BB@ dl;1( ) dl;2( ) dl;3( ) 1CCA : Then the embeddings of the elements of J (OK) are: 0@ 1A 2 Sl2(OK) ,! 0BBBBB@ dl( ) 0 dl( ) 0 0 Inl 0 0 dl( ) 0 dl( ) 0 0 0 0 Inl 1CCCCCA = 0@ A B C D 1A ; [ ; ] 2 O2 K ,! 0BBBBB@ Inl 0 0 dl( ) dl( ) Inl dl( ) dl( ) 0 0 Inl dl( ) 0 0 0 Inl 1CCCCCA = 0@ A0 B0 C 0 D0 1A : Set T = b Lc W . It should be noted that the actions of 0@ 1A (~ ; ~z) is the same as the action of 0@ tTA tT 1 tTBT T 1C tT 1 T 1DT 1A tT e ZT on the ~ ; ~z variables, and similarly for the embedded [ ; ] matrix. In fact the actions are exactly the same as those in chapter 5 in 5.6-5.9, since these embedding are simply l copies of the embeddings in chapter 5. So, by introducing a quadratic form the actions have not changed but it is still necessary to check that this matrix 0@ e A e B e C e D 1A = 0@ tTA tT 1 tTBT T 1C tT 1 T 1DT 1A (7.7) is in Sp4nl(Z) or particularly (4nl) 0 (4), whenever 0@ 1A is in a subgroup of Sl2(OK) and similarly for the embedded [ ; ] matrix. 47 In order to check that this embedded group matrix is actually in Sp4nl(Z) consider the entries of the matrices e A; e B; e C; e D which with a little computation are all traces of elements in K. One fact which is helpful here is that the entries of Q 1 are given in terms of determinants of minors divided by the determinant of Q. Therefore let g be the ideal generated by the principal minors of Q then all of the entries of Q 1 are divisible by gdet(Q) 1. The entries in e A; e D are all traces of elements from the ideal generated by 1 K the traces of these elements are by de nition all in Z. The entries of e B are all traces of elements in an ideal contained in the integers OK so they are rational integers. The entries of the matrix e C are all traces of elements from the ideal 2 K g2(det(Q)) 2 and therefore in order for all of these entries to be rational integers we require that be in Kdet(Q)2g 2. Now in order for this matrix to be in (4nl) 0 (4) it is also necessary that all of the entries of e C be divisible by 4, therefore if is in 4 Kdet(Q)2g 2 then the embedded matrix is in (4nl) 0 (4). So far, the construction has produced a symplectic theta function which has combinations of Jacobi theta functions of a quadratic form Q over the eld K as its Fourier coe cients and these may be separated into classes based on the index pair. In the last section it was shown that the embedded J (OK) acted correctly on the embedded version of the space H; and it is easy to check that the actions on the individual j ; zj ; 0 j are the same as in section 5.3. It is also easy to check that det(C e Z +D) = N (c + d)l as previously de ned. Therefore since Sp;Q is a symplectic modular form, by using 5.2 we see that for all M = 0@ A B C D 1A 2 (4nl) 0 (4), Sp;Q(M e Z) = (M)det(C e Z +D) 1 2 Sp;Q( e Z): Expanding this we have, X a2Z2nl e[2 i ta tT (M e Z)Ta] = (M)N (c + d) l 2 X a2Z2nl exp(2 i ta tT e ZTa): We expand out the right hand side of this equation as in the previous calculations, and look at the Fourier expansion 7.6 as in chapter 5. Since each 0 j ! 0 j (j)z2 j (j) j + (j) ; 8 1 j r1 (7.8) 480j ! 0j (uj +vj )( (j) j + (j)) 1 (j)(uj + vj ); 8 r1 < j r1 + r2; (7.9)We get the exponential part of the transformation formulas by matching the Fouriercoe cients in 7.6, and by replacing 0 as in 7.8, 7.9 where the index for the Jacobi thetafunction is given by tb tLLb = Q(b) and the index vector is now given by (Lb).Similarly, the embedded [ ; ] 2 O2K matrix takesj ! j ; zj ! zj + j (j) + (j); 8 1 j r1 + r20j ! 0j + (j)2 j + 2(j)zj + (j) (j) 8 1 j r10j ! 0j + (j) j (j) + 2(j)zj + (j) (j); 8 r1 + 1 j r1 + r2:Also the action of 0@ bA bBbC bD 1A = 0@ tTA0 tT 1tTB0TT 1C 0 tT 1 T 1D0T 1Aon tT eZT is the same as the action of [ ; ] on the variables (~ ; ~z). 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